Автор: Нурк Э.Р. Тельгмаа А.Э.
Теги: математика математика для детей издательство просвещение школьное образование учебник для школы
ISBN: 5-09-004623-9
Год: 1993
Текст
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XYJl.O>KeCTBeHHblii peJlaKTOp £. P. llawYK
TexHHlJecKHii peJlaKTOp C. C. flKywKuHa
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Op.o,eH3 TpYJ10BOro KpacHoro 3H3MeHH H3.D,3TenLCTRO (npOCBemelme) MHIIHcTepcTB8 neq3TH H HH4x>PM8UHH Poc-
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JJ;Han03HTHBbI H3rOTOBJIeHbI B CMoneHcKoM nonHrpa
KoM6HHaTe MHHHCTCPCTB3 neqaTH H HH4x>PM8IJ,HH Poc-
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ISBN 5-09-004623-9
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MOM .ll.pyr!
B 3TOM YQe6HOM roJJ.Y Ha nYTH K HOBblM 3HaHHHM Tbl npOJJ.OJI}KHlllb
H3Y4eHHe MaTeMaTHKH, Y3Haelllb MHOro HOBOro H HHTepecHoro. 51 X04Y
CJIY}KHTb Te6e XOPOWHM nOMOIll,HHKOM H COBeT4HKOM Ha :nOM nYTH.
4eM Mbl 6YJJ.eM 3aHHMaTbCH?
B npeJJ.blJJ.YllI,eM KJIacce Tbl Y}Ke OBJIaJJ.eJI HaBblKaMH Bbl4HCJIeHHH
C HaTypaJIbHblMH 4HCJIaMH, JJ.eCHTH4HbIMH JJ.P06HMH, a TaK}Ke nOJIY-
4HJI nepBoe npeJJ.cTaBJIeHHe 06 06blKHoBeHHblx JJ.P06HX. B npeJJ.cToH-
llI,eM Y4e6HoM roJJ.Y Tbl npoJJ.OJI}KHlllb H3Y4eHHe 06blKHoBeHHblx JJ.po-
6eH. Tbl HaY4HlllbcH JJ.eHcTBHHM C 3THMH JJ.P06HMH, Y3Haelllb, KaK
u 3 5 2 1
HaHTH, HanpHMep, npOH3BeJJ.eHHe ""4"""6 HJIH CYMMY "3+""4'
Tbl, KOHel/HO, elll,e He YMeelllb H3 MeHblllero 4HCJIa Bbl4eCTb
60JIblllee, HanpHMep HaHTH pa3HocTb 3-4. jJ,JIH 3Toro Te6e npeJJ.cToHT
n03HaKOMHTbCH C 4HCJIaMH HOBoro BHJJ.a.
TbI, HaBepHoe, XOpOlllO 3Haelllb, KaK 3anHcaTb pacnOJIO}KeHHe
waXMaTHblX <lmryp Ha JJ.OCKe HJIH KaK HaHTH CBoe MeCTO B TeaTpaJIb-
HOM 3aJIe. KaKaH }Ke 3JJ.eCb CBH3b C MaTeMaTHKoH? H 06 3TOM Tbl
Y3Haewb B npeJJ.cTOHllI,eM Y4e6HoM roJJ.Y.
jJ,eJIHTCH JIH 4HCJIO 387 954 Ha 9? jJ,JIH nOJIY4eHHH OTBeTa MO}KHO
6blJIO 6bl BbmOJIHHTb nHCbMeHHoe JJ.eJIeHHe. Ho He HaJJ.O! Ha MOHX
CTpaHHLlax Tbl HaHJJ.elllb npOCToe npaBHJIO, KOTopoe nOMO}KeT Te6e
6blCTpO HaHTH OTBeT Ha JJ.aHHbIH Bonpoc.
MHOro HOBoro Tbl Y3Haelllb, H3Y4aH reOMeTpH4eCKHH MaTepHaJI.
KaKHe eCTb B03MO}KHOCTH JJ.JIH pacnOJIO}KeHHH JJ.BYX npHMblX Ha
nJIOCKOCTH? C Y4aCTKa KaKOH nJIOlll,aJJ.H CMO}KeT KOpMHTbCH TpaBOH
K03a, eCJIH OHa npHBH3aHa Ha BepeBKe JJ.JIHHOH 10 M? Ha 3TH H
JJ.pyrwe BonpOCbl Tbl TO}Ke nOJIY4Hlllb OTBeTbl B 3TOM Y4e6HOM roJJ.Y.
BeCb MaTepHaJI Ha MOHX CTpaHHLlax pa3JJ.eJIeH Ha weCTb rJIaB,
Ka}KJJ.aH H3 KOTOpblX pa3JJ.eJIeHa Ha naparpa<f:>bI. B KOHLle Ka}KJJ.oro
naparpa<f:>a HMelOTCH 3aJJ.a4H JJ.BYX pa3JJ.eJIOB: A H 6. 3aJJ.a4H B pa3-
3
JJ.eJIe A 60JIee npocTble, a B pa3JJ.eJIe I) HeMHoro CJImKHee. ECJIH Tbl
3aXOLJelilb pelllaTb 3aJJ.aLJH elll,e CJIO}KHee, TO eCTb H TaKHe. OHH 0603Ha-
l/eHbl 3Be3JJ.OLJKOH.
Oc060e BHHMaHHe 06paTH Ha BonpOCbl H 3aJJ.a4H noJJ. 3arJIaBHeM
«jJ,JIH caMonpOBepKH». ECJIH TbI CYMeelilb pelllHTb Bce 3TH 3aJJ.aLJH
H OTBeTHTb Ha Bce BonpOCbl, TO Tbl Y4HJICH CTapaTeJIbHO. B 3TOM
CJIY4ae MO}Kelllb, He 60HCb TPYJJ.HOCTeu, npHcTynHTb K H3Y4eHHIO
HOBOH TeMbI.
3anOMHH: nYTb K COKpOBHIlI,HHLle 3HaHHH, oc06eHHO npH H3Y4e-
HHH MaTeMaTHKH, Tpe6yeT OT Te6H nOCJIeJJ.OBaTeJIbHOCTH H HaCTOH-
4HBOCTH.
5epem MeHH, 4T06bl H B CJIeJJ.YlOllI,eM Y4e6HOM roJJ.Y Mor 6bITb
XOpOlllHM nOMOIll,HHKOM JJ.pymM Y4eHHKaM.
TBOU YLf.e6H.UK.
1. J1.EJ1HMOCTb HATYPAJ1bHbIX QHCEJ1
nOBTopeHHe
A.
2
1. B KJIacce 30 4eJIOBeK. Bo BpeMH JIeTHHX KaHHKYJI 5 Bcex Y4a-
IlI.HXC5I KJIaCCa OTllbIXaJIH B cnopTHBHbIX JIarep5IX, a OCTaJIbHbIe
Y4alll,HeCH B Tel/eHHe 20 JJ.HeH - B JIarepe TpYJJ.a H OTJJ.blxa, rJJ.e
3apa60TaJIH B cpeJJ.HeM no 1,8 p, B JJ.eHb Ka}KJJ.bIH, nOCTaBb pa3YM-
Hble Bonpocbl H OTBeTb Ha HHX.
2. Bbl4HCJIH (y C THO) :
I) 32 + 14 2) 93 + 20
256+ 121 22+ 128
187-106 1000-18
250-130 518-409
3. BbI4HCJIH:
I) 54 310 + 23 690
450 760 + I 564 768
86,315+4,085
4. Bbl4HCJIH (y C THO):
I) 100.64 2) 10.4,52
6.103 0,64.1000
9600: 10 4,7: 10
5000: 100 56,4: 1000
5. BbI4HCJIH:
I) 4007.43 2)
88 044. 15
560.420
20.37,5
0,27.5,9
1,45.0,97
6. BbI4HCJIH:
I) 64200-53609+ 1860+64
17394-(5682+473)
51-(3,29.0,1 +6,241)
3) 0,6+0,3
1,2+5,3
0,9-0,4
0,24-0,13
4) 1,6+0,4
0,8+0,5
2-0,3
4-1,8
2) 1 000 000 - 24 306
8607333-7970315
201,1-7,35
3) 2.0,9
0,6.7
4,8:4
3,2:8
3) 1035:23
17 860: 94
14351:113
4) 0,6.0,9
0,07.0,5
6:0,3
8:0,04
4) 84,6:6
8,489:0,013
1,92:5
2) 339,52-129,92-71,35
207 605 - 409.506
(49 - 38,2) (0,82 + 1,68)
7. OTHOCHTeJIbHO KaKOU napbl 4HCeJI MO}KHO CKa3aTb, 4TO nepBoe
l/HCJIO JJ.eJIHTCH Ha BTopoe:
24 H 8, 18 H 4, 35 H 5, 48 H 3, 56 H 5, 85 H 1O?
5
8. Ha30BH Bce TaKHe HaTypaJIbHble 4HCJIa, Ha KOTopble 4HCJIO 45
JJ.eJIHTCH 6e3 OCTaTKa.
9. Bo BpeMH JIeTHHX KaHHKYJI Y4eHHKH 6 KJIacca n06bIBaJIH B Mec-
Tax 60eBOH CJIaBbI. jJ,JIH 3TOro HM npHlllJIOCb npoHTH 58 KM,
npoexaTb 247 KM noe3JJ.OM H 105 KM aBT06ycOM. nOCTaBb pa3YM-
Hble Bonpocbl H OTBeTb Ha HHX.
10. nJIOlll,aJJ.b 5eJIoro MOpH 90 TbIC. KM 2 , a nJIOlll,aJJ.b 4epHoro MOpH
420 TbIC. KM 2 . Bo CKOJIbKO pa3 nJIOlll,aJJ.b 5eJIoro MOpH MeHbwe
nJIOlll,aJJ.H 4epHoro MOpH? (OTBeT oKpyrJIH JJ.O JJ.ecHTblx.)
11. PaccToHHHe Me}KJJ.Y JJ.BYMH ropoJJ.aMH 270 KM. ABT06yc npOe3}Ka-
eT 3TO paccToHHHe C nOCTOHHHOH CKOpOCTblO 3a 4 4, JlerKo-
BaH MaWHHa npOe3}KaeT 75 KM B 4ac, CKOpOCTb KaKoH MalllHHbl
60JIblile H Ha CKOJIbKO?
12. Ha nepBoM nOJIe YP0}KaH 3epHa COCTaBHJI 40 Ll C OJJ.HOro reKTa-
pa. Ha BTOpOM nOJIe, nJIOlll,aJJ.b KOTOpOro 30 ra, YP0}KaH 3epHa
6b1JI 960 Ll. Ha KaKOM nOJIe YPO}KaH 3epHa C OJJ.HOro reKTapa 60JIb-
llle H BO CKOJIbKO pa3?
13. jJ,JIHHa nacT6HIlI,a npHMoyroJIbHOH <f:>OPMbl 580 M, a lllHpHHa Ha
130 M MeHblile. Bbl4HCJIH nJIOlll,aJJ.b nacT6HIlI,a (B reKTapax) Hero
nepHMeTp (B KHJIOMeTpax).
14. BbI4HCJIH: I) 20% OT 60; 2) 64% OT 1800; 3) 3,6% OT 72.
15. B aBTonapKe 180 aBTOMalllHH, H3 HHX 80% rpY30Bble. CKOJIbKO
rpY30BblX aBTOMalllHH B aBTonapKe?
16. Ha4epTH npHMYIO H OTMeTb Ha HeH T04KH A, B H C. H3Mepb
JJ.JIHHbl Bcex nOJIY4eHHblX oTpe3KoB.
17. OTpe30K KL paBeH 4,5 CM. Ha4epTH: I) OTpe30K MP, KOTOpblH
Ha 2,1 CM Kop04e OTpe3Ka KL; 2) OTpe30K CD, KOTOpblH B 1,2 pa-
3a JJ.JIHHHee oTpe3Ka KL.
18. Hal/epTH JIYl/ OA H OTMeTb Ha HeM T04KY B, TaKYlO, 4T06bl
OB=3,8 CM. Ha CKOJIbKO caHTHMeTpOB OTpe30K AB Kop04e
(HJIH JJ.JIHHHee) OTpe3Ka OA?
19. Ha4epTH yroJI, paBHblH 55°, 90° H 140°.
20. Ha4epTH TpeyrOJIbHHK ABC H H3Mepb ero yrJIbI. Bbl4HCJIH CYM-
MY yr JIOB 3TOro TpeyrOJIbHHKa.
21. jJ,JIHHa lllKa<f:>a, HMelOIlI,erO <f:>0PMY npHMoyroJIbHoro napaJIJIeJIenH-
neJJ.a, 1,8 M, lllHpHHa 0,5 M H BblCOTa 2 M. Bbl4HCJIH 06'beM wKa<f:>a.
22. Bblpa3H B MeTpax: 17 KM, 6,3 KM, 0,8 KM, 40 CM, 15 CM, 7 JJ.M,
4,3 JJ.M.
23. Bblpa3H B KBaJJ.paTHblx MeTpax: 6 ra, 1,8 ra, 0,5 ra, 2 KM 2 ,
0,4 KM 2 , 700 llM 2 , 1850 JJ.M 2 , 30000 CM 2 , 4000 CM 2 .
6
24. Bblpa3H B Ky6H4ecKHx JJ.eLlHMeTpax: 4 M 3 , 3,2 M 3 , 0,2 M 3 ,
7000 CM 3 , 34000 CM 3 , 4,8 JI, 0,7 JI.
25. Bblpa3H B KHJIOrpaMMax: 15 T, 0,45 T, 2 Ll, 0,3 Ll, 6500 r, 94 r,
245 r, 1250 r, 5 r.
6.
26. BbI4HCJIH:
I) 1657974:822.106-(50377+20338);
2) 4,32 (5,67 -2,07)+ 10,53:4,05-0,152;
3) (43.19-26928:33)(16 112:53-304).
27. Bbl4HCJIH 3Ha4eHHe Bblpa}KeHHH:
I) 4b-3c, eCJIH b=9 H c=O,8;
2) 28mn-5n+2m, eCJIH m=O,2 H n=O,3;
3) 2 (3x+5)-3 (4x-15), eCJIH x=6,5:
4) 3(2x-l)+2(3x+l), eCJIH x=I,1.
28. nJIOlll,aJJ.b apXHneJIara HOBaH 3eMJIH (COCTOHT H3 JJ.BYX KpynHblx
H MHO}KeCTBa MeJIKHX OCTpOBOB) npHMepHo paBHa 83000 KM 2 .
CeBepHblH OCTpOB apXHneJIara HMeeT nJIOlll,aJJ.b 48 900 KM 2 , a nJIO-
llI,aJJ.b IO}KHoro oCTpoBa Ha 15600 KM 2 MeHblile CeBepHoro.
Bo CKOJIbKO pa3 nJIOlll,aJJ.b MeJIKHX OCTpOBOB apXHneJIara MeHblile
nJIOll.laJJ.H CeBepHoro oCTpoBa?
29. CaMoe 60JIblilOe }KHBOTHOe 3eMHoro lllapa - KHT CHHHH, Macca
HOBOpO}KJJ.eHHoro KHTeHKa npHMepHO 3 T. B nepHoJJ. KOpMJIe-
HHH MaTepblO OH npH6aBJIHeT B Bece e>KeCYT04HO B cpeJJ.HeM
100 Kr. CKOJIbKO BeCHT KHTeHOK 4epe3 x CYTOK nOCJIe p0}KJJ.eHHH?
COCTaBb Bblpa}KeHHe H Bbll/HCJIH ero 3Ha4eHHe, eCJIH x = 30,
x= 100.
30. B 9 4 H3 JJ.BYX ropOJJ.OB, paCCTOHHHe Me}KJJ.Y KOTOpblMH 210 KM,
HaBcTpe4Y JJ.pyr JJ.pyry BbleXaJIH JJ.Ba aBTOM06HJIH. CpeJJ.HHH CKO-
pOCTb OJJ.HOro aBTOM06HJIH 50 KM , a JJ.pyroro 70 KM . nOcTaBb
lJ lJ
pa3YMHble Bonpocbl H OTBeTb Ha HHX.
3t. jJ,JIHHa 6acceHHa, HMelOIlI,erO <f:>0PMY npHMoyroJIbHOrO napaJIJIe-
JIenHneJJ.a, 25 M, lllHpHHa 10 M H rJIy6HHa 1,8 M. CKOJlbKO Bpe-
MeHH nOTpe6yeTcH, 4T06b1 3anOJIHHTb 6acceHH BOJJ.OH Ha 1,5 M,
eCJIH e}KeMHHYTHO B 6acceHH nOCTynaeT 2,5 M 3 BOJJ.bl?
32*. JlYl/H OA H OB 06pa3YlOT npHMoH yroJI. Hal/epTH JIY4 OD TaK,
4T06bl LBOD= 115°, H JIY4 OE TaK, 4T06bl LAOE=45°.
BW4HCJIH yroJI DOE (paccMoTpH Bce B03MO}KHWe CJIY4aH).
7
1.1. J].eJ1HTeJ1H H KpaTHbie I.JHCJ1a
HaHJJ.eM BCe HaTypaJIbHble 4HCJIa, Ha KOTOpble JJ.eJIHTCH LJHCJIO 12.
TaKHMH LJHCJIaMH 6YJJ.YT I, 2, 3, 4, 6 H 12. JIeHcTBHTeJIbHO,
12: I = 12, 12:2=6, 12:3=4, 12:4=3, 12:6=2 H 12: 12= l. HH
Ha OJJ.HO JJ.pyroe HaTypaJIbHOe l/HCJIO 12 He JJ.eJIHTCH, Bce 4HCJIa, Ha
KOTopble JJ.eJIHTCH 12, Ha3bIBalOTCH .ll.eJIHTeJIHMH 4HCJIa 12, 4HCJIO 7 He
HBJIHeTCH JJ.eJIHTeJIeM 4HCJIa 12. 06'bHCHH n04eMY.
JI1060e HaTypa/l.bHOe 'IUC/l.O, Ha lCoTopoe iJe/I.UTClI. iJaHHoe HaTY-
pa/l.bHOe 'IUC/l.O, Ha3blBaeTClI. iJe/I.UTe/l.eM iJaHHo2o 'Iuc/l.a.
HanpHMep, JJ.eJIHTeJIHMH 4HCJIa 10 6YJJ.YT 4HCJIa I, 2, 5 H 10,
a JJ.eJIHTeJIHMH 4HCJIa 19 JIHWb 4HCJIa I H 19.
B03bMeM KaKoe-HH6YJJ.b HaTypaJIbHOe 4HCJIO, HanpHMep 5, H
6YJJ.eM YMHmKaTb ero nOCJIeJJ.OBaTeJIbHO Ha I, 2, 3, 4, .., . nOJIY4HM
4HCJIa COOTBeTCTBeHHO 5, 10, 15, 20, ..., KOTopble 06JIaJJ.a1OT OJJ.HHM
061l1,HM CBOHCTBOM: Bce OHH JJ.eJIHTCH Ha 5. TaKHe l/HCJIa Ha3bIBalOTCH
KpaTHblMH l/HCJIY 5. 4HCJIO 21 He HBJlHeTCH KpaTHblM 4HCJIY 5.
06'bHCHH n04eMY.
JI1060e HaTypa/l.bHOe 'IUC/l.O, lCoTopoe iJe/l.UTClI. Ha iJalUWe H.aTY-
pa/l.bHOe 'IUC/I.O, Ha3blBaeTClI. ICpaTHblM. iJaHHOMY 'IUC/l.Y.
HanpHMep, 4HCJIa 3, 6, 9, ... 6Y.ll.YT KpaTHblMH 4HCJIY 3, a LJHCJIa
7, 14, 21, 28, ...- KpaTHblMH 4HCJIY 7. Bo061l1,e eCJIH 0603Ha4HTb
KaKoe-TO HaTypaJIbHOe 4HCJIO 4epe3 n, TO KpaTHblMH 3TOMY 4HCJIY
HBJIHIOTCH 4HCJIa
n.l, n.2, n.3, n.4, n.5,... ,
MHorOT04He B KOHLle 3TOH nOCJIeJJ.OBaTeJIbHOCTH 03Ha4aeT,
4TO cpeJJ.H KpaTHblX JJ.aHHOMY 4HCJIY HeT HaH60JIblilero 4HCJIa, n03TO-
MY HeB03MO>KHO nepeLJHCJIHTb Bce 4HCJIa, KpaTHble JJ.aHHOMY HaTypaJIb-
HOMY 4HCJIY.
A.
33. np04TH TeKCT naparpa<f:>a. 3anHlllH: I) iJe/l.UTe/l.U LJHCJIa 36;
2) nHTb 4HCeJI, KpaTHbtX 4HCJIY II.
34. YKa}KH Te napbl 4HCeJI, B KOTOpblX nepBoe 4HCJIO HBJIHeTCH JJ.eJIH-
TeJIeM BToporo:
2 H 6, 3 H 5, 3 H 9, 4 H 12, 7 H .28, 5 H 18, 10 H 30.
35. npoBepb, 6YJJ.eT JIH nepBoe 4HCJIO JJ.eJIHTeJIeM BToporo:
1) 13 H 611, 2) 32 H 277, 3) 10 H 9800,
14 H 516; 101 H 2727; 9 H 816,
8
36. Ha30BH Bce JJ.eJIHTeJIH 4HCeJI:
I) I, 2, 8, 12; 2) 3,9, 15,21;
3) 5, 26, 29, 30; 4) 13, 22, 27, 32.
37. HY}KHO pa3JJ.eJIHTb nopoBHY Me}KJJ.Y HeCKOJIbKHMH JJ.eTbMH 24
opexa. KaKoe 4HCJIO JJ.eTeH B03MO}KHO JJ.JIH 3Toro?
38. Ha30BH Te napbl 4 HCeJI , B KOTOpblX nepBoe 4HCJIO KpaTHO BTO-
POMY:
9 H 3, 10 H 4, 15 H I, 9 H 2, 24 H 6, 100 H 5, 32 H 9,
39. Ha30BH JJ.JIH Ka}KJJ.oro H3 cJIeJJ.YIOIlI,HX 4HCeJI TPH KpaTHblx:
3, 5, 8, 10, 12; 30, 80, 100, 1000.
40. npoBepb, 6YJJ.eT JIH nepBoe l/HCJIO KpaTHo BTOPOMY:
I) 288 H 8, 2) 2501 H 61, 3) 3920 H 112,
485 H 9; 2816 H 46; 1696 H 32.
41. Ha30BH Bce JJ.BY3Ha4Hble 4HCJIa, KpaTHble 4HCJIY 9.
42. KaKoH LlH<f:>POH OKaH4HBalOTCH 4HCJIa, KpaTHble 2, 5, 10?
6.
43. 06'bHCHH, n04eMY cpeJJ.H JJ.eJIHTeJIeH JJ.aHHoro 4HCJIa BcerJJ.a HaH-
JJ.eTCH HaHMeHbwee H HaH60JIblilee 4HCJIO.
44. 06'bHCHH, n04eMY cpeJJ.H KpaTHblx JJ.aHHoro 4HCJIa HaHJJ.eTcH
HaHMeHblllee, HO HeT HaH60JIblilerO 4HCJIa.
45. HaHJJ.H Bce JJ.eJIHTeJIH 4HCJIa 1110.
46. 3anHwH Te 3Ha4eHHH x, KpaTHble 4HCJIY 6, npH KOTOpblX Hepa-
BeHCTBO 48 < x < 94 BepHo.
47*. 3anHlllH 4eTblpex3Ha4Hble 4HCJIa, KpaTHble 423 H OKaH4HBalO-
IlI,HeCH LlH<f:>POH 5.
1.2. npH3HaKH .ll.eJ1HMOCTH
JIonYCTHM, 4TO HY}KHO Y3HaTb, JJ.eJIHTCH JIH, HanpHMep, 4HCJIO
38756 Ha 3. JIJIH 3TOro MO}KHO BbmOJIHHTb JJ.eJIeHHe 38756:3 H TeM
caMblM nOJIY4HTb OTBeT Ha nOCTaBJIeHHblH Bonpoc. Ho OTBeT MO}KHO
nOJIY4HTb ropa3JJ.0 npOlll,e, He BbmOJIHHH JJ.eJIeHHH. B 3TOM HaM nOMO-
rYT npH3HaKH .ll.eJlHMOCTH. PaCCMOTpHM JIHlllb HeKOTopble. CHa4aJIa
3aMeTHM, 4TO Ha iJaHHoe HaTypa/l.bHoe 'IUC/l.O n iJe/l.1l.TClI. Bce 'IUCM,
ICpaTHble n. Hu OiJHO iJpY20e 'IUl:/I.O Ha n He iJe/l.UTClI..
n p H 3 H a K H JJ. e JI HMO C T H HalO, 5 H 2
KpaTHblMH 4HCJIY 10 HBJIHIOTCH 4HCJIa
10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110,
9
3anHCb Ka}KJJ.OrO H3 3THX 4HCeJI OKaH4HBaeTCH LlH<f:>POH 0,
11TaK, HQ. 10 iJe/tJI.TCR. Bce Te HaTypaAbHble 'lUl:ILa. 3anUCb lCOTOpblX
OlCaH'lUBaeTCR. 'luq,pou 0; eCILu 3anUCb 'lUCILa OlCaH'lUBaeTCR. /U06ou
iJpyzou 'lUq,pOU. TO 'lUl:ILO He iJeILUTCR. Ha I O.
KpaTHblMH 4HCJIY 5 HBJIHIOTCH 4HCJIa
5, 10, 15, 20, 25, 30, 35, 40, ...
nOCJIeJJ.HeH LlH<f:>POH Ka}KJJ.oro H3 3THX 4HCeJI HBJIHeTCH HJIH 0,
HJIH 5, OTclOJJ.a BblTeKaeT, 4TO HQ. 5 iJe/tJI.TCR. Bce Te HaTypaAbHble
'lUCILa. 3anUCb ICOTOpblX OlCaH'lUBaeTCR. 'luq,pou 0 UILU 'lutPpou 5;
eCILu 3anUCb 'lUCILa OlCaH'lUBaeTCR. /U06ou iJpyzou 'luq,pou. TO 'lUl:/I.O
He iJe/UlTCR. Ha 5.
KpaTHblMH 4HCJIY 2 HBJIHIOTCH l/HCJIa
2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24,
3anHcb 4HCeJI, KpaTHblx 4HCJIY 2, OKaH4HBaeTCH OJJ.HOH H3 LlH<f:>p:
0, 2, 4, 6, 8. 3TH LlH<f:>Pbl Ha3bIBalOTCH qeTHblMH I\HcI»paMH. OCTaJIbHble
LlH<f:>Pbl, TO eCTb 1, 3, 5, 7 H 9, Ha3b1BaIOTCH HeqeTHblMH I\HcI»paMH.
CJIeJJ.oBaTeJIbHO, Ha 2 iJe/tJI.TCR. Bce Te HQ.TypaILbHble 'lUCILa. 3anUl:b
ICOTOpblX OlCaH'lUBaeTCR. 'leTHOU 'lutPpou; eCILu 3a1W£b 'lUl:/I.a OlCaH'lU-
BaeTCR. He'leTHOU 'luq,pOU. TO 'lUCILO He iJeILUTCR. Ha 2.
HaTypaJIbHble 4HCJIa, KOTopble JJ.eJIHTCH Ha 2, Ha3b1BaIOTCH qeT-
HblMH qHCJIaMH. Bce OCT
JIbHble HaTypaJIbHble 4HCJIa HeqeTHbie. Ha-
npHMep, 4HCJIa 58, 96, 100,38754 4eTHble, a 89, 311, 1003,57327 He-
4eTHble
A.
48. npoBepb no TeKcTY, nOMHHWb JIH Tbl:
I) KaKHe LlH<f:>Pbl Ha3bIBalOTCH 'ieTH.blMU, KaKHe H.e'ieTH.blMU;
2) npU3H.alCU iJeA.UMOCTU Ha 2, 5 H 10.
49. BbInHlllH 4HCJIa, KOTopble JJ.eJIHTCH: 1) Ha 5; 2) Ha 10:
430, 708, 95, 1300, 605, 400 I, 90, 175, 104, 800, 34 000.
50. HanHlllH TPH l/eTblpeX3Hal/HbIX 4HCJIa, KOTopble JJ.eJIHTCH: I) Ha 5;
2) Ha 10.
51. KaKHe H3 4HCeJI 37, 82, 136, 5023, 18008, 300421, I 000000,
554 4eTHble? KaKHe He4eTHble?
52. HanHlllH: 1) 4eTblpe Tpex3Ha4Hblx 4eTHblX lIHCJIa;
2) 4eTblpe Tpex3Ha4Hblx He4eTHblX 4HCJIa.
53. JIaHbl 4HCJIa:
I) 127, 568, 1932, 4939, 5300, 1026, 75443, 579314;
2) 375 654, 9999, 100 600, 394 497, 87 302 608.
KaKHe H3 HHX lleJIHTCH Ha 2? KaKHe He JJ.eJIHTCH?
10
54. B KOHLle ypoKa Y4aulHecH CJJ.aJIH TeTpaJJ.H JJ.J1H KOHTpOJIbHblX pa-
60T H TeTpaJJ.H JJ.J1H ynpa}KHeHHH, Bcero 47 TeTpaJJ.eH. Bce JIH
Y4alll,HeCH CJJ.aJIH 06e TeTpaJJ.H?
6.
55. BbIHCHH: I) Bce JIH 4HCJIa, KOTopble JJ.eJIHTCH Ha 5, JJ.eJIHTCH H
Ha 10;
2) Bce JIH l/HCJIa, KOTopble JJ.eJIHTCH Ha 10, JJ.eJIHTCH H Ha 5.
56. HaHJJ.H 3Ha4eHHH x, npH KOTOpblX HepaBeHcTBo x < 45 BepHo
H KOTopble npH 3TOM JJ.eJIHTCH: 1) Ha 5; 2) Ha 10,
57. 113 LlH<f:>p 0, 2, 5 H 7 COCTaBb no TPH Tpex3Ha4Hblx 4HCJIa, KOTO-
pble JJ.eJIHTCH: I) Ha 2; 2) Ha 5; 3) Ha 10,
58. HanHlllH 4eTblpe Tpex3Ha4Hblx 4HCJIa, KOTopble JJ.eJIHTCH H Ha 2,
H Ha 5, CJJ.eJIaH BbIBOJJ..
npH3HaKH JJ.eJIHMOCTH Ha 9 H 3
BblnHllleM lIHCJIa, KpaTHble 9:
9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99,
Mbl BHJJ.HM, LITO 4HCJIO, KpaTHoe 9, MO}KeT OKaH4HBaTbC.H JII060H
LlH<f:>POH. n03TOMY CYJJ.HTb 0 JJ.eJIHMOCTH Ha 9 no nOCJleJJ.HeH LlH<f:>pe
3anHCH 4HCJIa Mbl He MO}KeM. l..h06bl BblHCHHTb, JJ.eJIHTCH JJ.aHHoe.4HCJIO
Ha 9 HJIH HeT, Bbl4HCJIHeTCH cYMMa LlH<f:>p 3Toro 4HCJIa. HanpHMep,
cYMMa LlH<f:>p 4HCJIa 327 paBHa 3+2+ 7= 12, cYMMa LlH<f:>p 4HCJIa
20756 paBHa 2+0+ 7 +5+6=20.
3anHweM HeKoTopble l/HCJIa H noJJ. Ka}KJJ.bIM 4HCJIOM - CYMMY ero
LlH<f:>p:
72 ,
, 93, 108, 288, 323, 499, 873, 1003, 8883 , 99 873
9 9 12 9 18 8 22 18 4 27 36
MO}KHO npoBepHTb, 4TO Ka}KJJ.oe H3 nOJJ.4epKHYTbIX 4HCeJI JJ.eJIHTCH
Ha 9 H B TO }Ke BpeMH CYMMa LlH<f:>p 3THX LIHCeJI TO}Ke JJ.eJIHTCH Ha 9.
OCTaJIbHble (HenoJJ.LIepKHYTble) 4HCJIa He JJ.eJIHTCH Ha 9 (npoBepb!),
H cYMMa LlH<f:>p 3THX 4HCeJI TO}Ke He JJ.eJIHTCH Ha 9. MO}KHO y6eJJ.HTbcH,
4TO TaKaH 3aKoHoMepHocTb HMeeT MeCTO JJ.JIH JII060ro HaTypaJIbHOrO
l/HCJIa. C<f:>oPMYJIHpyeM npH3HaK JJ.eJIHMOCTH Ha 9: HQ. 9 iJe/tJI.TCR. Bce
Te HaTypaILbHble 'lUCILa. cYMMa 'luq,p ICOTOpblX iJeILUTCR. Ha 9; ec-
/Ul cYMMa 'lutPP 'lUCILa He iJe/UlTCR. Ha 9, TO aTO 'lUCILO He iJeILUTCR.
Ha 9.
HanpHMep, l/HCJIO 397566 JJ.eJIHTCH Ha 9, TaK KaK cYMMa ero
II
LlH<f:>p 3+9+ 7 +5+6+6=36 JJ.eJIHTCH Ha 9. I.JHCJIO 8972 He JJ.eJIHT-
CH Ha 9, TaK KaK CYMMa ero LlH<f:>p 8+9+ 7 +2=26 He JJ.eJIHTCH Ha 9.
npH3HaK JJ.eJIHMOCTH Ha 3 aHaJIOrH4eH npH3HaKY
eJIHMOCTH Ha
9: HQ. 3 iJe/tJI.TClI. Bce Te HaTgpaILbHble 'lUCILa. cgMMa 'luf/Jp ICOTOpblX
iJeILUTCll. Ha 3; eCILu cgMMa 'ltu/Jp 'lUC/I.a He iJeILUTClI. Ha 3. TO U CQMO
'lUCILO He iJeILUTClI. Ha 3.
npoBepb cnpaBeJJ.JIHBOCTb 3Toro npH3HaKa Ha npHMepax
A.
59. npoBepb no TeKcTY, nOMHHWb JIH Tbl npU3HaICU iJeA.UMOCTU Ha
9 H Ha 3,
60. HaHJJ.H CYMMY LlH<f:>p CJIeJJ.YIOIlI,Hx l/HCeJI:
I ) 135, 207, 396, 615, 3926, 9234;
2) 5688, 6703, 3006, 9001, 42732, 8500770.
KaKHe H3 HHX JJ.eJIHTCH Ha 9?
61. .LI.aHbI 4HCJIa:
I) 231,801, 1002,5765,5792,4501;
2) 9684, 4370, 9081, 21 708, 472634, 1 345000.
KaKHe H3 HHX JJ.eJIHTCH Ha 3?
62. MmKHo JIH H3 JJ.aHHbIX LlH<f:>p COCTaBHTb Tpex3Ha4Hoe 4HCJIO, JJ.e-
JIHllI,eeCH: a) Ha 3; 6) Ha 9:
I) I, 2, 5; 2) 2, 3, 7; 3) 0, 5, 4?
63. BMecTo 3Be3JJ.04KH nOCTaBb TaKYIO LlH<f:>PY, l/T06bl nOJIY4HJIOCb
4HCJIO, JJ.eJIHllI,eeCH Ha 9:
I) 1 * 00; 2) 578 * ; 3) 4 * I; 4) * 888.
64. MO>KHO JIH pa3
eJIHTb na4KY, COCTOHIlI,YIO H3 4731 ra3eTbl: I) Ha
TPH paBHble 4aCTH; 2) Ha JJ.eBHTb paBHblx 4aCTeH? (06'bHCHH
n04eMY. )
6.
65. BblHCHH: I) Bce JIH 4HCJIa, KOTopble
eJIHTCH Ha 3, JJ.eJIHTCH H
Ha 9; 2) Bce JIH 4HCJIa, KOTopble
eJIHTCH Ha 9, JJ.eJIHTCH H Ha 3.
66. KaKoH HaHMeHblileH H KaKoH HaH60JIbWeH MO}KeT 6bITb cYMMa
LlH<f:>p JJ.BY3Ha4Horo 4HCJIa?
67. HaHJJ.H 4HCJIa x, JJ.eJIHIlI,HeCH: a) Ha 3; 6) Ha 9, TaKHe, 4T06bl
HepaBeHcTBa 6blJIH BepHbIMH:
I) 27 <x<42; 2) 32<x<70; 3) 120<x< 150.
68. 113 JJ.aHHbIX LlH<f:>p COCTaBb, eCJIH 3TO B03MO}KHO, TpeX3Ha4Hoe
4HCJIO, JJ.eJIHllI,eeCH: a) Ha 3; 6) Ha 9:
I) 5, 8; 2) I, 8; 3) 3, 6.
12
69*. BMecTo 3Be3JJ.04eK nOCTaBb LlH<f:>Pbl TaKHe, 4T06b1 4HCJIO
4* I * JJ.eJIHJIOCb Ha 9. HaHTH Bce B03MO}KHble peweHHH.
70*. HanHwH nHTb JJ.BY3Ha4Hblx 4HCeJI, KOTopble JJ.eJIHTCH H Ha 2, H
Ha 3. Ha KaKoe elll,e 4HCJIO JJ.eJIHTCH Bce 3TH 4HCJIa?
71 *. 113 LlH<f:>p 3, 4, 5 H 6 COCTaBb Bce TpeX3Hal/Hble 4HCJIa, JJ.eJIH-
IlI,HeCH H Ha 3, H Ha 5.
CaMOCTOHTeJIbHaH pa60Ta 1
T e M a. npOcTI>le H COCTaBHble 'I HCJla.
1. HaHJJ.H Bce JJ.eJIHTeJIH Ka}KJJ.oro H3 l/HCeJI:
1, 2, 3, 4, 5, 6, 12, 17, 43, 60.
2. ECJIH Tbl C 3aJJ.aHHeM CnpaBHJICH, TO nOJIY4HJI:
HaTypaJIbHOe 4HCJIO I 2 3 4 5 6 12 17 43 60
KOJIHl/eCTBO JJ.eJIHTeJIeH I 2 2 3 2 4 6 2 2 12
3. I1TaK, l/HCJIO I HMeeT OJJ.HH JJ.eJIHTeJIb, 4HCJIO 12 - llleCTb JJ.eJIH-
TeJIeH, 4HCJIO 43 - JJ.Ba JJ.eJIHTeJIH. B MaTeMaTHKe BblJJ.eJIHIOT TaKHe
HaTypaJIbHble 4HCJIa, KOTopble HMelOT TOJIbKO JJ.Ba JJ.eJIHTeJIH: I H caMO
3TO 4HCJIO. HaTypaJIbHOe 4HCJIO Ha3bIBaeTCH npocTblM '1HCJlOM, eCJIH
OHO HMeeT TOJIbKO JJ.Ba pa3JIH4HblX JJ.eJIHTeJIH: eJJ.HHHLlY H caMO ce6H.
4HCJIO, HMelOllI,ee 60JIee JJ.BYX JJ.eJIHTeJIeH, Ha3blBaeTCH COCTaBHblM
'1HCJlOM. I.JHCJIO I He OTHOCHTCH HH K npOCTbIM, HH K COCTaBHblM
l/HCJIaM, TaK KaK HMeeT TOJIbKO OJJ.HH JJ.eJIHTeJIb.
4. npOJJ.YMaH elll,e pa3 TeKCT H CKa}KH, KaKHe 4HCJIa Ha3bIBalOTCH
npocTblMH, KaKHe COCTaBHbIMH.
5. KaKHe H3 4HCeJI, npHBeJJ.eHHble B n. I, npocTble H KaKHe
CocTaBHble?
6. KaKHe H3 4HCeJI 7, 9, II, 14, 19, 27, 29, 31 npocTble H KaKHe
COCTaBHble? (06'bHCHH n04eMY.)
1.3. Ta6J1Hu.a npocTblx '1HCeJ1
KaK onpeJJ.eJIHTb, HBJIHeTCH JIH JJ.aHHoe 4HCJIO (HanpHMep, 567)
npocTblM HJIH cocTaBHblM? .LI.JIH 3Toro HY}KHO BbIHCHHTb, HMeeT JIH 3TO
4HCJIO XOTH 6bl OJJ.HH JJ.eJIHTeJIb, OTJIHl/HbIH OT caMoro 4HCJIa H eJJ.H-
HHLlbl. ECJIH TaKoro JJ.eJIHTeJIH HeT, TO 4HCJIO npOCToe, B npOTHBHOM
CJIYl/ae OHO COCTaBHoe. Ho HaXO>KJJ.eHHe JJ.eJIHTeJIeH JJ.JIH 60JIblllHX
13
l/HCeJI - JJ.eJIO HeJIerKOe. n03TOMY JJ.JIH
ynpOlll,eHHH pa60Tbl COCTaBJIeHa Ta6-
JIHLla npOCTblX 4HCeJI. Ha cpOp3aLle
TBOerO Yl/e6HHKa HMeeTCH TaKaH Ta6-
JIHLla. B KOTOpOH HaH60JIbWee npOCTOe
4HCJIO 997. OJJ.HaKo 3TO He caMoe
60JIbWOe npOCToe 4HCJ10. .llpeBHerpe-
l/eCKHH MaTeMaTHK EBKJIHJJ. JJ.OKa3aJI
npHMepHo 2300 JIeT Ha3aJJ., 4TO npo-
CTblX 4HCeJI 6eCKOHe4HO MHOro, LITO
HaH60JIbWerO npOCToro l/HCJIa He cy-
mecTByeT.
HCTOpHQeCKHe CBe.ll.eHHH
.llpeBHerpel/eCKHH Yl/eHbIH 3paToccpeH, >KHBWHH HeCKOJIbKO n03JJ.-
Hee EBKJIHJJ.a, npeJ].JlO>KHJI CBOH cnoc06 JJ.JIH COCTaBJIeHHH Ta6JIHLlbl
npOCTblX 4HCeJI. 3TOT cnoc06 HOCHT Ha3BaHHe «peweTO 3paToccpe-
Ha». B 4eM OH 3aKJII04aeTCH? HaH.neM, HanpHMep, Bce npOCTble
4HCJIa OT I .no 20. .llJIH 3Toro BbInHWeM Bce 4HCJIa OT I JJ.O 20 B pH.n:
,K 2
3
l
)t)<$
11 yf 13 ;4 J.5 }6 17 ;t .!!!..J6
1 J9t1
3
\
It
14
.llaJIee 6YJJ.eM Bbl4epKHBaTb 4HCJ1a, KOTopble
He HBJIHIOTCH npOCTblMH, B nepBYIO 04epeJJ.b
Bbl4epKHeM I, TaK KaK 3TO He npOCToe 4HCJIO,
nepBoe npOCToe 4HCJIO 2. nOJJ.4epKHeM ero
H Bbl4epKHeM Bce l/HCJIa, KpaTHble 2, TO
eCTb l/HCJIa 4, 6, ..., 20. CJIeJJ.YlOllI,ee npoCToe
4HCJ10 3, nOJJ.4epKHeM ero H Bbll/epKHeM Bce
4HCJ1a, KpaTHble 3 (KoTopble OCTaJIHCb He
BbI4epKHYTbIMH), H T. JJ.. TaK Mbl «BblceeM»
Bce HHTepeCYlOllI,He Hac npOCTble 4HCJ1a:
2, 3,5,7, II, 13, 17, 19.
EOJIbWHe 3aCJ1Yrw B 06JIaCTH H3Y4eHHH
npOCTblX 4HCeJI npHHaJJ.J1e>KaT PYCCKHM H COBeTCKHM MaTeMaTHKaM.
n, J1. Ye6blweB (1821-1894) JJ.OKa3aJI, 4TO Me>KJJ.Y JII06blM HaTY-
paJIbHblM 4HCJ10M, 60JIbWHM I, H 4HCJIOM, BJJ.Boe 60JIbWHM JJ.aHHoro
(HanpHMep, 2 H 4, 3 H 6, 10 H 20 H T. JJ.,), BcerJJ.a HMeeTCH XOTH 6bl
OJJ.HO npoCToe l/HCJIO.
J;L M. BHHorpaJJ.oB (1891-1983) YCTaHOBHJI, LITO JII060e JJ.OCTa-
TOl/HO 60JIbWOe He4eTHoe l/HCJIO MO>KHO npeJJ.CTaBHTb B BHJJ.e CYMMbl
Tpex npOCTblX l/HCeJI, HanpHMep:
7 = 2 + 2 + 3, 9 = 3 + 3 + 3 = 2 + 2 + 5, 15 = 3 + 5 + 7 = 5 + 5 + 5.
A.
72. Ha30BH Bce npocTble I/.UClla, KOTopble pacnOJIO>KeHbl Ha 4HCJIO-
BOM JIY4e Me>KJJ.Y 4HCJIaMH: I) I H 10; 2) 10 H20; 3) 30 H 40.
73. npoBepb C nOMOlll,blO Ta6JIHLlbl npocTblX 4HCeJI, KaKHe H3 l/HCeJI
npocTble, a KaKHe COCTaBHble:
197,207,239,617,813,929,943.
74. 3paTOccpeH pOJJ.HJICH npHMepHo B 276 r. .no H. 3. H YMep npHMepHO
B 194 r. JJ.O H. 3. KaKHe rOJJ.bl, Bblpa>KeHHble npOCTblMH 4HCJIaMH,
npHXOJJ.HTCH Ha nepHOJJ. >KH3HH 3paToccpeHa?
6.
75. HaH.nH Bce npocTble 4HCJIa, npH KOTOpblX HepaBeHcTBo BepHo:
I) x< 12; 2) 23<x<49; 3) 150<x< 180.
76. Ha30BH npoCToe 4eTHoe 4 HCJIO. ECTb JIH elll,e npOCTble l/eTHble
l/HCJIa? nOl/eMY?
77. npoBepb Ha npHMepax, MO>KeT JIH cYMMa Tpex nOCJIeJJ.OBaTeJIbHbIX
HaTypaJIbHblX 4HCeJI 6blTb npocTblM l/HCJIOM. CJJ.eJIaH BbIBOJJ.bI.
78*. COBepUleHHbl.M l/HCJIOM Ha3blBaeTCH HaTypaJIbHOe 4HCJIO, KOTopoe
paBHo cYMMe JJ.eJIHTeJIeH 3Toro 4HCJIa, MeHbWHX caMoro 4HCJIa.
HanpHMep, 4HCJ10 6 - cOBepweHHoe l/HCJIO, TaK KaK 6 = 3 +
+2+ 1. HMeeTcH elll,e OJJ.HO COBepweHHoe 4HCJIO MeHbwe 100
HaHJJ.H 3TO 4HCJIO.
.D.JlH nOBTopeHHH
79.
PaCnOJIO>KH JJ.P06H B nopH.nKe B03paCTaHHH:
9 1 17 5 13
18'18'18'18'18.
17 94 36 5 234 9 77
30 ' 12' "'9' 13' 100' 14' "'8'
197
19
80. KaKHe H3 JJ.p06eH
15
81.
HBJIHIOTCH: I) npaBHJIbHbIMH; 2) HenpaBHJIbHbIMH? B Henpa-
BHJIbHblX JJ.P06HX BblAeJIH LleJIYIO H JJ.P06HYIO l/aCTH.
Ka>KJJ.oe H3 H6JIOK pa3pe3aJIH Ha 4eTblpe paBHble l/aCTH, CKOJIbKO
4eTBepTblx cOJJ.ep>KHTCH B OJJ.HOM H6JIOKe? ABYX H6JIOKax? nHTH
H6JIOKax?
Bbll/HCJIH (y C THO) :
I) 27+49+53 2)
3,8+1,5+2,5
57 +46+24
3 7
12+12
82.
84-50+ II
6,9-2,6-2,3
9,7 -1,8-3,7
11 5
---
16 16
3) 15 . 32 . 2
0,4: 18.5
2.0,05.97
45 23
---
50 50
83.
Bbll/HCJIH:
I) 3424:32+51.80; 2) 43,08-1,632:0,08.
MOpKOBb B cpeJJ.HeM COJJ.ep>KHT 6% caxapa. CKOJIbKO rpaMMOB
c,axapa COJJ.ep>KHTCH B 200 r MOpKOBH?
84.
1.4. Pa3nO>KeHHe COCTaBHblX I.fHCeJ1
Ha npocTble MHO>KHTeJ1H
Ka>KJJ.oe COCTaBHoe 4HCJIO MO>KHO npeJJ.CTaBHTb B BHJJ.e npOH3Be-
AeHHH XOTH 6b1 ABYX MHO>KHTeJIeH, OTJIHl/HbIX OT I. HanpHMep,
24 = 6.4, ECJIH cpeJJ.H nOJIYl/eHHbIX MHO>KHTeJIeH HMelOTCH COCTaBHble
4HCJIa, TO OHH TO>Ke MOrYT 6blTb npeJJ.CTaBJIeHbl B BHJJ.e npOH3BeJJ.eHHH
JJ.BYX MHO>KHTeJIeH. B JJ.aHHOM npHMepe
24= 6. 4 =(3 .2).(2.2)=2.2.2.3.
CXeMaTHl/HO 3TO MO>KHO H306pa3HTb TaK:
24
/ "
4
/ "
2 2
6
/ "-
2 3
ECJIH COCTaBHoe l/HCJIO npeJJ.CTaBJIeHO B BHAe npOH3BeJJ.eHHH,
Bce MHO>KHTeJIH KOTOpOro TOJIbKO npOCTble l/HCJIa, TO roBOpHT: COC-
T8BHOe 'IHCJIO p83J10)l(eHO H8 npOCTble MHO)l(HTeJIH. HanpHMep:
18=2.3.3, 54=2.3.3.3,
900 = 2 . 2 . 3 . 3 . 5 . 5.
npH pa3JIO>KeHHH 60JIbWHX l/HCeJI Ha npocTble MHO>KHTeJIH MO>KHO
nOJIb30BaTbCH cxeMoH, npeJJ.cTaBJIeHHOH B CJIeJJ.YlOllI,eM npHMepe:
16
n p H M e p. Pa3JIO>KHM Ha npocTble MHO>KHTeJIH l/HCJ10 420.
o 6 'b H C H e H H e.
420 2 3anHweM l/HCJIO 420 H cnpaBa OT Hero npoBeAeM
210 2 BepTHKaJIbHYIO l/epTY. nOJJ.6epeM HaHMeHbwHH npoc-
105 3 TOH JJ.eJIHTeJIb (2) 3Toro QHCJIa H 3anHweM ero cnpaBa
35 5 OT JJ.aHHoro 4HCJIa. 4acTHoe 420:2=210 3anHweM
7 7 CJIeBa OT l/epTbl noJJ. JJ.aHHbIM 4HCJ10M. C 4HCJ10M 210
I nOCTynHM TaKHM >Ke 06pa30M: 210:2= 105. .llaJIee
nOJIYl/aeM 105:3=35,35:7=7,7:7=1.
Ka>KJJ.bIH CJIeJJ.YlOlliHH npoCToH JJ.eJIHTeJIb nHweM noJJ. npeJJ.blJJ.YIlI,HM
JJ.eJIHTeJIeM H Ka>KAOe cJIeJJ.YlOllI,ee l/aCTHOe - no A npeJJ.blJJ.YIlI,HM 4aCT-
HbIM. Pa3JIO>KeHHe l/HCJIa Ha npocTble MHO>KHTeJIH 3aKaHl/HBaeTCH,
KorJJ.a B JIeBOM cToJI6HKe nOJIYl/HM 4aCTHoe I. CTOJI6HK 4HCeJI cnpaBa
OT l/epTbl COCTOHT H3 npocTblx MHO>KHTeJIeH, npoH3BeJJ.eHHe KOTOpblX
paBHo 420, TO eCTb 420=2.2.3.5.7.
COCTaBJIHH Bce B03MO>KHble npoH3BeAeHHH H3 nOJIY4eHHblx npoc-
TblX MHO>KHTeJIeH no 2, no 3 H T. A., nOJIY4HM Bce OCTaJIbHble JJ.eJIH-
TeJIH l/HCJIa. HanpHMep, TaKHMH JJ.eJIHTeJIHMH l/HCJIa 420 6YAYT:
2.2=4,2.3=6,2.7= 14,2.2.3= 12,2.2.7=28 H T. JJ..
A.
85. 06'bHCHH, LITO 3Hal/HT 4HCJIO pa3flO
UTb Ha npOCTble .MHO
UTe-
flU, npHBeAH npHMepbl.
86. (y C THO.) Pa3JIO>KH Ha npOCTble MHO>KHTeJIH l/HCJIa:
8, 10, 12, 14, 17, 18, 20, 25, 27, 31.
87. Pa3JIO>KH Ha npocTble MHO>KHTeJIH l/HCJIa:
48, 130, 60, 88, 96, 660, 250, 300, 72, 256, 1000.
88. KaKHe H3 4HCeJI 63, 71, 85, 101, 127, 160, 181, 204 cOCTaBHble?
KaKHe npocTble? COCTaBHble 4HCJ1a pa3JIO>KH Ha npocTble MHO-
>KHTeJIH.
89. HanHwH l/HCJ10, KOTopoe MO>KHO pa3JIO>KHTb: I) Ha JJ.Ba pa3JIHl/-
HblX npocTblX MHO>KHTeJIH; 2) Ha TPH pa3JIHl/HbIX npocTblX MHO-
>KHTeJIH; 3) Ha TPH OJJ.HHaKOBbIX npocTblX MHO>KHTeJIH.
90. HaHJJ.H Bce JJ.eJIHTeJIH Ka>KJJ.oro H3 l/HCeJI: 28, 32, 50, 60.
91. Ha KaKHe l/HCJIa JJ.eJIHTCH: I) npoH3BeJJ.eHHe 3.5. 13; 2) npoH3-
BeJJ.eHHe 2.2.7.11?
6.
92. Pa3JIO>KH Ha npocTble MHO>KHTeJIH 4HCJIa: 1500, 741, 2464,
15360.
17
93. HaHJJ.H Bce 'JJ.eJIHTeJIH Ka>KJJ.oro H3 l/HCeJI: 510, 1155, 1368.
94. l.J:HCJlO paCKJIaJJ.blBaeTcH Ha JJ.Ba JlBY3Hal/HbIX npocTblx MHO>KHTe-
JIH, pa3HocTb KOTOpblX 2. HaHJJ.H Bce TaKHe l/HCJIa.
95*. 3aMeHH 3Be3JJ.0l/KH LlH4>paMH, TaKHMH, l/T06bl paBeHcTBo 6blJIO
BepHblM:
I) 77* = * .5.7.11; 2) 3*5= * .3.5.7,
1.5. HaH6oJ1bWHH 06ll.1,HH .ll.enHTenb
BbInHWeM Bce JJ.eJIHTeJIH l/HCJIa 32:
h
h
16 H 32-
H Bce JJ.eJIHTeJIH 4HCJIa 24:
h
3, h6,
12 H 24.
06lliHMH JJ.eJIHTeJIHMH (OHH nOJJ.l/epKHYTbl) 4HCeJI 32 H 24 6YJJ.YT 1,2,4
H 8, HaH60JIbWHH H3 KOTOpblX 8.
Hau60llbUlUM o6UfuM iJelluTelleM iJ8gx HaTgpallbHblX 'Iucell Ha3bl-
8aeTClI. CaMoe 6ollbUloe HaTgpallbHoe 'IUCIlO. Ha ICoTopoe iJellUTClI.
ICClJICiJoe U3 iJaHHblx 'Iucell.
HaH60JIbWHH 061l1,HH JleJIHTeJIb JJ.BYX l/HCeJI a H b 0603Hal/a1OT
TaK: HOll (a, b). n03TOMY MO>KeM 3anHcaTb, LITO HOll (32, 24) =8,
B paCCMOTpeHHOM npHMepe HOll (32, 24) Mbl JIerKO nOJJ.06paJIH.
A eCJIH l/HCJIa 60JIbWHe? B TaKOM CJlY4ae pa3JIO>KHM HX Ha npOCTble
MHO>KHTeJIH H 6YJJ.eM PYKOBOJJ.CTBOBaTbCH CJIeJlYlOlliHM npaBHJIOM:
Hau60llbUlUU o6UfUU iJelluTellb iJ8gx 'Iucell pa8eH npoU38eiJeHulO
o6UfUX npOCTblX MH03ICuTelleii. iJaHHblx 'Iucell.
n p H Me p. HaHJJ.eM HOll (630, 252).
630
315
105
35
7
I
2 252
3 126
3 63
5 21
7 7
I
2 0 6 'b H C H e H H e.
2 Pa3JIO>KHM 3TH l/HCJIa Ha npOCTble
3 MHO>KHTeJIH H nOJJ.l/epKHeM Bce Te MHO-
3 >KHTeJIH, KOTopble HBJIHIOTCH 061l1,HMH B
7 060HX pa3JIO>KeHHHx: 2, 3, 3 H 7. npo-
H3BeJJ.eHHe 061l1,HX MHO>KHTeJIeH JJ.aeT
o T BeT: HOll (630, 252) =
=2.3.3.7= 126.
ItfTaK, JJ.JIH HaXO>KJJ.eHHH HaH60JIbWerO 061l1,ero JJ.eJIHTeJIH JJ.BYX
l/HCeJI:
I) pa31l03ICUM iJaHHble 'IUClla Ha npocTble MH03ICUTeIlU;
2) HauiJeM (noiJ'IepICHeM) 8ce o6Ufue npocTble MH03ICUTeAU 8
nOllg'leHHblx pa31l03ICeHU1I.X;
18
3) Haii.iJeM npoU38eiJeHue o6UfUX npocTblx MHo3ICuTelleii..
no 3TOMY >Ke npaBHJIY MO>KHO HaHTH HaH60JIbWHH 061l1,HH JJ.eJIH-
TeJlb JJ.JIH Tpex H 60JIee 4HCeJI. HanpHMep, 72=
'1.2'Q.:3, 84=
=2.2.7.3, 180=2.2.3.3.5,
-- - ---
3Hal/HT, HOll (72, 84, 180) =2.2.3= 12.
ECJlH JJ.aHHble 4HCJla He HMelOT 061l1,HX npOCTblX MHO>KHTeJIeH,
TO HaH60JIbWHM'061l1,HM JJ.eJIHTeJIeM 3THX l/HCeJI 6YJleT l/HCJIO l. HaTY-
paJIbHble l/HCJIa, HaH60JIbWHH 061l1,HH JJ.eJIHTeJIb KOTOpblX paBeH I, Ha-
3bIBalOTCH B3aHMHO npOCTblMH 4HcnaMH. HanpHMep, l/HCJIa 10 H 21
B3aHMHO npocTble, TaK KaK 10=2.5, 21 =3.7 H HOll (10, 21) = l.
ECJIH OJJ.HO HaTypaJIbHOe l/HCJIO JJ.eJIHTCH Ha JJ.Pyroe, TO MeHbwee
4HCJIO H HBJIHeTCH HaH60JIbWHM 061l1,HM JleJIHTeJIeM JJ.aHHbIX l/HCeJI.
HanpHMep, 32: 16=2, 3Hal/HT, HOll (32, 16) = 16.
A.
96. npOl/TH TeKCT naparpa4>a, OTBeTb Ha Bonpocbl H npHBeJJ.H npH-
Mepbl:
I) KaKoe l/HCJIO Ha3blBaeTCH Hau60llbUlUM 06U{UM aellurelleM
JJ.BYX l/HCeJI?
2) KaK HaHTH HaH60JIbWHH 061l1,HH JleJIHTeJIb JJ.BYX l/HCeJI?
97. HaHJJ.H (y C THO) JJ.eJIHTeJIH Ka>KJJ.oro H3 l/HCeJI, 061l1,He JJ.eJIHTeJIH
H HaH60JIbWHH 061l1,HH JJ.eJIHTeJIb l/HCeJI:
1) 4 H 12; 2) 6 H 15; 3) 4 H 10; 4) '8 H 18.
98. Bbl6epH H3 CJleJJ.YIOIlI,HX nap 4HCeJI B3aHMHO npocTble. Ha30BH HX
HaH60JIbWHH 061l1,HH JJ.eJIHTeJIb:
I) 15 H 21; 2) 6 H 7; 3) 18 H 34; 4) 23 H 46; 5) 24 H 25.
99. HaHJJ.H 061l1,He JJ.eJIHTeJIH H HaH60JIbWHH 061l1,HH JJ.eJIHTeJIb l/HCeJI:
I) 15 H 60; 2) 36 H 78; 3) 18 H 48; 4) 84 H 112; 5) 54 H 90.
100. HaHJJ.H HaH60JIbWHH 061l1,HH JJ.eJIHTeJIb l/HCeJI (HJIH HX pa3JIO>Ke-
HHH) :
I) 2.7.11.13 H 3.5.11.17;
2) 3.13.23.29 H 3.23.31;
3) 16 H 24;
4) 100 H 40;
5) 72 H 128;
6) 96 H 36;
7) 135 H 105;
8) 360 H 840;
9) 120 H 720.
101. B nepBOM TYpHCTCKOM oTPMe 24, a BO BTOpOM 18 TYpHCTOB.
1) HY}J{HO nOCTpOHTb nepBblir OTpHJJ. B KOJIOHHY, 'lTOObi B Ka}J{)J.OM
PH.D:Y ObIJIO TYpHCTOB nopoBHY. CKOJIbI<.O HMeeTCH B03Mo}JtHocTeir?
19
?
.
2) HYJKHO nOCTpOIITb BTOPOH
OTpjlJJ. B KOJIOHHY, QT06bI B KaJK-
.D:OM PMY ObIJIO TyplICTOB no-
POBHY. CKOJIbKO IIMeeTCH B03-
MOJKHOCTeiI?
3) HYJKHO nOCTpOIITb nepBbIH II
BTOpoiI OTP5I.IJ:bI BMeCTe B OJJ.HY KO-
JIOHHY (OJJ.IIH OTpHJJ. pHJJ.OM C JJ.PY-
rlIM) TaK, QTOObI B Ka)!().J:OM pHJJ.}
ObIJIO TYplICTOB nopOBHY. CKOJIb-
KO IIMeeTCjI B03MOJKHOCTeiI?
6.
102. 06'bHCHH, nOl/eMY cpeJJ.H 061l1,HX JJ.eJIHTeJIeH JJ.aHHbIX l/HCeJI BcerJJ.a
HaHJJ.eTcH HaHMeHbwee H HaH60JIblllee 4HCJIO.
103. HaHJJ.H HaH60JIbWHH 061l1,HH JJ.eJIHTeJIb 4HCeJI:
I) 1028 H 1152; 2) 18, 24 H 36; 3) 15, 45 H 165; 4) 12, 24
36 H 42.
104. B weCTblX KJIaCCaX 36 MaJIbl/HKOB H 48 JJ.eBOl/eK. CKOJIbKO
cYllI,eCTByeT B03MO>KHOCTeH C03JJ.aTb rpynnbl Yl/aIll,HXCH TaK,
4T06b1 BO Bcex rpynnax 6blJIO no oJJ.HHaKoBoMy l/HCJIY JJ.eB04eK H
no oJJ.HHaKoBoMY l/HCJIY MaJIbl/HKoB? KaKoe MO>KeT 6blTb HaH60JIb-
wee l/HCJIO TaKHX rpynn?
105*.113 LlH4>P 2, 4 H 6 COCTaBb BceB03MO>KHble TpeX3Hal/Hble l/HCJIa,
B KOTOpblX Bce LlH4>Pbl pa3JIHl/Hble. HaHJJ.H HaH60JIbWHH 061l1,HH
JJ.eJIHTeJIb 3THX l/HCeJI.
1.6. HaHMeHbwee o6w.ee KpaTHoe
BbInHWeM l/HCJIa, KpaTHble 2:
2, 4,
8, 10,
, 14, 16, j!!, 20, 22, 24 , 26,
H l/HCJIa, KpaTHble 3:
3,
9,
, 15, j!!, 21, 24 , 27, ..,
061l1,HMH KpaTHblMH (OHH nOJJ.l/epKHYTbl) l/HCeJI 2 H 3 6YJJ.YT 6, 12,
18, 24, ..., HaHMeHbillee H3 KOTOpblX 6.
HaUMeHbUluM o6UfUM ICpaTHblM iJ8YX IUlTypa/l.bHblx 'luce/l. Ha3bl-
8aeTCll. HaUMeHbUlee IUlTypa/l.bHoe 'lUC/l.O, ICOTopoe iJe/l.UTClI. Ha lCaJlCiJoe
U3 iJaHHblx 'luce/l..
HaHMeHbwee 061l1,ee KpaTHoe JJ.BYX l/HCeJI a H b 0603Ha4alOT
HOK (a, b). CTIeJJ.oBaTeJIbHO, MO>KHO 3anHcaTb, LITO HOK (2, 3) =6,
20
n p H M e p I. HaH.D:eM HOK (36, 42).
4T06bl HaHTH HaHMeHbwee 061l1,ee KpaTHoe .D:BYX l/HCeJI, pa3JIO>KHM
3TH 4HCJIa Ha npOCTble MHO>KHTeJIH. nOJIYl/HM 36=2.2.3.3 H 42=
= 2.3.7. HaHMeHbwee 061l1,ee KpaTHoe JJ.OJI>KHO JleJIHTbCH H Ha 36,
H Ha 42. n03TOMY OHO JJ.OJI>KHO cOJJ.ep>KaTb Bce npocTble MHO>KHTeJIH
H nepBoro H BToporo l/HCJIa. PaccMoTpHM pa3JIO>KeHHe OJJ.Horo H3
3THX l/HCeJI, HanpHMep 42=2.3.7, H BbIHCHHM, KaKHe npocTble MHO-
>KHTeJIH JJ.Pyroro 4HCJIa B 3TOM pa3JIO>KeHHH OTCYTCTBYIOT. TaKHMH
MHO>KHTeJlHMH 6YJlYT 2 H 3, .lleHcTBHTeJIbHO, B pa3JIO>KeHHH 42=2X
X 3.7 HMeeTCH OJJ.HH MHO>KHTeJIb 2 H OJJ.HH MHO>KHTeJIb 3, a B pa3JIO>Ke-
HHH 36 = 2.2.3.3 JJ.Ba MHO>KHTeJIH 2 H JJ.Ba MHO>KHTeJIH 3. CJIeJJ.o-
BaTeJlbHO, JJ.JIH nOJIYl/eHHH HOK (36, 42) HY>KHO pa3JIO>KeHHe
42=2.3.7 JJ.onOJIHHTb HeJJ.OCTaIOIlI,HMH MHO>KHTeJIHMH 2.3. no-
J1Yl/HM:
HOK (36,42) =2.3.7.2.3=252.
-.---
42
I1TaK, JJ.JIH HaXO>KJleHHH HaHMeHbwero 061l1,ero KpaTHoro JJ.BYX
l/HCeJI:
I) pa3/1.03ICUM iJaHHble 'lUC/l.a Ha npocTble MH03ICUTe/l.U;
2) pa3/1.03ICeHUe OiJH020 U3 HUX iJonO/l.HUM TeMU MH03ICUTe/l.1l.MU
pa3/1.03ICeHU1I. iJpY2020 'lUC/l.a, ICOTOpblX HeT 8 pa3/1.03ICeHUU nep802o;
3) 8bl'lUC/l.UM npou38eiJeHue nO/l.Y'leHHblX MH03ICUTe/l.eu.
n p H M e p 2. HaHJJ.eM HOK (462, 420).
462 2 420 2
231 3 210 2 HOK (462, 420) =
77 7 105 3 = 2 . 3 .7. II .2.5 = 4620
II II 35 5
I 7 7 462
I
ECJIH OJJ.HO l/HCJIO JleJIJ.ITCH Ha JJ.Pyroe, TO 60JIbWee l/HCJIO H HBJIHeTCH
HaHMeHbWHM 061l1,HM KpaTHblM 3THX 4HCeJI. HanpHMep, HOK (8, 16) =
=16,
HaHMeHbwHM 061l1,HM KpaTHblM JJ.BYX B3aHMHO npocTblX l/HCeJI 6y-
JJ.eT npOH3BeJleHHe 3THX l/HCeJI. HanpHMep, HOK (8, 15) =8.15=
= 120. 06'bHCHH n04eMY!
HaHMeHbwee 061l1,ee KpaTHoe MO>KHO HaHTH He TOJIbKO JJ.JIH JJ.BYX
l/HCeJI, HO H JJ.JIH Tpex H 60JIee 4HCeJI,
21
n p H M e p 3, HaHJJ.eM HOK (28, 53, 60),
nOJIY4HM:
28=2.2.7
63=3.3.7 HOK (28,63,60)=2.2.7.3.3.5=1260
60=2.2.3.5
A.
106. npOl/TH TeKCT naparpacpa, OTBeTb Ha Bonpocbl H npHBeJJ.H
npHMepbl:
I) KaKoe l/HCJIO Ha3bIBaeTCH HaU.MeHbUlU.M o6U{u.M KpaTHbt.M
JlBYX l/HCeJI?
2) KaK HaHTH HaHMeHbwee 061l1,ee KpaTHoe JJ.BYX l/HCeJI?
107. Ha30BH TPH 4HCJIa, KpaTHble 2 H 10.
108. HaHJJ.H (y C THO) l/eTblpe 061l1,HX KpaTHblX H HaHMeHblilee 061l1,ee
KpaTHOe l/HCeJI:
I) 2 H 4; 2) 3 H 5; 3) 4 H 6; 4) 5 H 10; 5) 20 H 50.
109. HaHJJ.H HaHMeHbwee 061l1,ee KpaTHoe 4HCeJI (HJIH HX pa3JIO>Ke-
HHH) :
I) 6 H 10; 2) 9 H 24; 3) 6 H 35; 4) 15 H 18; 5) 16 H 24;
6) 36 H 60; 7) 23 H 69; 8) 28 H 21; 9) 2.2.2.5 H 2.3.3.
110. HanHwH TPH napbl B3aHMHO npocTblX l/HCeJI. HaHJJ.H JJ.JIH Ka>KJJ.oH
napbl HaHMeHbwee 061l1,ee KpaTHoe.
111. KaKoe HaHMeHbwee 4HCJIO MeTpoB MaTepHaJIa JJ.OJI>KHO 6bITb B
PYJIOHe, l/T06b1 ero MO>KHO 6blJIO npoJJ.aTb 6e3 OCTaTKa no 3 M HJIH
no 4 M?
6.
112. 06'bHCHH, n04eMY JII06ble JlaHHble HaTypaJIbHble l/HCJIa HMelOT
6eCKOHel/HO MHoro 061l1,HX KpaTHblx. nOl/eMY cpeJJ.H HHX HMeeTCH
HaHMeHbwee l/HCJIO (HaHMeHbwee 061l1,ee KpaTHoe)?
113. HaHJJ.H HaHMeHbwee 061l1,ee KpaTHoe l/HCeJI:
I) 50 H 180; 2) 270 H 360; 3) 220 H 231; 4) 3,4 H 6; 5) 4,
6 H 18; 6) 10, 25 H 30; 7) 8, 15 H 23.
114. B JJ.eTCKOM BeJIOCHneJJ.e BeJJ.ymaH weCTepHH (cKpenJIeHHaH C
neJJ.aJIHMH) HMeeT 44 3y6Lla, a BeJJ.OMaH weCTepHH (cKpenJIeH-
HaH C 3aJJ.HHM KOJIeCOM BeJIOCHneJJ.a) HMeeT 20 3y6LlOB. Onpe-
JJ.eJIH HaHMeHbwee l/HCJIO 060POTOB, KOTopoe CJJ.eJIaeT BeJJ.Y-
llI,aH weCTepHH, l/T06bl wecTepHH 3aHHJIH nepBOHal/aJIbHOe no-
JIO>KeHHe. CKOJIbKO 060pOTOB 3a 3TO BpeMH CJJ.eJIaeT BeJJ.OMaH
weCTepHH?
22
115*. (3 a JJ. a l/ a - w YT K a.) MaJIb-
l/HKH C naJIbl/HK peWHJIH opra-
HH30BaTb KOMaHJJ.Y, KOTopaH ox-
paHHJIa 6bl COKpOBHIlI,HHLlY, 3a-
TpYJJ.HeHHH Hal/aJIHCb, Koma
BblHCHHJIOCb, LITO MO>KeT B03HHK-
HYTb He06xoJJ.HMOCTb pa36HTb
KOMaHJJ.Y Ha OTpHJJ.W HJIH no 12,
HJIH no 15 l/J1eHOB B Ka>KJJ.OM,
MaJIb4HKH C naJIb4HK peWHJIH
3TY CJIO>KHYIO 3aJJ.al/Y - HaWJIH HaHMeHbwee l/HCJIO l/JIeHOB,
H3 KOToporo 6bl COCTOHJIa JJ.e>KypHaH KOMaHJJ.a. nonp06YH H
Tbl cnpaBHTbcH C 3THM,
AnH nOBTopeHHH
116. HaHJJ.H rpaJJ.ycHYIO Mepy yrJIa, KOTOpblH COCTaBJIHeT:
I) 1
2
2
npHMoro yrJIa; 2) "3 npHMoro yrJIa; 3) "'3 npHMoro yrJIa;
1 3 2
4) ""6 pa3BepHYToro yrJIa; 5) "'4 pa3BepHYToro yrJIa; 6) "'9
pa3BepHYToro yrJIa. nOCTpOH 3TH yrJIbl C nOMOlll,blO TpaHcnop-
THpa,
3
C06paJIH 24 Kr HroJJ., "'4 Bcex HroJJ. HCnOJIb30BaJIH Ha BapeHbe.
CKOJIbKO KHJIOrpaMMoB HroJJ. HCnOJIb30BaJIH Ha BapeHbe?
PewH ypaBHeHHe:
I) 5963+x=6398 2) x-42=5,1 3) x: 100=0,4
14,89-y=6,9 15z=303 u: 10,4=2,4
YnpocTH Bblpa>KeHHe:
I) 85-35 2) 9/+4/-/
6x+4x 16y-5y-3y
117.
It 8.
1t9.
3) 17x+33x
45a - a
4) 4,8x-3,lx+0,7x
O,16a+ 1,2a-O,85a
OJJ.HH MeWOK C KapTocpeJIeM BeCHJI 47,6 Kr. KorJJ.a K HeMY noJJ.-
JIO>KHJIH BTOpOH MelllOK, TO BeCbl nOKa3aJIH 93,5 Kr, a KorJJ.a
nOJJ.JIO>KHJIH TpeTHH MewOK, TO BeCbl nOKa3aJIH 152,7 Kr. nYCTble
MeWKH BeCHJIH COOTBeTCTBeHHO 0,9 Kr, 0,8 Kr H 0,9 Kr. nOCTaBb
pa3YMHble BonpOCbl H OTBeTb Ha HHX.
120.
23
AnH caMonpOBepKH
121. KaKHe l/HCJIa Ha3bIBalOTCH JJ.eJIHTeJIHMH 4HCJIa?
122. 3anHwH Bce JJ.eJIHTeJIH 4HCJIa 56 H l/HCJIa 72,
123. CpeJJ.H l/HCeJI 1,2,3,5,7,9, 10 H II HaHJJ.H AeJIHTeJIH Ka>KJJ.oro
H3 4HCeJI 12, 17, 44, 90 H 98.
124. Y 3HHbl 18 LlBeTHblX KapaHJJ.aweH, Ka>KJJ.oro LlBeTa nopoBHY,
CKOJIbKO OAHOLlBeTHblX KapaHJJ.aweH MO>KeT 6blTb Y 3HHbl? BbIH-
BH Bce B03MO>KHOCTH.
125. KaKoe l/HCJIO Ha3blBaeTCH KpaTHblM l/HCJIY?
126. HaHAH JJ.BY3Hal/Hble l/HCJIa, KpaTHble 26.
127. HanHwH TpeX3Hal/HOe l/HCJIO, KpaTHoe 5.
128. 06'bHCHH, nOl/eMY l/HCJIO 6984 KpaTHo 18.
129. OPOPMYJIHPYH npH3HaKH JJ.eJIHMOCTH Ha 2, 3, 5, 9 H 10. npH-
BeJJ.H npHMepbl Ha Ka>KAbIH npH3HaK.
130. KaKHe H3 l/HCeJI 38, 75, 2340, I 074393, 65400 JJ.eJIHTCH:
I) Ha 2; 2) Ha 3; 3) Ha 5; 4) Ha 9; 5) Ha 1O?
131. 51BJIHeTcH JIH l/HCJ10 9 JJ.eJIHTeJIeM l/HCJ1a 604 799? (nOl/eMY?)
132. KaKHe l/HCJIa Ha3blBalOTCH npocTblMH? cOCTaBHbIMH? npHBeAH
npHMepbl npocTblX H COCTaBHblX l/HCeJI.
133. Pa3JIO>KH Ha npocTble MHO>KHTeJIH l/HCJIa 42, 76, 88, 495.
134. KaKoe l/HCJIO Ha3blBaeTCH HaH60JIbWHM 061l1,HM JJ.eJIHTeJIeM (HaH-
MeHbWHM 061l1,HM KpaTHbIM) JJ.BYX HaTypaJIbHblX l/HCeJI?
135. Bbll/HCJIH:
I) HOJl (12, 18); 2) HOJl (21, 105); 3) HOJl (54, 72);
4) HOK (8, 18); 5) HOK (14,42); 6) HOK (25,40).
136. Ha copeBHoBaHHHx no HaCTOJIbHOMY TeHHHcy Yl/aCTBOBaJIH paB-
Hble no cOCTaBY KOMaHJJ.bl, Bcero 145 MaJIbl/HKOB H 87 JJ.eBOl/eK,
Bo Bcex KOMaHJJ.ax 6b1JIO OJJ.HHaKOBoe 4HCJIO MaJIbl/HKOB H OJJ.H-
HaKOBoe 4HCJIO JJ.eBOl/eK. CKOJIbKO KOMaHJJ. Yl/aCTBOBaJIO B copeB-
HOBaHHHx? CKOJIbKO MaJIbl/HKOB
H CKOJIbKO JJ.eBOl/eK 6blJIO B Ka>K-
JJ.OH KOMaHJJ.e?
137. MaJIeHbKaH Kop06Ka BMelll,aeT 24
KapaHJJ.awa, a 60JIbWaH 30
KapaHJJ.aweH. HaHJJ.H HaHMeHb-
wee l/HCJ10 KapaHJJ.aweH, KOTO-
poe MO>KeT 6bITb pa3JIO>KeHO KaK
B MaJIeHbKHe Kop06KH, TaK H B
60JIbW He.
24
--
2. 06ll1.HE CBOACTBA 06blKHOBEHHbiX
J].P06EA.
CJ10)l(EHHE H Bbl....HTAHHE
2.1. 06b1KHOBeHHaSl .ll.p06b
KaK qaCTHOe OT )I.eJ1eHHSI
MaTepHaJI 3Toro naparpa4>a H3Yl/H caMOCTOHTeJIbHO, HCnOJIb3YH
yKa3aHHH K 3aJJ.aHHIO 138.
Tbl Y>Ke 3HaeWb, 4TO 06blKHOBeHHaH JJ.P06b nOKa3blBaeT, Ha
CKOJlbKO paBHWX 4aCTeH pa3JJ.eJIeHa OJJ.Ha LleJIaH H CKOJIbKO TaKHX
l/aCTeH B3HTO. HanpHMep,
nOKa3bIBalOT, LITO LleJIaH (eJJ.HHHLla)
pa3JJ.eJIeHa Ha 4 paBHble l/aCTH H TaKHX l/aCTeH Bcero B3HTO 3 (pHC. 2.1).
nOKa>KeM, LITO 06blKHOBeHHYIO Ap06b MO>KHO paCCMaTpHBaTb KaK
l/aCTHOe OT JJ.eJIeHHH JJ.BYX HaTypaJIbHblX l/HCeJI. HanpHMep, HY>K-
HO pa3JJ.eJIHTb 8 H6JIOK nopoBHY Me>KJJ.Y l/eTblpbMH JJ.eTbMH. JlJIH
:noro HaHJJ.eM 4aCTHoe OT JJ.eJIeHHH, T. e. 8: 4 = 2. Ka>K.!lbIH pe6eHoK
nOJIYl/HT no 2 H6JIOKa. npeJJ.nOJIO>KHM, 4TO Tenepb HY>KHO pa3JJ.eJIHTb
nopOBHY 3 H6JIOKa Me>KJJ.Y l/eTblpbMH JJ.eTbMH. JlJIH 3Toro Ka>KJJ.oe H6JIO-
1
KO pa3JJ.eJIHM Ha 4 paBHble l/aCTH. 3Hal/HT, ll/acTb - 4 H6JIOKa. TaKHX
KYCOl/KOB Bcero 12 (pHC. 2.2), n03ToMY Ka>KJJ.bIH pe6eHoK nOJIYl/HT no
3 KycOl/Ka, TO eCTb
H6JIOKa, ECJIH 3anHcaTb peweHHe 3TOH 3aJJ.a-
3
4H, TO nOJIY4HM 3:4=4' TO eCTb
06blKHOBeHHYIO JJ.p06b MO>KHO pac-
CMaTpHBaTb KaK 4aCTHoe OT JJ.eJIeHHH.
Bo06w.e lCaJlCiJylO 06blICH08eHHYlO
\...
\\\.\\\\\.
1
\\\\\.\\\\\
3 _I
4
-I
I
PHC. 2.1
KonR
Bepa
HMH8
PHC. 2.2
25
iJP06b M03ICHO paCCMaTpU8aTb teate 'laCTHOe OT iJelleHWI., 8 teOTOpOU
'lUCIlUTellb iJpo6u - iJellUMoe, a 3HaMeHaTellb iJpo6u - iJelluTellb.
YepTa JJ.P06H 0603Hal/aeT JJ.eHCTBHe JJ.eJIeHHe. 3Hal/HT,
'b a
a. =b.
3aMeTHM, LITO 3HaMeHaTellb iJpo6u He M03ICeT 6blTb pa8eH Hymo,
TaK KaK Ha HYJIb JJ.eJIHTb HeJIb3H. Ka>KJJ.oe HaTypaJIbHOe l/HCJIO TaK>Ke
MO>KHO 3anHcaTb B BHJJ.e JJ.P06H.
npHMep l.
I ) 1=- 1 1 = 2 2 = 3 3 =... , . 2 ) 2-2-
-
- 3) 3 3
- 1 - 2 - 3 - ..., =,=
_6_9
-""2-3
YepTa JJ.P06H - 3TO 3HaK JJ.eJIeHHH, n03TOMY C ee nOMOlll,blO
MO>KHO 3anHcaTb 60JIee CJIO>KHble Bblpa>KeHHH, KOTopble Ha3bIBalOT
J1.p06HblMH Bblpa)l(eHHHMH.
n p H M e p 2.
I) 0 0 , ' 5 8 =0,8:0,5= 1,6; 2) 1,2+0,9_
=3,
7: 10 0,7
A.
138. I. np04TH BHHMaTeJIbHO a63aLl TeKcTa, KOTOpblH Hal/HHaeTCH
CJIOBaMH «Tbl Y>Ke 3Haewb...», H 06'bHCHH, 4TO nOKa3blBaeT
524
JJ.P06b: 9' "3' 5'
2. npOl/TH CJIeJJ.YlOlliHH a63aLl TeKCTa H BblJJ.eJIH OCHOBHoe. KaK
7 1 5
nO-JJ.PyrOMY MO>KHO 3anHcaTb JJ.P06H 12' "3' 6?
3. np04TH a63aLl TeKCTa, KOTOpblH Ha4HHaeTCH CJ10BaMH «3a-
MeTHM, l/TO..,», H npoJJ.YMaH npHMep l. B BHJJ.e KaKHX JJ.p06eH
MO>KHO 3anHcaTb HaTypaJIbHble l/HCJIa 5, 10, 99?
4. npOl/TH nOCJIeJJ.HHH a63aLl TeKCTa H pa36epHcb B npHMepe 2,
5. Bbll/HCJIH 3Ha4eHHe Bblpa>KeHHH:
I) 8+ 10 , 2) 2.3+8
2.9' 20-13
139. 4TO 03Hal/a1OT JJ.P 06H 4 9 17 H
';)
15' 23 ' 60 100 .
140. 3anHwH KaK l/aCTHOe OT JJ.eJIeHHH npH nOMOIll,H 3HaKa «:» JJ.P06b:
I 13 2 33
8""' 15' 25 H 100 '
26
141. 3anHwH 4aCTHble B BHJJ.e 06b1KHOBeHHbIX J1.p06eH:
I) 2:3, 4:5, 1:7, 6:9, 4:7, 5: 100;
2) 3: I, 6:2, 7:3, 10:5, 20:4, 35:7.
142. Bblpa3H HaTypaJIbHble 4HCJIa 12, 15, 20 B BHJJ.e JJ.p06eH, 3HaMe-
HaTeJIHMH KOTOpblX HBJIHIOTCH l/HCJIa: I) 2; 2) 3; 3) 4; 4) 10.
143. 4eMY paBHbl J1.P06H:
10 0 25 30 0 45 0 100 500 ';)
2' 7' 5' 6' 16' "3' 200' 10' 5 '
144. Bbl4HCJIH:
I )
. 2 ) 24-6 3 ) 10.40+60 . 4 )
8+4 '3 23'
6) 0,56 , 7) 12,1 +4,9 . 8) 1,8
0,07 ' 1,7' 5,4-0,6 .
85 . 5 3,2 ,
120:6-15 ' ) 0,8'
6.
145. 3anHwH: I) l/HCJIO 3 B BHJJ.e JJ.P06H, 3HaMeHaTeJIb KOTOpOH 33;
2) 4HCJIO 7 B BHJJ.e JJ.p06H,3HaMeHaTeJIb KOTOpOH 777; 3) l/HCJIO
II B BHJJ.e JJ.P06H, 3HaMeHaTeJIb KOTOpOH 1210,
146. npHBeJJ.H no TPH npHMepa JJ.JIH CJIY4aH: I) JJ.P06b paBHa CBoe-
MY 3HaMeHaTeJIIO; 2) J1.p06b paBHa cBoeMY 4HCJIHTeJIIO.
147. Bbll/HCJIH:
I) 15,6+4.4 . 2) 2 (0,75+2,8) . 3) 9,4.0,75-1,85
(0,6-0,2).25' 10:0,1' 2,5,\,3
2.2. OCHOBHoe CBoikTBO .ll.P06H
npH JJ.eJIeHHH HaTypaJIbHblX l/HCeJI H JJ.eCHTHl/HbIX JJ.p06eH Mbl n03Ha-
KOMHJIHCb C OCHOBHblM CBOHCTBOM l/aCTHoro: JJ.eJIHMOe H JJ.eJIHTeJIb
MO>KHO YMHO>KHTb Ha OJJ.HO H TO >Ke HaTypaJIbHOe l/HCJ10, l/aCTHOe npH
3TOM He H3MeHHTCH. TaK KaK 06blKHOBeHHYlO JJ.P06b MO>KHO paCCMaT-
pHBaTb KaK l/aCTHOe OT JJ.eJIeHHH, TO 3TO CBOHCTBO J1.0JI>KHO 6blTb
npHMeHHMO H K 06blKHOBeHHblM JJ.P06HM. Ha pHcYHKe 2.3 JJ.P06H
()
fj - fD . / £) 3 ", f9 5 !.--
4 6 '- 10 __
, \
1 2 4 3
2 - ""4 - B 6"
PHC. 2.3
5
10
27
243 5
2 ' 4' 8' 6' 10 H306pa>Ka1OT OJJ.HY H TY >Ke cpHryPY (noJIY-
Kpyr), 3Hal/HT, OHH paBHbI. 3anHllleM:
I 2 4 3 5
2-4"=8=6=10'
2 4
PaCCMOTpHM, HanpHMep, paBeHCTBO 4=8' B 3TOM paBeHCTBe
H3 JIeBOH 4aCTH nOJIY4HM npaBYIO, eCJIH 4HCJIHTeJIb H 3HaMeHaTeJIb
6 2 2 Jl u 4 2.2
JlpO H 4 YMHO>KHTb Ha " eHCTBHTeJIbHO, 8= 4.2 .
06paTHO: H3 JJ.P06H
4
8 MO>KHO nOJIY4HTb paBHYIO eH JJ.P06b
2 4
4' eCJIH 4HCJIHTeJIb H 3HaMeHaTeJIb JJ.P06H 8 pa3JJ.eJIHTb Ha 2, TO
2 4:2
eCTb 4"= 8:2 .
I-haK, 8eIlU'lUHa iJpo6u He U3MeHUTCR. eCllu 'lUCIlUTellb U 300-
MeHaTellb iJpo6u YMH03ICUTb UIlU pa3iJelluTb Ha OiJHO U TO 3ICe HaTY-
pallbHOe 'lUCIlO. 3TO H eCTb OCHOBHoe CBOHCTBO JJ.P06H.
06'bHCHH Tenepb, nOl/eMY paBeHCTBa
3 9 5 25 30 3 55 11
4" 12'"6 30 ' 40 4' 25 -5
BepHbI.
A.
148. 06'bHCHH, B l/eM 3aKJIIOl/aeTCH OCHOBHoe CBOii.CTBO apo6u. npH-
BeJlH npHMepbl.
149. Pa3JJ.eJIH l/HCJIHTeJIb H 3HaMeHaTeJIb Ka>KJJ.oH JJ.P06H Ha 5:
5 30 15 100 90 125 85 245
10 ' 40 ' 35 ' 80' 120 ' 1000 ' 500 ' 200
3anHWH COOTBeTCTBYlOllI,He paBeHcTBa.
150. Pa3JJ.eJIH l/HCJIHTeJIb H 3HaMeHaTeJIb Ka>KJJ.oH JJ.P06H Ha HX
HaH60JIbWHH 061l1,HH JJ.eJIHTeJIb:
3 6 6 10 15 16 4 15
9' 12 ' 18' 15' 20 ' 18' 16' "24"
8
15t. 3anHWH TPH JJ.P06H, paBHble 24 ,3HaMeHaTeJIH KOTOpblX MeHbwe
3HaMeHaTeJIH JJ.aHHOH JJ.P06H.
28
2 YMHO>Kb l/HCJIHTeJIb H 3HaMeHaTeJIb Ka>KJJ.oH JJ.P06H Ha 3:
15 .
1 3 8 5 10 I 11 5
2'4'5'6'"3'10'2'1'
3anHWH COOTBeTCTBYlOllI,He paBeHCTBa.
1
153. 3anHwH TPH JJ.P06H, paBHble 7"'
154. 3aMeHH Ka>KJJ.YIO H3 CJIeJJ.YlOlliHx Jlp06eH JJ.P06blO, 3HaMeHaTeJIb
KOTOpOH paBeH 24:
121355711
2' "3' 4"' 4' 6' 8' 12' 12
155. 3aMeHH Ka>KJJ.YIO H3 CJIeJJ.YIOIlI,Hx JJ.p06eH JJ.P06blO, 3HaMeHaTeJIb
KOTOpOH paBeH 100:
11397231129
2' 4' 5' 10' 20 ' 20 ' 25' 50 '
6.
156. BepHbl JIH paBeHcTBa (oTBeT 060CHYH):
28 7 9 54 4 120 4 ) 66 =
;I
I) 42"=21; 2) 10= 60 ; 3) 20 = 300 ; 88 4'
157. Ka>KJJ.bIH l/HCJIHTeJIb H Ka>KJJ.bIH 3HaMeHaTeJIb Bcex JJ.aHHbIX JJ.po-
6eH MO>KHO pa3JJ.eJIHTb Ha OJJ.HO H TO >Ke 4HCJIO. BbInOJIHH JJ.e-
JIeHHe Ha 3TO 4HCJIO H 3anHWH COOTBeTCTBYlOllI,He paBeHCTBa:
24 15 39 450 75
36 ' 42' 21' 180 ' 90'
158. 3aMeHH x TaKHM l/HCJIOM, l/T06bl paBeHcTBo 6blJIO BepHblM:
12 3 I x 60 3 x 48
1) -;- =4; 2) 5= 45 ; 3) 100 =-;-; 4) "3= 72 '
159. Ha4epTH 4HCJIOBOH JIYl/, OTMeTb Ha HeM TOl/KH, KOTOpblM COOT-
1 3 1 6 2 4 1
BeTCTBYIOT l/HCJIa 8' 4' 2' 8' 8' 8' 4' 3anHWH noJJ.
Ka>KJJ.oH T04KOH elll,e JJ.Be JlP06H, KOTopble H306pa>KaIOTCH TOH
>Ke TOl/KOH,
.D.JlH nOBTOpeHHH
160. Bbll/HCJIH:
I) HOJl (25, 40); 2) HOJl (84, 105); 3) HOJl (23, 46, 69);
4) HOK (16, 48); 5) HOK (21, 35); 6) HOK (8, 20, 30).
29
161.
Bbll/HCJIH (y C THO) :
I) (12.7+ 12.3):4
8.42-8.2+80
0,95.10+0,5
Bbll/HCJIH:
2) 13.28-3.28+20
(92.51 +8.51): 100
180: 100-0,8
162.
I ) 2.17,05 2 ) 1,32+1,509 . 3 ) (4,2+15,6):0,3
132,6-98,5' 3.(3,85-2,7)' 187 + 143
163. 3anHwH B BHJJ.e 06blKHoBeHHoH JJ.P06H:
0,4; 0,7; 0,03; 0,009; 0,23; 0,701; 1,5; 3,67.
164. 3anHwH B BHJle JJ.ecHTHl/HOH JJ.P06H:
8 9 37 7 159 43 6
10 ' 10' 100 ' 100 ' 1000 ' 1000 ' 1000 '
165. 4HCJIO a YMeHbWHJIH Ha 13 H l/HCJIO 522 pa3JJ.eJIHJIH Ha no-
JIYl/eHHYIO pa3HocTb. COCTaBb Bblpa>KeHHe H Bbll/HCJ1H ero 3Ha-
l/eHHe, eCJIH a = 73, a = 100.
166*. B l/HCJIe 7030605 Bce HYJIH 3aMeHH OJJ.HOH H TOH >Ke LlHlPpOH,
TaKoH, l/T06bl nOJIYl/eHHOe l/HCJIO JJ.eJIHJIOCb Ha 9. 3anHwH Bce
B03MO>KHble LlH4>PbI.
2.3. COKpaUJ.eHHe .ll.p06eH
.lleJIeHHe l/HCJIHTeJIH H 3HaMeHaTeJIH JJ.P06H Ha OJJ.HO H TO >Ke
HaTypaJIbHOe 4HCJIO Ha3bIBaeTCH COKpallJ,eHHeM JlP06H.
3 3:3 1
n p H M e p I. 6= 6:3 =""2' .llP06b COKpalll,eHa Ha 3,
25 25:5 5
n p H M e p 2. 15= 15:5 ="'3' .llP06b cOKpaw.eHa Ha 5,
06Wl/HO JJ.eHcTBHe JJ.eJIeHHe (l/HCJIHTeJIH H 3HaMeHaTeJIH Ha OJJ.HO
H TO >Ke l/HCJIO) He YKa3blBaeTCH H COKpalll,eHHaH JJ.P06b 3anHCblBaeT-
CH cpa3Y 3a 3HaKOM paBeHCTBa,
12 3 yP 3
n p H Me p 3. 16=""4 HJIH
=""4'
.llP06b COKpalll,eHa Ha 4,
COKpaTHTb JJ.P06b MO>KHO, eCJIH ee l/HCJIHTeJIb H 3HaMeHaTeJIb
HMelOT 061l1,HH JJ.eJIHTeJIb, OT JIHl/HbIH OT l. ECJIH l/HCJIHTeJIb H 3HaMe-
HaTeJIb JJ.P06H B3aHMHO npOCTble, TO JJ.p06b cOKpaTHTb HeJIb3H. Ta-
KaH JJ.P06b Ha3WBaeTCH HeCOKpaTHMOH JlP06bW. HanpHMep, JJ.P06H
5 4 11 u
6' 7'"'8 HeCOKpaTHMble. 4T06bl H3 JJ.aHHOH JJ.P06H nOJIYl/HTb HC-
30
COKpaTHMYIO JJ.P06b, HY>KHO l/HCJIHTeJIb H 3HaMeHaTeJIb pa3JJ.eJIHTb Ha
!IX Hall60JIbWHH 061l1,HH JJ.eJIHTeJIb,
COKpaw.aTb JJ.P06b MO>KHO JJ.BYMH cnoc06aMH:
I) nOCTeneHHO, nOJJ.6HpaH nOJJ.XOJJ.HIlI,He 061l1,He JJ.eJIHTeJIH l/HCJIHTe-
nfl II 3HaMeHaTeJIH, nOKa He nOJIYl/HTCH HeCOKpaTHMaH JJ.p06b;
2) cpa3Y, pa3JJ.eJIHB l/HCJIHTeJIb H 3HaMeHaTeJIb Ha HX HaH60JIbWHH
o6illHH JJ.eJIHTeJIb.
42
n p H M e p 4. COKpaTHM JJ.P06b 70 .
42 21 3
n e p B bl H C no C 0 6. 70 35 -""'5 (CHal/aJIa COKpaTHJIH Ha 2,
3aTeM Ha 7).
B TOp 0 H C n 0 C 0 6. HO.ll (42, 70) = 14, CJIeJJ.OBaTeJIbHO,
E.=2. (l/HCJIHTeJIb H 3HaMeHaTeJIb pa3JJ.eJIHJIH Ha 14).
70 5
A.
167. npoBepb no TeKcTY. 3HaeWb JIH:
I) LITO 3Hal/HT COKpaTUTb ap06b;
2) Ha KaKoe l/HCJIO HY>KHO cOKpaTHTb JJ.P06b, 4T06bl nOJIYl/HTb
necoKpaTUMYIO JJ.P06b.
168. BbIHCHH, Ha KaKoe HaH60JIbWee 4HCJIO JleJIHTCH l/HCJIHTeJIb H
3HaMeHaTeJIb Ka>KJJ.oH H3 JJ.aHHbIX JJ.p06eH. COKpaTH (y C THO):
2 3 4 6 4 5 10 15 6 10
""4'6'6'9'W'I5' 30 ' 20 '
' 100 '
169. BbInHWH H3 JJ.aHHbIX JJ.p06eH Te, KOTopble MO>KHO CDKp.aTHTb, H
BbInOJIHH COKpalll,eHHe:
5 9 4 8 12 3 6 9
6' 6' 9'
, 18' 10' 24 ' 24 '
170. CoKpaTH JJ.P06H:
1 ) 12 18 36 48 14 16 18 21
15 , 54 , 42 ' 42 ' 35 ' 10' 15 ' 7'
2) 36 72 45 12 9 17 22 24
64 , 81 , 60 ' 45' 30 ' 51 . 66 ' 56 '
3) 21 60 250 500 120 60 140 420
300 ' 100 ' 1000 ' 2000 ' 150 ' 144 ' 770 ' 600 '
171. 3anHwH B BHJJ.e 06blKHoBeHHoH JJ.P06H H, eCJIH B03MO>KHO, co-
KpaTH:
0,2; 0,4; 0,5; 0,7; 0,06; 0,08; 0,09; 0,25; 0,36; 0,55; 0,75;
0,84; 0,125; 0,375; 0,875.
31
6.
172. .llaH OTBeT B BHJJ.e HecoKpaTHMoH Ap06H:
1) KaKYIO l/aCTb KHJIOrpaMMa COCTaBJIHIOT 80 r? 400 r? 750 r?
2) KaKYIO l/aCTb MeTpa COCTaBJIHIOT 25 CM? 50 CM? 90 CM?
3) KaKYIO l/aCTb KBaJJ.paTHoro MeTpa COCTaBJIHIOT 8 JJ.M 2 ? 55 JJ.M 2 ?
72 JJ.M 2 ?
173. COKpaTH JJ.P06H:
147 26 206 114 1008
210 ' 390 ' 618 ' 171' 1224'
174. 3anHwH B BHJJ.e 06blKHoBeHHoH JJ.P06H H cOKpaTH:
0,645; 0,075; 0,364; 0,0025; 0,0725.
175. BHal/aJIe JJ.P06b cOKpaTHJIH Ha 2, nOTOM Ha 3, 3aTeM Ha 5,
Ha KaKoe l/HCJIO MO>KHO 6blJIO cpa3Y cOKpaTHTb JJ.p06b?
176. COKpaTH, 3aTeM Bbll/HCJIH:
21.16 36 12.5 45.11
I). 28; 2) 18.14 ; 3) 15.3 ; 4) 22.18 '
2.4. npHBe.ll.eHHe .ll.p06eH K o6UJ.eMY 3HaMeHaTeJ1l0
OCHOBHoe CBOHCTBO JJ.P06H n03BOJIHeT 3aMeHHTb JJ.P06H C pa3Hbl-
MH 3HaMeHaTeJIHMH JJ.P06HMH, 3HaMeHaTeJIH KOTOpblX paBHbl. B 3TOM
CJIYl/ae Mbl rOBopHM, LITO JJ.p06H C pa3HblMH 3HaMeHaTeJIHMH MO>KHO
npHBeCTH K o6w.eMY 3HaMeHaTeJIIO.
3 5
n p H M e p I. npHBeJJ.eM JJ.P06H 4" H""6 K 061l1,eMY 3HaMeHaTeJIIO.
061l1,HH 3HaMeHaTeJIb 3THX JJ.p06eH JJ.OJI>KeH JJ.eJIHTbCH H Ha 4, H Ha 6,
TO eCTb OH HBJIHeTCH 061l1,HM KpaTHblM l/HCeJI 4 H 6. Ho TaKHX 061l1,HX
KpaTHblX 6ecKoHe4Ho MHoro: 12, 24, 36, ... . 4T06bl HOBblH (0611lHH)
3HaMeHaTeJIb 6blJI l/HCJIOM HaHMeHbWHM, 6epeM B Kal/eCTBe 061l1,ero
3HaMeHaTeJIH HaHMeHbwee 061l1,ee KpaTHoe AaHHblX 3HaMeHaTeJIeH, TO
eCTb l/HCJIO 12. 3aTeM HaHJJ.eM AOnOJlHHTeJlbHblH MHO>KHTeJIb JJ.JIH
Ka>KJJ.oH H3 JJ.aHHbIX JJ.p06eH, TO eCTb 4HCJIO, Ha KOTopoe HY>KHO YMHO-
>KHTb l/HCJIHTeJIb H 3HaMeHaTeJIb JJ.P06H, l/T06bl nOJIYl/HTb JJ.P06b CO
3HaMeHaTeJIeM 12. .llJIH 3Toro HY>KHO HOBblH 3HaMeHaTeJIb 12 JJ.eJIHTb
Ha 3HaMeHaTeJIH JJ.aHHbIX JJ.p06eH: 12:4=3 H 12:6=2, CJIeJJ.oBa
3
TeJIbHO, JJ.onOJIHHTeJIbHbIM MHO>KHTeJIeM JJ.P06H 4" 6YJJ.eT l/HCJIO 3,
5
a JJ.P06H ""6 l/HCJIO 2. .llonOJIHHTeJIbHble MHO>KHTeJIH 3anHweM HaJJ.
COOTBeTCTBYlOllI,HMH 4HCJIHTeJIHMH. nOJIYl/HM:
32
3 2
3'-' 9 5'-' 10
"4 -12 H 6"" =12'
HTaK, JJ.aHHble JJ.P06H Mbl npHBeJIH K 061l1,eMY 3HaMeHaTeJIIO.
4T06bl npHBeCTH JJ.P06H K 061l1,eMY 3HaMeHaTeJIIO:
I) HaiiiJeM (HaUMeHbUlee) o6Ufee ICpaTHoe 3HaMeHare./l.eU iJaH-
tU/lX iJpooeii, ICOTopoe U 6yiJeT (HaUMeHbUlUM) o6UfUM 3HaMeHa-
re/U!M;
2) HaiiiJeM iJ.Ml ICaJlCiJoii iJpo6u iJono./l.HUTe./l.bHblii MH03ICUTe./l.b. l/,.Ml
aTOZO pa3iJe./l.UM o6Ufuii 3HaMeHtlTe./l.b Ha 3HaMeHaTe./l.U iJaHHblx
iJpo6eii;
3) yMH03ICUM '1UC./I.UTe./l.b U 3HaMeHaTe./l.b ICa:JICiJoii iJpo6u Ha ee
iJono./l.HUTe./l.bHblii MH03ICUTe./l.b.
npHBOJJ.HTb K 061l1,eMY 3HaMeHaTemo MO>KHO He TOJIbKO JJ.Be
.ll.p06H, HO H TPH, l/eTblpe H T. A.
1 2 3
n p H M e p 2. npHBeJJ.eM JJ.P06H 2' 5' 4" K o(5ll1,eMY 3HaMe-
HaTeJlIO. HaHMeHbwHH 061l1,HH 3HaMeHaTeJIb 20, TaK KaK 3TO HaHMeHb-
wee l/HCJIO, KOTopoe JJ.eJIHTCH Ha JJ.aHHble 3HaMeHaTeJIH. nOJIYl/HM:
10
1'-' 10
2"= 20 '
5
3'-' 15
"4 - 20 .
4
2'-' 8
"5 = 20 '
ECJIH HaHMeHblllHH 061l1,HH 3HaMeHaTeJIb nOJJ.06paTb TPYJJ.HO, TO
HaAO pa3JIO>KHTb 3HaMeHaTeJIH Ha npOCTble MHO>KHTeJIH.
5 7
n p H M e p 3. npHBeJJ.eM K 061l1,eMY 3HaMeHaTeJIIO 36 H 48 '
Pewe H H e. 36=2.2.3.3, 48=2.2.2.2.3,
HOK (36,48)=2.2.3.3.2.2=144,
4 3
5'"' 20 7'-' 21
36 = 144 H 48 = 144 .
A.
177. PaccMoTpH npHMep I H BHHMaTeJIbHO np04TH npaBuAo npuBe-
5 2
aeHUR. apo6eu K 06U{eMY 3HaMeHareAIO. .llP06H ""6 H ""9 npHBeJJ.H
K 061l1,eMY 3 HaMeHaTeJIIO, 06'bHCHHH Ka>KJJ.bIH nYHKT npaBHJI
.
178. 4eMY paBHHeTcH HaHMeHbwHH 061l1,HH 3HaMeHaTeJIb JJ.p06eH,
eCJIH: I) 3HaMeHaTeJIH B3aHMHO npocTble l/HCJla; 2) HaH60JIb-
lllHH 3HaMeHaTeJIb JJ.eJIHTCH Ha Ka>KJJ.b1H H3 OCTaJIbHbIX?
33
179.
npHBeJJ.H JJ.P06H K 061l1,eMY 3HaMeHaTeJIIO (y C THO):
I ) I I 3 3 I I 4 I I
""6 H 2 2) 8 H 4 3)2 H 3 4) 7 H 2 5) 4 H 3
371121 1435
""'5 H 20 I5 H ""'5 "3 H ""'5 4 H 9 ""'5 H 6
..!..H
H
....!....
1 3 1 2
9 27 7 21 3 H 4 2 H 5 4 H ""'5
npHBeJJ.H JJ.P06H K 061l1,eMY 3HaMeHaTeJIIO:
I) _ 6 1 H 4 3 2) ..i... H ....!.... 3)
H 3
9 6 12 3
7 13 5 2 5 4
20 H 30 6 H 15 "9 H 15
10 13 5 11 3 7
9 H 12 18 H 24 16 H 24
npHBeJJ.H JJ.P06H K 061l1,eMY 3HaMeHareJIIO:
I ) I 1 5 2 1 5 2 1 5
2' 3 H""6 2)"3'"9 H 18 3) 3 ' 6 H 12
231 357 573
3 ' 4 H 12 3' 8 H 12 6' 9" H 4""
374 254 513
""'5' 10 H 15 3 ' 6 H 7 12' 7 H 2
180.
3 5
4) 4 H 18
1
6
6
35
3
8 H
11
14 H
181.
6.
182. npHBeJJ.H JJ.P06H
11 13
1) 36 H 60
7 3
52 H 260
1 13
45 H 120
K 061l1,eMY 3HaMeHaTeJIlO:
2) 5 7 1 3) 5
24 ' 96 H 16 6 '
9 17 1 2
80 ' 360 H 30 9 '
7 9 5 5
24' 40 H 36 36'
4 1 2
9' 2 H 3
7 3 1
24 'IO H 6
7 1 3
8' 144 H 4""
J],nA nOBTopeHHA
183. BblJJ.eJIH LleJIYIO H JJ.P06HYIO 4aCTH:
18 37 45 124 57 260
""'5' 4' 12' 6'25' 35"'
184. PaCnOJIO>KH 4HCJIa B nopHJJ.Ke B03paCTaHHH:
18 11 1 15 22 7
23 ' 23 ' 23 ' 23 ' 23 ' 23 '
185. HaHJJ.H: I) 25% OT 428; 2) 40% OT 1060; 3) 5% OT 13,
186. 113 1200 pa604Hx 3aBOJJ.a 45% pa60TalOT B JIHTeHHOM Llexe.
CKOJIbKO 4eJIOBeK pa604Hx pa60TaeT B JIHTeHHOM Llexe?
187. Bbl4HCJIH:
I) 1500-632,4-59,6-43; 2) 58,956: 1.7 +2,06.4,7.
34
18 8 *. 3aMeHH 3Be3JJ.04KH TaKHMH LlH<t>paMH, 4T06b1 Bbl4HCJIeHHe
6b1JIO BepHblM:
I) * 2 *
X *7
2 2 * 8
* 6 * 0
1 * 4 6 *
2) * 8 * * *
3*8
*
Tlf5 8
* * * *
* * *
504
---0
3) * 2 * 5 *
***
T'(f'i *
* 9 * *
* 5 *
* 5 *
----0-
2.5. CpaBHeHHe 06b1KHOBeHHblx .ll.p06eH
113Yl/H MaTepHaJI 3TOro naparpa<t>a caMOCTOHTeJIbHO, HCnOJIb-
3YH YKa3aHHH K 3aJJ.aHHIO 189.
Tbi Y>Ke YMeewb cpaBHHBaTb JJ.P06H C paBHblMH 3HaMeHa-
TeJlHMH: U3 iJ8YX iJpo6etl C pa8HblMU 3HaMeHaTelUlMU Ta iJP06b
60/lbUle, 'lUC/lUTe/lb ICOTOpOtl 60/lb , Ule BblHCHHM, KaK cpaBHHBaTb
.lI.P06H C pa3HblMH 3HaMeHaTeJIHMH.
npeJJ.nOJIO>KHM, 4TO y Hac JJ.Be OJJ.HHaKOBbie nJIHTKH WOKOJIaJJ.a,
OJJ.Ha pa3JJ.eJIeHa Ha 4 paBHble 4aCTH, a JJ.pyraH Ha 5. Tbi MO>KeWb
3 u 3 u M 6
B3HTb 4 OJJ.HOH nJIHTKH HJIH ""'5 JJ.pyroH. O>KeT blTb, Tbi X04eWb
KYC04eK n060JIbwe? KaKOH >Ke B03bMeWb? 3JJ.eCb HaM HaJJ.O
cpaBHHTb JJ.P06H C paBHblMH 4HCJIHTeJIHMH, HO C pa3HblMH 3HaMeHa-
TeJIHMH. npH cpaBHeHHH TaKHX JJ.p06eH 3aMeTHM: eCJIH LleJIOe pa3JJ.e-
.1HTb Ha 60JIbWee 4HCJIO 4aCTeH, TO BeJIH4HHa OJJ.HOH 4aCTH 6YJJ.eT
MeHbwe (pHC. 2.4). CJIeJJ.oBaTeJIbHO, U3 iJ8YX iJpo6etl C pa8HblMU
'lUC/lUTe/UlMU 60/lbUle Ta, y ICOTOpOtl 3HaMeHaTe/lb MeHbUle. 3Ha4HT,
3 3 5 5 8 8
4">""'5' TaK >Ke 7>8' Tl<"9 H T. JJ..
ECJIH HY>KHO cpaBHHTb JJ.Be JJ.pO-
6H C pa3HblMH 4HCJIHTeJIHMH H pa3-
HblMH 3HaMeHaTeJIHMH, HanpHMep
3 4
7 H g' TO CJIeJJ.yeT npHBeCTH HX
K 06meMY 3HaMeHaTeJIIO, 3aTeM npH-
MeHHTb npaBHJIO cpaBHeHHH JJ.p06eH C
paBHblMH 3HaMeHaTeJIHMH. 06mHH 3Ha-
MeHaTeJIb JJ.aHHblX JJ.p06eH 63, CJIeJJ.OBa-
TeJIbHO,
9 7
3'"' 27 4'"' 28 27 28 +
-=;- = 63 H "9 = 63 . TaK KaK 63 < 63 ' Wff/ff.a
TO H l....<.!..
7 9.
1
T
W$ff$M'ffffff$.a
1
T
I7ffff/M'ffff.a
1
T
Wffff/ff//A
PMC. 2.4
35
A.
189. l. np04TH a63aLl TeKCTa, KOTOpblH Ha4HHaeTCH CJIOBaMH «Tbl
Y>Ke YMeewb...», H 3anOMHH npaBHJIO cpaBHeHHH JJ.p06eH C paB-
6 I 584
HblMH 3HaMeHaTeJIHMH, PaCnOJIO>KH JJ.P06H 7' 7' 7' 7' 7"
3
H 7 B nopHJJ.Ke B03paCTaHHH.
190.
2. np04TH CJIeJJ.YIOUlHH a63aLl. 0 4eM TaM rOBopHTCH? CpaB-
HeHHe KaKHX JJ.p06eH H306pa>KeHO Ha pHCYHKe 2.4? n04eMY
2- > 2-;>
4 5 .
3. npHJJ.YMaH. npHMepbl Ha cpaBHeHHe JJ.p06eii C paBHblMH
4HCJIHTeJIHMH. 3anoMHH npaBHJIO cpaBHeHHH TaKHX JJ.p06eH.
4. np04TH nOCJIeJJ.HHH a63aLl TeKCTa H 3anOMHH, KaK cpaB-
HHBalOTCH JJ.P06H C pa3HblMH 4HCJIHTeJI5!MH H pa3HblMH 3Ha-
n 3 4;>
MeHaTeJIHMH. 04eMY 7< (I'
CpaBHH JJ.P06H:
4 4 16 16 7 7 13 13
I) 15 H 19; 2) 25 H 21; 3) 9 H 19; 4) 16 H 14'
CpaBHH JJ.P06H:
1 3 5 7 9 47 11 7
I) 2 H 4; 2) 6 H 9; 3) 20 H 100 ; 4) 24 H 16'
CpaBHH JJ.P06H H 3aMeHH 3Be3JJ.04KY 3HaKOM > HJIH <:
7 3 3 9 12 12 3 5
1)8*8 2)17*17 3)19* 35 4 )8*12
6 6 5 9 16 18 11 1
25 *17 7*14 31*31 18*3
3 4 4 7 11 34 34
4*""'5 2*9 10*15 55 * 45
PaCnOJIO>KH JJ.P06H B nopHJJ.Ke B03paCTaHHH:
3 9 13 1 7 7 7 7 7 7
I) 10' 10' 10' 10' 10; 2) 8' 15' IT' 6' 9;
13155
3) 2' 4' 12' 6' 8'
.LI.HMa JJ.eJIaeT 3 wara 3a 5 c, a feHa 5 warOB TaKoH >Ke JJ.JIHHbl
3a 8 c. KTO XOJJ.HT 6blcTpee?
OJJ.HH pa604HH 3a 4 4 H3rOTaBJIHBaeT 3 H3JJ.eJIHH, a JJ.PyroH
3a 5 4 4 TaKHX >Ke H3JJ.eJIHH. KaKoH pa604HH H3roTaBJIHBaeT
3a OJJ.HH 4ac H3JJ.eJIHH 60JIbwe?
Ha4epTH 4HCJIOBOH JIY4, npHHHB 3a eJJ.HHHLlY oTpe30K 8 CM
191.
192.
193.
194.
195.
196.
36
OTMeTb Ha 4HCJIOBOM JIY4e T04KH, KOTOpblM COOTBeTCTBYIOT
313517 3
4HCJIa 8' 2' 4' 8' 4' 8' anHWH 3TH 4HCJIa B nopHJJ.Ke
y6b1BaHHH.
6.
197.
CpaBHH JJ.P06H (npeJJ.BapHTeJIbHO COKpaTH HX):
6 15 . 2 10 6, 3)
. 4 )
. 5 ) 24 H.!2..
I) 8" H 20 ' ) 15 H 18' 42 H 28 ' 30 H 35 ' 36 24'
PaCnOJIO>KH 4HCJIa B nopHJJ.Ke y6b1BaHHH:
2 7 2 1 17
1 3 '6'3'2'12'
Bblpa3H JJ.eCHTH4Hble JJ.P06H 06blKHOBeHHblMH H paCnOJIO>KH Bce
JJ.P06H B nopHJJ.Ke B03paCTaHHH:
3 1 14
0,45, 20 ' 4' 0,64, 25 '
198.
199.
.lI.JIA nOBTopeHHA
200.
Bbl4HCJIH (y C THO), eCJIH B03MO>KHO, nOJIY4eHHYIO JJ.P06b COK-
paTH:
I ) 2.+2. 2 )
_2- 3) ...!....+...!.... 4)
5 5 7 7 4 4
3 5 2 1 1 3
8+8 ""'5-""'5 10+10
2 5 9 7 3 1
9+9 14-14 8+8
Bbl4HCJIH:
I ) 0,28 (19,5-10,5)+0,34 ; 2) 10,8-3,2 0,04.
2,5: 10 2,5
113 JJ.BYX ropOJJ.OB OJJ.HOBpeMeHHO HaBCTpe4Y JJ.Pyr JJ.pyry BblWJIH
KM
JJ.Ba noe3JJ.a. CKOpOCTb OJJ.HOro noe3JJ.a 54,4 --;j"""' CKOpOCTb
JJ.pyroro 60,8 KM , 4epe3 2,5 4 OHH BCTpeTHJIHCb. nOCTaBb pa3YM-
lJ
5 1
6-6
7 1
8-8
11 5
12-12
201.
202.
203.
Hble BonpOCbI H OTBeTb Ha HHX.
Bbl4HCJIH nepHMeTp <t>Hrypbl
(pHC. 2.5), BbInOJIHHB He06xo-
JJ.HMble H3MepeHHH.
\uj
PMC. 2.5
37
204. Ka>KJJ.bIH HIlI,HK B wTa6eJIe
(pHC. 2.6) HMeeT <t>OPMY npHMO-
yrOJIbHoro napaJIJIeJIenHneJJ.a, H3-
MepeHHH KOTOpOro I M, 0,5 M H
0,6 M. Bbl4HCJIH 060eM wTa6eJIH,
PHC. 2.6
2.6. CJIO>KeHHe H Bbll.JHTaHHe .ll.p06eH
c pa3HblMH 3HaMeHaTeJ1HMH
Tbl Y>Ke YMeeWb CKJIaJJ.blBaTb H Bbl4HTaTb JJ.P06H C paBHblMH
3HaMeHaTeJIHMH:
a b a+b a b a-b
-+-=-' ---=-(a>b HJIH a=b),
m m m'm m m
4T06b1 CJIO>KHTb (Bbl4ecTb) JJ.P06H C pa3HblMH 3HaMeHaTeJIHMH:
I) npu8eiJeM aTU iJpo6u " (HaUMeHbweMY) 06tqeMY 3HaMeHa-
Te./IIO;
2) 8blnOllHUM C/IO:JICeHUe ( 8blfluTaHue) no npa8ullY CIIO:JICeHWl
( 8blfluTaHuR) iJpo6eii. C pa8HblMU 3HllMeHaTe/UlMU.
n p H M e p I, HaHJJ.eM CYMMY
+ 1 7 0 '
HaHMeHbwHH 061l1,HH 3HaMeHaTeJIb JJ.aHHblX JJ.p06eH 20. .LI.onoJI-
HHTeJIbHblH MHO>KHTeJIb nepBoH JJ.P06H paBeH 5 (20:4=5), BTOpOH
JJ.P06H 2 (20: 10=2). 3anHweM:
7& _ 15 14 15+ 14 29 9
4 +10 - 20 + 20- 20 -20- 1 20 '
06b14HO nOJJ.4epKHYTaH 4aCTb He 3anHCblB3eTCH. TorJJ.a BW4HC-
JIeHHe 6YJJ.eT HMeTb BHJJ.:
6 2
E +2..."" _ 15+14 _29_ 1 .J!.
4 10 20 20 20
n 2 H v 5 3
pH M e p . aHJJ.eM pa3HOCTb ""6-4"'
HaHMeHbWHH 061l1,HH 3HaMeHaTeJIb JJ.aHHblX JJ.p06eH 12. .LI.aJIee
nOJIY4HM:
38
5& _ 3<!, _
_
10-9 1
6 4 - 12 12 =--"'12=12 '
UJiM Kop04e:
5& 3<!, 10-9 1
6 -4= --"'12-12'
ECJIH cpeJJ.H JJ.aHHblx HMelOTCH CMewaHHble 4HCJIa, TO HCnOJIb-
3yeM nOJJ.XOJJ.Hll!He CBOHCTBa JJ.eHCTBHH.
BHHMaTeJIbHO npOCJIeJJ.H 3a peweHHeM CJIeJJ.YIOIlI,HX npHMepOB:
npHMep 3. 3++5
=(3+5)+( t + t )=8+ 4i;9 =
13 1
=8+ 12 =9\"2'
06bI4HO npOMe>KYT04Hble Bbl4HCJIeHHH BbinOJIHHIOT YCTHO H pewe-
HHe 3anHCbIBalOT Kop04e:
1
3
4+9 13 1
3"3 +54 =812=8\"2=9\"2'
B paCCMOTpeHHOM npHMepe HCnOJIb30BaJIHCb nepeM.eCTUTellbHblU
H COl.f.eTaTellbHblU 3aKOHbl CII03ICeHUfl
2 3
5 1 ( 5 \ 1 ( 5 Co' 1 Co' ) _
n p H Me p 4. 3 6 - 4 = 3+ 6 ) -4=3+ 6"-4 -
10-3 7 3 7
=3+
=3+12= 12'
HJlH Kop04e:
5& ,<!' 10-3 7
3--- = 3-=3-.
6 4 12 12
B 3TOM npHMepe HCnOJIb30BaHO npaBHJIO: I.f.To6bl Bbll.f.eCTb I.f.UCIIO
U3 CYM.M.bl, M.03ICHO Bbll.f.eCTb ezo U3 OiJHOZO CllazaeM.OZO U K pe3Yllb-
TaTY npu6aBUTb iJpyzoe ClluzaeM.oe.
n p H M e p 5. 4 : - 2
= 4 : - ( 2 +
) = (4 : - 2 ) -
=
4
3
16-15 -2 +
=2
=2 5 - 4 =2+ 20 20 20'
HJIH Kop04e:
4 ,!, 3
16 15 1
2 2 - _ 2 -
4 5 - 4 = 20 20 .
39
3JJ.eCb HCnOJIb30BaHO npaBHJIO: I.f.T06bl Bbll.f.eCTb CYM.M.Y U3 I.f.UC/la,
M.03ICHO U3 HeZO Bbll.f.eCTb OiJHO C/lazaeM.oe, a 3aTeM. U3 pe3Y/lbTaTa
Bbll.f.eCTb iJpyzoe C/luzaeM.oe.
n 373 4
pH M e p 6 1--=---=-
. 7 7 7 7 '
TaK KaK 4HCJIO I MO>KHO Bblpa3HTb JJ.P06blO C JII06b1M paBHblM 4HCJIHTe-
JIeM H paBHblM eMY 3HaMeH3TeJIeM.
n 5 ( 6 ) 5 I
pHMep 7.5- 6 = 4+""6 -""6=4""6'
06'bHCHH 3TO peweHHe, onHpaHcb Ha npHMep 6.
npHMep 8. 6 2 1 ..:!, _2 2& =4+ 3-4 =3+-2- + 3-4 =
3 6 6 6
3 9-4 5
+
=36' HJIH Kop04e:
I ..:!, 2 & 3
6- -2- =4 -4 =3 9-4 =3
2 3 6 6 6'
B nOCJIeJJ.HeM npHMepe JJ.p06HaH qaCTb YMeHbiliaeMoro MeHbille
JJ.P06HOH 4aCTH Bbl4HTaeMoro, B 3TOM CJIY4ae HY>KHO H3 LleJIOH 4aCTH
YMeHbwaeMoro B3HTb OJJ.HY eJJ.HHHLlY H Bblpa3HTb ee B BHJJ.e JJ.P06H,
B JJ.aHHoM cJIyqae : ' 3Ha4HT, HOBbiM YMeHbwaeMbiM 6YJJ.eT 3 : .
A.
205. npOJJ.YMaH npHMepbl I H 2. 06'bHCHH, KaK C/103ICUTb U Bbll.f.eCTb
iJpo6u C pa3HblM.U 3HaM.eHaTe/lflM.U.
206. Bbl4HCJIH:
207.
I)
+
2)
+
3 ) 7 2
8 9 4 14 12-9
7 3 4 II 13 1
12+8 9+\5 \5-2
1 2 7 12 9 11
""6+5 \0+ 25 20 - 30
Bbl4HCJIH (y C THO), 06'bHCHHH
I) ....!....+
2 ) ....!.... +
3 ) 7 I
2 3 4 5 8-2
....!....+....!....
+....!.... J..!._
5 2 3 2 14 7
I 1 7 2 5 1
6+"3 20 +5 6-"3
5 1 11 1 13 2
12+4 30 +6 20 -5
40
II 7
4) 16-12
6 3
---
7 II
23 7
---
24 18
Ka>KJJ.blH war peweHHH:
4) J..!._..!.. 5 ) -.!._....!....
18 9 5 2
9 1 I 2
16-4 5+"3
8 I 7 1
\5-5 \5+"3
II 2 II 1
12-"3 12-""6
208. BbI4HCJIH:
I ) ....!....+....!....+
2)
248
4 4 3
5-\5+\0
357
8+6-12
+
+....!.... 3)
462
2 3 1
-----
3 5 15
13 1 3
-----
14 4 7
3 5 I
-+---
4 8 12
5 1 1
-----
6 2 8
941
IT+ 33 +"3
209. B nepBblH JJ.eHb npoJJ.aJIH "3' a BO BTOpOH -4- nOCTynHBwero
B Mara3HH BHHorpaJJ.a. KaKYIO 4aCTb BHHorpaJJ.a npoJJ.aJIH 3a
JJ.Ba JJ.HH?
O M v 2 v 3
21. aWHHHCTKa B nepBblH JJ.eHb nepene4aTaJIa 5' BO BTOpOH \0
v 4 v 6 K 6
H B TpeTHH 15 BceH pa OTbi. aKYIO 4aCTb pa OTbi nepene4a-
TaJIa MaWHHHCTKa 3a TPH JJ.HH? '--
211. Bepa B nepBblH JJ.eHb np04HTaJIa
KHHrH, a BO BTOpOH JJ.eHb
Ha + MeHbwe. KaKYIO 4aCTb KHHrH np04HTaJIa Bepa BO BTOpOH
JJ.eHb? YcneJIa JIH OHa np04HTaTb KHHry 3a JJ.Ba JJ.HH? .
1 I
212. B OJJ.HOM naKeTe 2 Kr KOH<t>eT, a B JJ.pyroM Ha 5 Kr MeHbwe.
CKOJIbKO KHJIOrpaMMOB KOH<t>eT B JJ.BYX naKeTax BMecTe?
213. npOJJ.YMaH npHMep 3. 06'bHCHH, KaK CKJIaJJ.blBaIOTCH CMewaHHble
JJ.P06H,
214. BbI4HCJIH:
7 5
3) 2 18 +5 12
7 3
3 15 +8 4
215.
I ) 3
+ 3J..!. 2 ) 57 + 3 1
5 15 12 4
5
+6
9
+12
3 6 6 2
3 4 8 3 13 3
4 4 +5 5 8\5+14\0 I 36 + 6 8
Bbl4HCJIH (y C THO), 06'bHCHHH Ka>KJJ.blH war CJIO>KeHHH:
I 1 I 5 1 1 1
I) 3+ 1 2 2) 1"3+5 3) 12+62 4) 18+ 2 2
5+2
+
+4....!.... 3
+1
3 42 93 52
3 23 13 II
7 5 +4 5 5 +\0 10 2 + 8 4 2 + 9
MeTaJIJIH4ecKYIO Tpy6y pa3pe3aJIH Ha JJ.Be 4aCTH, JJ.JIHHbI
1 1
TOpblX 4 4 M H 3 2 M. KaKoH JJ.JIHHbl 6b1JIa TPy6a?
KO-
216.
41
217. TYPRCTbJ BO BpeMji rroxoJJ.a 3a rrepBbIH qac rrpOlliJUI 3 1 7 0 KM, a 3a
BTOPOH Ha I} KM 60JJbllIe. CKOJJbKO KRJJOMeTpOB rrpOlliJIR TYPH-
CTbJ 3a JJ:Ba qaca?
218. npOJJ.YMaH npHMepbl 4 H 5, 06oHCHH, KaK BbinOJIHHeTCH
Bbl4HTaHHe, eCJIH cpeJJ.H JJ.aHHblX eCTb CMewaHHble JJ.P06H.
219. Bbl4HCJIH:
4 3 5 7 6 I
I) 9""'5-310 2) 8 6 -8 12 3) 17 7 - 13 21
5 1 9 4 2 1
7 12 -6 6 27 10 -17 15 11"3- 7 6
3 5 23 7 5 5
25 IT- 4 22 15 24 - 10 12 9 8- 9 12
220. Bbl4HCJIH (y C THO), 060HCHHH Ka>KJJ.bIH war BbI4HTaHHH:
8 2 1 7 4 1 1
I) 4 9 -2 2) 7 3 -3 3 3) 210-""'54) 5 2 -2 4
3 4 8 6 7 1 5 2
--I 5--5- 6--- 3--2-
5 11 11 15 3 6 3
7...!....-6 4
-3
8...!.!_...!.... I...!..-I...!....
12 15 15 12 2 5 10
51 9 I
221. IlI,HK C TOBapOM BeCHT 23 10 Kr, a nycTOH HIlI,HK BeCHT 1 2 Kr.
CKOJIbKO BeCHT TOBap?
222. 4T06b1 n06bIBaTb B TeaTpe, TaHe nOTpe60BaJIOCb 3
4.
2 6
Ha JJ.opory TYJJ.a H 06paTHO Y Hee YWJIO I 34. CKOJIbKO Bpe-
MeHH JJ.JIHJIOCb TeaTpaJIbHOe npe
CTaBJIeHHe?
223. npOJJ.YMaH npHMepbl 6, 7 H 8. 06oHCHH, KaK BbinOJIHHeTCH
Bbl4HTaHHe, eCJIH YMeHbwaeMoe HaTypaJIbHOe 4HCJIO HJIH ero
JJ.p06HaH 4aCTb MeHbwe JJ.P06HOH 4aCTH Bbl4HTaeMoro.
224. 3anHwH 4HCJIO 1 B BHJJ.e JJ.P06H co 3HaMeHaTeJIeM: I) I; 2) 3;
3) 5; 4) 8; 5) 10; 6) 12.
225. Bbl4HCJIH (y C THO), 060HCHHH Ka>KJJ.bIH war Bbl4HTaHHH:
I 5 1
I) 1-- 2) 1-- 3) 1-- 4)
3 7 10
I 3 5
1-- 1-- 1--
5 8 11
226. Bbl4HCJIH (y C THO), 060HCHHH Ka>KJJ.blH war Bbl4HTaHHH:
5 1
2) 2-- 3) 6-- 4 )
6 10
7-
7-2-
8 13
1
1--
20
7
1--
30
I) 3-...!..
5
5-
3
67
12- 100
13-2-
25
42
227.
Bbl4HCJIH:
1 ) 4
-2
2)
7 7
6...!....-5...!..
9 9
8
-3.!Z.
24 24
1
10-3 -
4
1 3
12 -- 7 -
3 4
3 3
24 --4 -
10 5
5 7
3 ) 8--2-
6 8
9
-5...!.!
42 14
I 8
2 \5- 45
228.
3 1
nYCTaH 6aHKa BeCHT 4 Kr, a HanOJIHeHHaH MeJJ.OM 6 2 Kr.
CKOJIbKO KHJIOrpaMMOB MeJJ.a B 6aHKe?
HY>KHO 3aac<t>aJIbTHpOBaTb wocce Me>KJJ.Y TpeMH CeJIaMH 1, B H
C (pHC. 2.7). PaccToHHHe Me>KJJ.Y A H B paBHo 6""'5 KM,
B
229.
A
PMC. 2.7
a Me>KJJ. Y B H C Ha I
KM MeHbwe, 4eM paCCTOHHHe Me>K-
10
JJ.Y A H B. CKOJIbKO KHJIOMeTpOB lllocce HY>KHO 3aac<t>aJIbTH-
pOBaTb? I
230. B nepBblH 4ac JIbI>KHHK npoweJI "3 Bcero paccToHHHH, KOTO-
poe OH JJ.OJI>KeH npoHTH, BO BTOpOH
Bcero nYTH, a B TpeTHH
oCTaBwYlOcH 4aCTb nYTH, KaKYIO 4aCTb Bcero paccToHHHH
npoweJI JIbI>KHHK B TpeTHH 4ac? 1
231. YTpeHHHH HCiJJ.OH MOJIOKa OT KOpOBbi COCTaBHJI 8 2 JI, Be4ep-
HHH 6 1 3 0 JI, a B 06eJJ. HaJJ.OH MOJIOKa 6b1JI Ha
JI MeHbwe,
4eM yTpOM. KaKOB HaJJ.OH MOJIOKa 6b1JI 3a JJ.eHb?
1 2 B
232. B JJ.BYX MeWKax 15 2 Kr MyKH, npH4eM B OJJ.HOM 7""'5 Kr. Ka-
KOM MeWKe MYKH 60JIbWe H Ha CKOJIbKO?
1 3
233. OJJ.Ha CTOpoHa TpeyrOJIbHHKa 4""'5
M, JJ.PyraH - Ha 110 JJ.M
I u
Kop04e, a TpeTbH - Ha 2 JJ.M JJ.JIHHHee nepBOH
nepHMeTp TpeyrOJIbHHKa.
2 1 1
234. Bbl4HCJIH 3Ha4eHHe Bblpa>KeHHH a+3 3' eCJIH a=2; a= 1 3 ;
Bbl4HCJIH
2 4
a=5-' a=I- 9 '
7 '
43
235.
Bbl4HCJIH 3Ha4eHHe Bblpa>KeHHH x- 2 :
, eCJIH x=3; x=4 +;
x=62-. x=IOE
15' 30'
PeWH ypaBHeHHe:
I) x+ : = I
[+ I....!....= I....!....
12 8
+ _29
100 z- 50
236.
2) x- I 2-=
3 )
20 10
4 5
Y-9=6
1 1
s- 32 =12
2 3
I --X=-
5 10
5 1
T- x =14
5 5
--[-
9 -18
6.
237. Bbl4HCJIH (y C THO), HCnOJIb3YH 3aKOHbI CJIO>KeHHH:
J) 1+5 4 1 2 3 3 5
5 6+5+6 2) T+8+T+8+
I ++
+
+ I : I 3 7 + 9 4 +....!....+
+
2 9 7
238. Bbl4HCJIH:
239.
I) 42-+1..!..!.. 2) 23 + ..!..!..
30 36 42 56
9 2 3 13 3 35
15- 36 5 IT -4 66
I
_
5 11
12 144 6 18 +2 21
Bbl4HCJIH:
I ) 3 2 1 3
10+ 1 5+2+5
2....!....+1....!.... 5 7
4 6 +9+12
7 5 I I
8+12+ 1 9+ 3 4
Ha KaKoe 4HCJIO HY>KHO YBeJIH4HTb
nOJIY4HTb:
3 11 9
) 34 - 170
22-+5....!....
13 6
8 ..!..!..- 7 E.
20 18
240.
2) .!.+ I
-
7 3 21
5J!...-2
-2....!....
10 5 2
19-
-
6 7
6
T' 4T06bl
13
I) 14;
2
2) 1 21 ;
3 ) I....!....",)
10'
241. Ha KaKoe 4HCJIO H Y >KHO Y MeHb w u Tb 2 8 3 , 6
n 4TO bI nOJIY4HTb:
I ) 5 2 ) 11 3) I
")
16; 12; 20'
242. KaKoe 4HCJIO 60JIbWe 3
4
Ha
CTOJIbKO,
Ha
CKOJIbKO
MeHbwe
",)
6'
44
5 1
243. KaKoe 4HCJIO 60JIbWe 16 Ha CTOJIbKO, Ha CKOJIbKO 18 MeHb-
1
we 14?
3
4
3
244. KaKoe BpeMH nOKa3b1BaIOT 4aCbI 3a 24 4 JJ.O nOJIYH04H? 3a
3 5
I "5 4 JJ.O nOJIYH04H? 4epe3 6 4 nOCJIe nOJIYH04H?
245. 113 JJ.BYX rOpO.D,OB OJJ.HOBpeMeHHO HaBCTpe4Y JJ.Pyr JJ.Pyry BblWJIH
JJ.Ba noe3JJ.a. PaCCTOHHHe Me>KJJ.Y 3THMH ropOJJ.aMH nepBblH noe3JJ.
npOXOJJ.HT 3a 3 4, a BTOpOH - 3a 4 4, KaKYIO 4aCTb Bcero
nYTH COCTaBHT paCCTOHHHe Me>KJJ.Y noe3JJ.aMH 4epe3 I 4 nOCJIe
HX BblXOJJ.a?
T v 3 1 v
246. ypHCTbI 3a nepBblH 4ac npOWJIH 4 KM, 3a BTOpOH 4ac Ha
3 6 v I v
4 KM OJIbWe, a 3a TpeTHH Ha 2 KM MeHbwe, 4eM 3a BTOpOH
l/ac. nOCTaBb pa3YMHble BonpOCbl H OTBeTb Ha HHX,
247. OJJ.HH KpaH 3anOJIHHeT BaHHY 3a 15 MHH, JJ.PyroH - 3a 10 MHH.
KaKaH 4aCTb BaHHbl 6YJJ.eT 3anOJIHeHa 3a I MHH, 3a 2 MHH,
3a 5 MHH, eCJIH OTKpbITb 06a KpaHa?
248. OJJ.HH KpaH 3anOJIHHeT 6aK 3a 6 MHH, a JJ.PyroH - 3a 12 MHH.
KaKaH 4aCTb 6aKa OCTaHeTCH He3anOJIHeHHOH, eCJIH OTKpbITb
06a KpaHa Ha I MHH?
249. JIerKoBoH aBTOM06HJIb npoxoJJ.HT JJ.aHHoe paccToHHHe 3a 2 MHH,
a rpY30BoH - 3a 15 MHH. nOcTaBb pa3YMHble BonpocbI H OTBeTb
Ha HHX.
250. KOJIX03 BbmOJIHHJI B nepBblH MeCHLl 10%, BO BTOpOH 14% ro-
JJ.OBoro nJIaHa npoJJ.a>KH MHca rocYJJ.apcTBY. KaKYIO 4aCTb
roJJ.OBoro nJIaHa BbinOJIHHJI KOJIX03 3a JJ.Ba MecHLla? KaKYIO
4aCTb OCTaJIOCb BbinOJIHHTb?
251. B Mara3HH 3aBe3JIH OBOIllH. + Bcex OBOlll,eH COCTaBJIHJIa
CBeKJIa, 15% - orypLlbl, 35% - nOMHJJ.opbl, a OCTaBWYIOCH
4aCTb COCTaBJIHJIa MOpKOBb, KaKYIO l/aCTb OT Bcex 3aBe3eHHblX
B Mara3HH npoJJ.YKTOB COCTaBJIHJIa MOpKOBb?
252. Bce MaJIb4HKH KJIacca npHHHJIH Y4acTHe B WKOJIbHblX copeBHO-
1 I
BaHHHX: ""34aCTb BOWJIH B q,YT60JIbHYIO KOMaHJJ.Y, 5 4aCTb -
1
B 6aCKeT60JIbHYIO, 64acTb COCTH3aJIaCb no npbl>KKaM B JJ.JIHHY,
OCTaJIbHble Y4alll,HeCH KJlaCCa - no 6ery. Ha KaKYIO 4aCTb 6e-
45
46
E
C
F E
0 0 C
H G
B B
PHC. 2.8 PHC. 2.9
A
rYHOB 6blJIO 60JIbille (MeHbwe), 4eM <t>YT60JIHCTOB? 6ac-
KeT60JIHCTOB?
253. CoCT3Bb Ii pewH 33JJ.a4Y Ha CJIO>KeHHe (BblIJHTaHHe) 06b1K-
HOBeHHblX JJ.p06eH.
254. H3 pHcYHKe 2.8 AE + DC= I
AB - I 3 DE=
5 JJ.M, - 4 JJ.M
-I 1 BC 3
- 4 JJ.M H = I TO JJ.M. Bbl4HCJIH nepHMeTp <t>HrYPbl.
255*. Ha pHc Y HKe 2.9 AB=7l..- M BC 2 1 1
2 ' = -M H DE=I-M
44'
Bbl4HCJIH nepHMeTp <t>HrYPbl.
Bbl4HCJIH 3H34eHHe Bblp3>KeHHH a - 3 ++ b, eCJIH:
I) a=4
H b=5
' 2) a _ 5 4 b I 7
8 3 ' - 5 H = \2'
nepHMeTp TpeyrOJIbHHKa p JJ.M. 0JJ.H3 CTopOHa TpeyrOJIbHHKa
p3BH3 12
JJ.M 3 3
5 ' JJ.pyr3H - H3 2 20 JJ.M Kop04e. COCTaBb
Bblp3>KeHHe JJ.JIH Bbl4HCJIeHHH JJ.JIHHbI TpeTbeH CTOpOHbl. Bbl-
4HCJIH ee, eCJIH: I) p=28
; 2) p=35,
256.
257.
258.
AnA caMonpoBepKH
3anHwH JJ. P 06b K3K 4ac
THoe OT JJ.eJIeHHH npH nOMOIll,H 3H3K3
«: »:
15 5
5' 5' 4' 8'
Qt>OPMYJIHPYH OCHOBHoe CBoikTBO JJ.P06H.
COKp3TH JJ.P06H:
20 8 15 32 35
50 ' 18' 40 ' 48 ' 63 '
259.
260.
261. npHBeJJ.H JJ.P06H K 061l1,eMY 3H3MeHaTeJIIO:
8 I 9 7 I 17
I) 15 H 24 ; 2) 16 H \2; 3) 3 H 50 '
262. HeKoTOpbIH Y'laCTOK rralIIHR 3aCe5IH pmKblO, OBCOM, IIlIIeHRIJ;eH,
rpe'lRXOH H rrpOCOM. POJKb 3aHHMaeT rrpH6nH3HTenbHO 5 1 0 'laCTb
_ 3 27-
BceH rrITOma)J.H, OBec - 4 'laCTH, rrlIIeHHIJ;a - 1000 qaCTeH, rpe'lHXa -
3 11 - - B
25 'laCTH H rrpoco - 500 'laCTeH BceH rrnomaJJ.H. bIIlHIIIH 3TH
KyITbTYPbI B rropH)J.Ke B03paCTaHH5I HX rrnOrn;a)J.eH.
263. Bbl4HCJIH (y C THO) :
1 5 1 1
I) \2+12 2) "3+5
4 5 7 1
9+9 18+2
2
+
I
+
5 5 2 4
14 2
3) 15-15
5 1
4) ---
8 4
1 1
---
3 7
3
-
6 3
19 9
---
20 20
1
5--
5
264. BbI4HCJIH:
I) 3 I 2) 1 4 3) 9 1 4) 5 3
8+12 7 3 +2"27 16-6" 5--2-
6 4
6 5 11 2 7 2 1 I
T+2f 3 18+ I 24 --- 9--8-
8 3 14 7
3
+
4 4 2
-2.. 6
-4E
4 5 29+ 5 15 36 9 30 20
265. PewH ypaBHeHHe:
1 1
I) 26"+x=5 2 ;
4 2
2) 215- z = 1 3 ;
7
Bblpa>KeHHH c- 9 ,
7 1
3) y-6 8 =216'
1
eCJIH c=3; c=2 2 ,
266. Bbl4HCJIH 3H34eHHe
267. TYPHCTCKHH rroxoJJ. COCT05IIT H3 )J.BYX 3TarrOB. Ha rrepBoM 3Tarre TY-
pHCTbI )J.OITJKHbI 6bIITH rrpOHTH 3
KM, a Ha BTOpOM - Ha x KM
60nblIIe, COCTaBb BbIpaJKeHHe JJ.ITH HaxmK)J.eHH5I paCCT05IHH5I, KO-
Topoe )J.OITJKHbI rrpOHTH TYPHCTbI. BbI'lHCITH ero 3Ha'leHHe, eCITH
1 3
x="2;x= 1 10 '
268. OJJ.HO 4HCJIO 5+, a JJ.pyroe - Ha 2
60JIbWe, HaHJJ.H CYMMY
3THX 4HCeJI.
269. Ha BblCTaBKe xYJJ.O>KeCTBeHHblx pa60T npeJJ.cTaBJIeHa >KHBO-
47
270.
nHCb, CKYJIbnTypa H rpa<t>HKa. + BCeX pa60T COCTaBJIHer
2
cKYJIbnTyp3, 5- >KHBOnHCb, OCT3BWYIOCH 43CTb - rpa<t>HKa.
KaKYIO 43CTb Bcex p360T COCT3BJIHeT rpa<t>HK3?
OJJ.Ha M3WHHa npHBe3JIa Ha CKJI3JJ. 3 : T K3pTO<t>eJIH, 3 JJ.PyraH
1
H3 2 T MeHbwe. CKOJIbKO TOHH K3pTO<t>eJIH npHBe3JIH H3 CKJIa.n.
06e M3IlIHHbI BMecre?
B OJJ.HOM KYBWHHe -!. JI MOJIOK3 BO BTO p OM I I
5 ' ""4 JI H B TpeTbeM
I
H3 2 JI MeHbwe, 4eM BO BTOpOM. H3 CKOJIbKO JIHTpOB MOJIOKa
60JIbWe (HJIH MeHbwe) B TpeTbeM KYBWHHe, 4eM B nepBoM?
Bbl4HCJIH:
I) 738 (19,6-19,559)'2
10(32,7+4,2)' 2) 0,82
271.
272.
-
3. nPE06PA30BAHHE .ll.P06EIiI.
YMHO)l(EHHE 06blKHOBEHHbiX
.ll.PO 6E iii
CaMOCTOATeJIbHaA pa60Ta 2
T e M a. 06palll,eHHe 06b1KHOBeHHblx APo6eH B .necATH'IHble.
TbI y>Ke YMeewb 06p31l1,aTb JJ.eCHTH4Hble JJ.P06H B 06b1KHOBeH-
Hble. 3TO 04eHb npocTo, H3npHMep:
1 25 I 2 1
0,1 =10,0,25= 100 =""4,3,2=3 10 =3 5 '
75 3
0,075 = 1000 = 40 .
Tenepb HaY4Hcb 06paTHoMY npe06pa30BaHHIO - 06palll3Tb 06bIKHO-
BeHHYIO JJ.P06b B JJ.ecHTH4HYIO.
1. BcnoMHH, K3K 33nHC3Tb 06blKHoBeHHYlO JJ.P06b co 3HaMeHa-
TeJleM 10, 100, 1000, ... B BHJJ.e JJ.ecHTH4HOH. 3TO TO>Ke HeCJIO>KHO, H3-
npHMep:
7 27 641 9
10=0,7, 100 =0,27, 10=64,1, 7 100 =7,09.
2. 06p3TH B JJ.ecHTH4HYIO JJ.P06b:
1 3 43 37 7 3 71
) 10; 2) 100 ; 3) 1000 ; 4) 1000 ; 5) 5 10 ; 6) 1O'
o T BeT bI JJ.JIH npOBepKH HaHJJ.H cpeJJ.H CJIeJJ.YIOIllHX 4HCeJI:
0,43; 0,08; 0,3; 0,037; 0,007; 5,03; 7,1; 5,3.
3. TaK KaK 06bIKHoBeHHYIO JJ.P06b MO>KHO paccM3TpHBaTb K3K
l/aCTHoe OT JJ.eJIeHHH 4HCJIHTeJIH Ha 3HaMeHaTeJIb, TO 06P3THTb ee
B JJ.eCHTH4HYIO MO>KHO nYTeM JJ.eJIeHHH 4HCJIHTeJIH Ha 3HaMeHareJIb.
06paTH TaKHM 06pa30M CJIeJJ.YIOIllHe 06b1KHOBeHHble JJ.P06H B
.n.eCHTH4Hble: I)
; 2)
; 3) ;0 ; 4) +; 5)
; 6)
6 '
o T BeT bI JJ.JIH npOBepKH (npOBepb YMHO>KeHHeM) HaHJJ.H cpeJJ.H
CJIeJJ.YIOIlI,HX 4HCeJI: 1,625; 0,625; 1,4; 0,16; 0,125; 0,15; 1,34;
0,304; 0,4375; 0,34.
49
4. 06paTH B JJ.ecHTHl/HYIO JJ.P06b:
1 7 2 3 3 5
32",1 10 , 3 25 ,1 50 , ITH 316'
o T BeT bl JJ.JIH npOBepKH HaHJJ.H cpeJJ.H CJIeJJ.YIOIlI,HX 4HCeJI:
3,3125; 3,07; 1,06; 1,8; 3,5; 3,08; 3,9; 1,7; 1,75.
5. 06blKHoBeHHble JJ.P06H 06paTH B JJ.ecHTH4Hble, 33TeM BbJ-
4HCJIH:
7
I) 10+ 1,12
097-
, 100
2)
+3,8
09-
, 4
I-
2
6
3) 1 25 -0,31
1
4 5 + 1,16
9
2 10 -1,9
5
38+0,105
58-.!2.
, 20
33
50 +2,04
o T BeT bI JJ.JIH npOBepKH H3HJJ.H cpeJJ.H CJIeJJ.YIOIllHX 4HCeJI:
0,65; 5,2; 1,82; 5,36; I; 4,9; 0,5; 0,44; 0,8; 7; 4,95; 0,93;
6,8; 3,73; 2,7; 4,2.
6,25++
3. I. nepHO.ll.HI.JeCKaSi .ll.eCSiTHI.JHaSi .ll.p06b
B npeJJ.blJJ.YllI,eH C3MOCTOHTeJIbHOH pa60Te Tbi 06p31l1,aJI 06bIKHO-
BeHHble JJ.P06H B JJ.eCHTH4Hble JJ.eJIeHHeM 4HCJIHTeJIH Ha 3HaMeH3TeJIb,
nonp06yeM 06P3THTb T3KHM >Ke cnoc060M JJ.P06b 1 7 1 B JJ.ecHTH4HYIO.
7 I II 7
70 I 0,6363... I1TaK, nOJIY4HJIH, 4TO IT= 0,6363... T04KH
66 B KOHLle 4HCJIa nOKa3b1B31OT, 4TO JJ.eJIeHHe
40 He 3aKOH4HJIOCb. nOJIY4HJI3Cb 6eCKOHe'lHaA
33 J].eCATH'IHaA JJ.P06b (BO Bcex npeJJ.blJJ.YIlI,HX
70 npHMepax nOJIY43JIH KOHe'lH
e JJ.p06H).
66 B
HOH 6ecKoHe4HoH JJ.ecHTH4HOH JJ.P06H
40 pH.D.OM CTOHIlI,He LlH<t>Pbl 6 H 3 nOBTOpHIOT-
33 CH nOJJ.pHJJ. 6ecKoHe4Hoe 4HCJIO p33. OHH
7 COCTaBJIHIOT nepHoJ]. 6ecKoHe4HoH JJ.eCHTH4-
HOH JJ.P06H. .lI.P06H, cOJJ.ep>KaIll,He nepHoJJ.,
Ha3bIBalOTCH nepHoJ].H'IeCKHMH. I1TaK, 06blKHoBeHHaH JJ.p06b .!..... 06-
11
paTHJIaCb B nepHoJJ.H4ecKYIO JJ.ecHTHl/HYIO JJ.P06b 0,636363... . Ee ne-
pHOJJ. paBeH 63.
50
L.
1 -
7: 11=0,63 63636363636363...
)
8
))»
' _ ;1
---
-;
'J.,
r
B CTapWHX KJIaCCaX 6YJJ.eT nOKa3aHO, 4TO JII06yIO 06blKHOBeHHYIO
JI.P06b MO)l{HO 06paTHTb JIH60 B KOHe4HYIO JJ.eCHTH4HYIO JJ.P06b,
J1H60 B nepHOJJ.H4eCKYIO JJ.eCHTH4HYIO JJ.P06b. TpeTbeH B03MO)l{HOCTH
HeT,
KOHe4HaH JJ.eCHTHl/HaH JJ.P06b nOJIY4aeTCH TOJIbKO B TOM CJIY4ae,
eCJlH 3HaMeHaTeJIb HeCOKpaTHMOH 06blKHOBeHHOH JJ.p06H He COJJ.ep)l{HT
JI.PyrHx npOCTblX MHO>KHTeJIeH, KpoMe 2 H 5. Bo Bcex JJ.pyrHx CJIY-
LJaHX nOJIY4aeTcH nepHoJJ.H4ecKaH JJ.eCHTH4HaH JJ.P06b, TaK, Ha-
5
npHMep, JJ.P06b 6"" 06palll,aeTCH B nepHOJJ.H4ecKylO JJ.ecHTH4HYIO
JI.P06b, TaK KaK cpeJJ.H npOCTblX MHO>KHTeJIeH 3HaMeHaTeJIH HMeeTCH
MHO)l{HTeJIb 3 (6 = 2.3), Bbl4HCJIeHHe nOKa3b1BaeT, 4TO JJ.eHcTBHTeJIbHO
=0,8333... (nepHoJJ. 3TOH JJ.P06H 3).
11
D,P06b 20 06palll,aeTCH B KOHe4HYlO JJ.eCHTH4HYIO JJ.P06b, TaK
KaK B 3HaMeHaTeJIe 20=2.2.5 HeT JJ.pyrHx npocTblX MHO>KHTeJIeH,
KpoMe 2 H 5. 11 JJ.eHcTBHTeJIbHO,
=0,55.
A.
273. npOl/TH TeKCT naparpa<t>a H CKa>KH, KaKaH JJ.eCHTH4HaH JJ.P06b
Ha3bIBaeTCH nepUOaUl.f.eCKot1?
274. Ha30BH nepHoJJ.bl JJ.ecHTH4HOH JJ.P06H:
0,777..., 1,2727..., 0,123123..., 5,9090..., 0,234234...
51
275.
06p3TH 06b1KHOBeHHYIO JJ.P06b B JJ.eCHTH4HYIO (eCJIH B OTBeTe
nOJIY4aeTCH nepHOJJ.Hl/eCKaH JJ.eCHTH4HaH JJ.P06b, TO JJ.eJIeHHe
npeKp3TH nOCJIe Bbl4HCJIeHHH ee nepHOJJ.3):
...!...., 2-.!.. 1
...!.... 7 5
3 5' 3' 40' 6' 8' 3 9 ' 9' 12
6.
276.
KorJJ.3 06b1KHOBeHH3H JJ.P06b 06palll,3eTCH B KOHe4HYlO JJ.eCHTH4-
HYIO JJ.P06b, KOrJJ.3 B nepHOJJ.H4eCKYIO JJ.eCHTH4HYIO JJ.p06b?
BblJJ.eJIH H3 JJ.3HHblX 06b1KHOBeHHbIX JJ.p06eH Te, KOTOpble 06p31l1,a-
IOTCH B nepHOJl.H4eCKYlO JJ.eCHTH4HYIO JJ.P06b. 06paTH 3TH Jl.PO-
6H B JJ.eCHTH4Hble (JJ.eJIeHHe npeKp3TH nOCJIe nOJIY4eHHH BTO-
poro nepHOJJ.3):
-!. 5 43 11 101
125' 32' 33' 27' 160 ' 54' 999 '
277.
JJ.JIR nOBTopeHHR
278. Bbl4HCJIH:
I) 3+-(1+++); 2) (10
-3-k )-(2++ 1+).
KHpa KynHJI3 6YM3ry H
ee H3pacxoJJ.oBaJI3 H3 06epTblBaHHe
3
KHHr, a IO - H3 06epTblBaHHe TeTp3JJ.eH. K3K3H 43CTb 6YM3rH
OCTaJI3Cb HeH3p3CXOJJ.OB3HHOH?
06p3TH JJ.eCHTH4Hble JJ.P06H B 06b1KHOBeHHble, 3aTeM Bbl4HCJIH:
I) 0,5++ 2) :
-0,6 3) 8 5 +0,3 4) 1,5+2-!.
15
07-...!.... 14 2
'6 15- 0 ,6 0,8- 3 3,15+2+
0,75+ 9 4 -.!.. + 025 24- 1
2
7' '9 I ,84 - I "3
06p3TH 06b1KHOBeHHble JJ.P06H B JJ.eCHTH4Hble, 33TeM Bbl4HCJIH:
I) ++0,85 2)
-0,505 3) 1,34+
4) 3
-2,35
0,9-+
+0,68 2.4-
2 ;0 + 4 ,9
0,14+ : +-0,125
+7,03 5,21-4 :
OKpyrJIH:
I) JJ.O COTeH: 4753; 382311; I 750 109;
279.
280.
281.
282.
52
283.
2) JJ.O eJJ.HHHLl: 5,74; 10,59; 1,183; 0,6;
3) JJ.O JJ.eCHTblX: 5,63; 3,447; 0,387; 6,25; 2,038;
4) JJ.O COTblX: 14,0005; 0,078; 1,048; 7,326; 0,0061.
Bbl4HCJIH (oTBeT oKpyrJIH JJ.O TblCH4HblX):
l) 3,7: 1.2 2) 5,6:4,7 3) 8,9:2,3
92,1: 1,42 2,01 :4,1 3,75:0,83
4) 42,5:23,7
0,93:0,41
3.2. Jl.ecHTHI.JHOe npH6J1H>KeHHe
06b1KHOBeHHoH .ll.P06H
npH Bbl4HCJIeHHHX 4aCTO npHXOJJ.HTCH 06p31l1,aTb 06b1KHoBeHHYIO
JI,p06b B JJ.ecHTH4HYIO. A 4TO JJ.eJI3Tb, eCJIH 06b1KHOBeHH3H JJ.P06b 06-
paru3eTcH B 6ecKoHe4HYlO JJ.ecHTH4HYIO JJ.p06b? B 3TOM CJIY43e 6ec-
KOHe4HaH JJ.eCHTH4H3H JJ.P06b oKpyrJIHeTcH. nOJIY4eHH3H KOHe4H3H
JI,eCHTH4H3H JJ.P06b H33b1B3eTCH Jl.eCRTH'IHblM npH6JJH)KeHHeM 06b1K-
HOBeHHoH JJ.P06H. 1..JHCJIO, nOJIY4eHHoe npH oKpyrJIeHHH, TeM T04Hee,
4eM 60JIbWe JJ.eCHTH4HblX 3H3KOB B npH6JIH>KeHHH.
151
n p H M e p I. Bbil/HCJIeHHH nOKa3b1BaIOT, 4TO TiQ= 1,3727272... .
.LI.ecHTH4Hble npH6JIH>KeHHH 3TOH JJ.P06H T3KOBbI:
151
liD:::::::: 1,4 (OKpyrJIeHHOe JJ.O JJ.eCHTblX);
151
liD:::::::: 1,37 (oKpyrJIeHHOe JJ.O COTblX);
151
liD:::::::: 1,373 (OKpyrJIeHHOe JJ.O TblCH4HblX),
1..JT06b1 H3HTH Jl.eCHTH4Hoe npH6JIH>KeHHe, oKpyrJIeHHoe JJ.O 3a-
JJ.aHHoro pa3pHJJ.3, Hy>KHO BbinOJIHHTb JJ.eJIeHHe JJ.O CJIeJJ.YlOruero P33-
pHJJ.a H nOJIY4eHHblH pe3YJIbT3T oKpyrJIHTb.
ECJIH HCnOJIb3yeTcH MHKpOK3JIbKYJIHTOp, TO MO>KHO nOJIY4HTb B OT-
BeTe 8 HJIH JJ.3>Ke 60JIbWe LlH<t>p. nOJIY4eHHblH OTBeT npH He06xoJJ.H-
MOCTH OKpyrJIHeTCH.
3
n p H M e p 2. npH JJ.eJIeHHH l/HCJIHTeJIH Ha 3H3MeHaTeJIb JJ.P06H 7"
Ha 3KpaHe nOHBJIHeTCH 0,4285714. Tenepb MO>KeM 33nHC3Tb, 4TO:
3
]:::::::: 0,4 (oKpyr JIeHO JJ.O JJ.eCHTblX);
3
]::::::::0,43 (oKpyrJIeHO JJ.O COTblX) H T. JJ..
53
HCTOpHileCKHe CBe.ll.eHHH
Pe3YJIbTaTbl H3MepeHHH (JJ.JIHHbl, nJIOlll,aJJ.H) He BcerJJ.a BbIpa>Ka-
IOTCH HaTypaJIbHbIMH 4HCJIaMH, n03TOMY Y>Ke B JJ.aBHHe BpeMeHa no-
HBHJIaCb He06xoJJ.HMOCTb BBeJJ.eHHH JJ.P06HbIX 4HCeJI. nepBaH JJ.P06b,
KOTOPYIO HCnOJIb30BaJIH, 6bIJIa, BHJJ.HMO, «nOJIOBHHa» ( -4- ) . 3aTeM no-
HBHJIHCb JJ.P06H C eJJ.HHHLleH B 4HCJIHTeJIe, TaK Ha3bIBaeMbIe eOUH.UIi-
I I I
H.bte JJ.P06H: "3' 4' 5'''' ECJIH paC4eTbI npHBOJJ.HJIH K nOHBJIe-
HHIO JJ.pyrHx JJ.p06eH, TO HX 3aMeHHJIH CYMMOH eJJ.HHH4HbIX JJ.p06eH, Ha-
2 1 I
npHMep: 5="3+15'
CTapeHIllHM MaTeMaTH4eCKHM JJ.OKYMeHTOM HBJIHeTCH «MOCKOB-
CKHH nanHpyc», KOTOpbIH ISWJI HanHcaH B JI.peBHeM ErHme OKOJIO
4eTblpex TbICH4 JIeT Ha3aJJ. H xpaHHTCH B MOCKOBCKOM MY3ee H306pa3H-
TeJIbHbIX HCKYCCTB. 113 3Toro JJ.OKYMeHTa BbIHCHHJIOCb, 4TO JJ.JIH 060-
3Ha4eHHH JJ.p06eH paHbwe HCnOJIb30BaJIH oc06bIe 3HaKH. l.JepTa
JJ.P06H CTaJIa HCnOJIb30BaTbCH 3Ha4HTeJIbHO n03}f{e.
JI.ecHTH4HbIe }f{e JJ.P06H nOHBHJIHCb n03>Ke, MeHee JJ.BYX TbICH4
JIeT TOMY Ha3aJJ.. BnepBbIe npaBHJIa BbI4HCJIeHHH C JJ.eCHTH4HbIMH
JJ.P06HMH 6bIJIH 060cHoBaHbI cpeJJ.Hea3HaTcKHM Y4eHbIM YKJlHJJ.HCH
(IX B.). CHCTeMaTH4eCKoe H3JIO>KeHHe Y4eHHH 0 JJ.eCHTH4HbIX JJ.P06HX
6bIJIO JJ.aHO JJ.PyrHM MaTeMaTHKOM CpeJJ.HeH A3HH KaWH (XV B.) B pa-
60Te, HOCHBweH KpaCHBoe Ha3BaHHe «KJII04 K HCKYCCTBY C4eTa»
nO-BHJlHMOMY He3aBHCHMO OT KaWH 6eJIbrHHCKHH Y4eHbIH CHMOH
CTeBHH (1548-1620) pa3BHJI Y4eHHe 0 JJ.eCHTH4HbIX JJ.P06HX. OH
HanHCaJI He60JIbwyIO KHHry noJJ. Ha3BaHHeM «JI.ecHTaH», Eha KHHra
COCTaBJIHJIa Bcero JIHWb ceMb cTpaHHLl, OJJ.HaKO cOJJ.ep>KaJIa BCIO Teo-
pHIO JJ.eCHTH4HbIX JJ.p06eH.
A.
284. np04TH BHHMaTeJIbHO TeKCT naparpa<t>a H CKa}f{H, KaK HaHTH
JJ.eCHTH4Hoe npH6JIH}f{eHHe JJ.P06H
, oKpyrJIeHHOe JJ.O COTbIX.
285.
fA I
n3BeCTHO, 4TO 2""6=2,1666... H 55 =0,7818181... HaHJJ.H
JJ.eCHTH4Hoe npH6JIH>KeHHe Ka>KJJ.oH H3 JJ.aHHbIX 06bIKHoBeHHbIx
JJ.p06eH, oKpyrJIeHHOe JJ.O: I) JJ.eCHTbIX; 2) COTbIX; 3) TbICH4HbIX.
54
286. HaHJJ.H JJ.eCHTH4HbIe npH6JIH>KeHHH JJ.p06eH, oKpyrJIeHHbIe JJ.O Co-
5 7 5 49 7
TbIX: 14' 13' 26' TIT' 2)2'
287. OlSpaTH 06bIKHoBeHHbIe JJ.POISH B JJ.eCHTH4HbIe, oKpyrJIeHHbIe JJ.O
JJ.ecHTbIx, 3aTeM BbI4HCJIH:
1 2) 4 3) 4 4) 5
1) "3+0,4 0,8+9 5,2+7 2-- I 8
6 '
8 17 16 2
13- 0,3 23 - 0,2 2,1- 17 2,5 + 3 9
6.
288. 3aMeHH 06bIKHoBeHHbIe JJ.P06H HX JJ.eCHTH4HbIMH npHMH>KeHHH-
MH, oKpyrJIeHHbIMH JJ.O COTbIX. BbI4HCJIH, oKpyrJIHB OTBeT JJ.O JJ.eCH-
TbIX (B Ka>KJJ.OM CJIY4ae nOJJ.YMaH, KaK npolll,e pewHTb npHMep):
2 1 2 I 47 5
I) "3+7+0,85 2) "3+5,07-"3 3) 50 + 3 ,49-""6
5 1
--- + I 42
6 5 '
5 1
--- + 09
8 6 '
1 347
2"'8- 1000 + 0 ,3
289. HaHJJ.H KOpHH ypaBHeHHH (oKpyrJIH JJ.O COTbIX):
I) 37x=82; 2) 56x= 103; 3) 8x=47; 4) 52y=475,
290. nJIOlll,aJJ.b npHMoyrOJIbHOrO Y4acTKa 3eMJIH 3720 M 2 , a JJ.JIHHa
OJJ.HOH CTOpOHbl 86 M. Bbl4HCJIH JJ.JIHHY JJ.pyroH CTOpOHbl (OTBeT
oKpyrJIH JJ.O eJJ.HHHLl).
291. Bbl4HCJIH nJIOlll,aJJ.b OCHOBaHHH npHMoyroJIbHOrO napaJIJIeJIenHne-
JJ.a, eCJIH ero 06"beM 123 JJ.M 3 , a BblCOTa 2,3 JJ.M (oTBeT
oKpyrJIH JJ.O JJ.ecHTbIx).
292*. Bbl4HCJIH CYMMY Bcex eJJ.HHH4HbIX JJ.p06eH, 3HaMeHaTeJIH KO-
TOpblX - HepaBHbIe npoCTbIe 0JJ.H03Ha4Hble 4HCJIa,
JJ,JlH nOBTopeHHH
293. Bbl4HCJIH (y C THO) :
I) (2. 3,2 - 5). 10
(387: 100-0,87):2
2) 36: 10+0,4+6:2
1,86.100+ 15-100:2
294.
nOJl 4HTaJIbHOrO 3aJIa 6H6JIHOTeKH HMeeT pa3Mepbl 9,6 M,
5,4 M, a BbICOTa 3aJIa paBHa 4,5 M. Ha CKOJIbKO MeCT pac-
C4HTaH 3aJI, eCJIH no HopMe Ha OJJ.Horo 4eJIOBeKa JJ.OJI>KHO
npHxoJJ.HTbCH 3 M 3 B03JJ.yxa?
55
295. COKpaTH (y C THO):
6 6 12 20 9 16
36' 9' 14' 100 ' 45' 40'
296. BbI4HCJIH:
I 1 7
I) 5- 5 +2"3-15;
3)
-(0,1 + ; ) ;
5) 3,7-2 + 6,8+2'1,2 ,
2.0,85 0,92'
2) 12...!...- ( 4
+ I...!... ) +...!....
4 18 2 36'
5 7
4) 1,25+ 16 - 24;
6) 2,8-1,2 9,9
1: 100 O,T.
297.
06paTH B HenpaBHJIbHYIO JJ.P06b:
;3...!... 2
1 11 1 5 23 5
4' 7' 6i-, 112,9 2 ,4 6 , 1 30 , 518'
298.
Pe6po
(pHC.
06'beM
Ka>KJJ.oro Ky6HKa B
3,1) paBHO 3 JJ.M.
wTa6eJIH,
wTa6eJIe
BbI4HCJIH
PHC. 3.1
3.3. YMHO>KeHHe 06b1KHOBeHHblx .ll.p06eH
Tbl Y>Ke YMeeWb pewaTb 3aJJ.a4H, B KOTOpbIX npHXOJJ.HTCH YMHO-
>KaTb HaTypaJIbHble 4HCJIa H JJ.eCHTH4Hble JJ.P06H. HanpHMep, Tbl YMe-
eWb Bbl4HCJIHTb nJIOlll,aJJ.b npHMoyrOJIbHHKa no ero CTopOHaM, Haxo-
JJ.HTb JJ.JIHHY nYTH, 3HaH CKOpOCTb H BpeMH JJ.BH>KeHHH, 3HaeWb, KaK
HaHTH CTOHMOCTb KynJIeHHoro TOBapa, eCJIH H3BeCTHbl KOJIH4eCTBO
3Toro TOBapa H CTOHMOCTb OJJ.HOH ero eJJ.HHHLlbl (Kr, M, WT. H JJ.p.).
C<t>OPMYJIHPyeM Tenepb npaBHJIO YMHO>KeHHH 06bIKHOBeHHbIX JJ.p06eH
TaKHM 06pa30M, 4T06bl 3TH H JJ.pyrHe 3aJJ.a4H MO>KHO 6b1JIO peWHTb
YMHO>KeHHeM H B CJIY4ae, eCJIH cpeJJ.H JJ.aHHbIX HMelOTCH 06bIKHOBeH-
Hble JJ.P06H.
n p H M e p I. CTOpOH.bt npRMOY20JtbH.UKa : M U
M. Bbt'iUCJtUM
e20 nJtoU{aOb (8 K8aOpaTH.btX MeTpax).
llJIH peweHHH 3aJJ.a4H BbIpa3HM CHa4aJIa H3MepeHHH npHMoyrOJIb-
HHK8 n.eCHTH4HbIMH JJ.P06HMH. nOJIY4HM:
3 7
"'4 M=0,75 M, 5 M= 1,4 M H 5=0,75.1,4= 1,05 (M 2 ).
llaJIee 06paTHM nOJIY4eHHYlO JJ.ecHTH4HYIO JJ.P06b B 06bIKHOBeHHYIO:
56
2 5 2 105 2 21 2 '='
1.05 M = 1 100 M = 100 M = 20 M . vTOT >Ke pe3YJIbTaT MO>KeM "no-
JlYl/HTb ropa3JJ.0 npOlll,e, 6e3 npeJJ.BapHTeJIbHOrO 06palll,eHHH JJ.aH-
tlb!X 06bIKHOBeHHbIx JJ.po/SeH B .neCHTH4HbIe. MbI BHJJ.HM, 4TO 4HCJIH-
TeJIb pe3YJIbTaTa (
) paBeH npOH3BeJJ.eHHIO 4HCJIHTeJIeH (3.7), a
3HaMeHaTeJIb - npoH3Be.neHHIO 3HaMeHaTeJIeH (4.5) JJ.aHHbIX JJ.p06eH,
21 u 3 7
OOJIY4eHHaH JJ.P06b 20 H eCTb npOH3BeJJ.eHHe JJ.p06eH 4 H 5' 3anHweM:
3 7 _3.7_21
4'5- 4.5 - 20 .
Bo061l1,e npOH3BeJJ.eHHe JJ.BYX O/SbIKHOBeHHbIX JJ.p06eH - 3ro iJpo6b,
'lucJtureJtb ICoropou paBeH. npou3BeiJeH.uro flucJtureAeu, a 3H.aMeH.areJtb
paBeH. npou3BeiJeH.uro 3H.aMeH.areAeU iJaHHblx iJpo6eu. B 6YKBeH-
HOM BHJJ.e 3TO npaBHJIO 3anHCblBaeTCH TaK:
a c a.c
lJ'd"= b.d '
ECJIH B03MO>KHO, TO CJIeJJ.yeT BbInOJIHHTb COKpalll,eHHe pe3YJIb-
TaTa, JIY4we COKpaTHTb 4HCJIHTeJIH H 3HaMeHaTeJIH JJ.O Bbl4HCJIeHHH
HX npOH3BeJJ.eHHH:
5 8 5.Jf 20 2 2
npHMep 2. 6'3=73=9= g'
2 3 1',i'
n pHMep 3. 3'2-
=1.
ECJIH cpeJJ.H MHO>KHTeJIeH HMelOTCH CMewaHHbIe 4HCJIa, TO HX Ha-
n.0 06paTHTb B HenpaBHJIbHble JJ.P06H, a 3aTeM npHMeHHTb npaBHJIO
YMHO>KeHHH JJ.P06H Ha JJ.P06b.
2 3 _ J..!..:L. -l..!..- 2 2..
npHMep 4.33"'4"- ;i1.4 -4- 4'
ECJIH cpeJJ.H MHO>KHTeJIeH HMeeTCH HaTypaJIbHOe 4HCJIO, TO ero 3a-
MeHHIOT JJ.P06blO CO 3HaMeHaTeJIeM I,
3 2 3 2.3 6 1
npHMep 5. 2'5=J'5="'j"':5=5=15'
HJIH Kop04e:
3 2.3 6 I
2.-=-=-=1-
5 5 5 5 .
ECJIH XOTH 6bI OJJ.HH H3 MHO>KHTeJIeH paBeH HYJIIO, TO H npOH3-
BeJJ.eHHe paBHO HYJIIO. 06paTHO: eCJIH npOH3BeJJ.eHHe paBHO HYJIIO,
TO XOTH 6bI OJJ.HH MHO>KHTeJIb paBeH HYJIIO.
57
n 3 3
pH Me p 6. 5.0=0'5=0.
n p H M e p 7. ECJIH 3.( x-+ )=0,
I
TO x- 2 =0,
H, CJIe-
JJ.OBaTeJIbHO, x=....!....,
2
ECJIH H3 JJ.BYX MHO>KHTeJIeH OJJ.HH - 06bIKHOBeHHaH JJ.P06b, a JJ.PY-
rOH 0603Ha4eH 6yKBOH. TO 6YKBa 3anHCWBaeTCH 3a JJ.P06blO Ha
ypOBHe 4epTbI JJ.P06H. 3HaK YMHO>KeHHH MO>KHO He nHcaTb, HanpHMep,
3anHCb
a 03Ha4aeT npOH3BeJJ.eHHe
. a.
n p H M e p 8. BbI4HCJIHM 3Ha4eHHe npoH3BeJJ.eHHH
x, eCJIIf
3
x=5'
5 3
I 'II 1
nOJIY4HM -.-=
=-
6 5 2
'
I 2
MO>KHO y6eJJ.HTbCH, 4TO Bce paHee H3Y4eHHble 3aKOHbl YMHO>Ke.
HHH (nepeMeCTHTeJIbHbIH, C04eTaTeJIbHblH, pacnpeJJ.eJIHTeJIbHbIH) OC-
TalOTCH BepHbIMH H JJ.JIH YMHO>KeHHH 06bIKHOBeHHbIX JJ.polSeH. BcnOMHH
H C<t>OPMYJIHPYH 3TH 3aKOHbl!
A,
299. PaccMoTpH npHMepbl 2 H 3. 061>HCHH YMHO>KeHHe JJ.p06eH 2
3
9
HW'
300. BbI4HCJIH
1 1
I) 2'"3
1 1
4 5
1 I
6'2
301. BbI4HCJIH:
1 2
I) 2'"3
4 3
7'2
9 2
7'"3
(y C THO):
2 2
2) "3'5
4 2
9'"3
5 1
6'2
I 3
3) 5'4
2 4
"3'5
1 4
9'7
3 5
2) 10'6
7 15
5'14
5 4
12'5
1 6
3) 6'7
3 4
4'"3
3 5
5'"3
58
3 3
4) 10'8
1 7
2'12
6 6
U'5
25 27
4) 36 ' 50
7 45
15' 49
19 18
24 '19
302. PaccMoTpH npHMepbl 4, 5 H 6. 061>HCHH YMHO>KeHHe 4HCM:
123
I) 1 2 H 2"3; 2) 5 H W'
303. Bbl4HCJIH (y C THO) :
1 1 1 1 3) 1 4) 5
I) 1-.- 2) 2-.- 3.- -.3
2 2 5 7 5 16
1 1 1 1 4 7
-.2- 1-.- 0.- -.2
5 2 6 4 7 25
1 1 1 4 2 2... 16
-.1- 1-.- 5.1-
3 2 4 5 5 4
304.
BW4HCJIH (y C THO), npHMeHHH
TeJIbHblH 3aKOHbl YMHO>KeHHH:
I ) 4.2.......!.... 2)
.....!.....
5 4 2 4 7
1 1
2.3.2 12. 12 . 4
2 1 3 2 1
_.-.-- -.6.-
3 2 2 9 6
nepeMeCTHTeJIbHblH H C04eTa-
1 7
3) 9'\5.9
2...4.....!....
4 3
.....!.....12
12 10
305. Bbl4HCJIIf (y C THO), npHMeHHH pacnpeJJ.eJIHTeJIbHblH 3aKOH yM-
HO>KeHHH:
I) 5(1++)
1O(6+
)
4 (5-+)
2) 7 (
+ 2)
8(10++)
3(2-
)
306.
BbI4HCJIH:
3 5
1 ) --.1-
5 6
1 3
4"3'10
7 1
-.1-
8 7
5 8
5) 18'15
9
50.-
100
8 I
39 . 3 4
3
2) -. 16
8
5
2.-
6
27.
1 I
6) 7 2 .1 25
11 7
21'22
5
16.--
24
3) 2+.3
2..!.. 15
5
1
2-.0
5
7) 9 25
50 27
5
21. 42
I I
115. 1 24"
3) (++8 ).5
(7++ ).6
(5-
).5
4 ) 1....!.....2..!.
4 5
3....!.....4
2 3
I 2....!..... 32..
3 5
5 9
8) 16'22
7 18
18' 35
1 33
3-.-
3 100
59
(3
-2
)( ; ++)
+..i.. ( 3...!...- I...!... )
6 5 3 4
06paTH JJ.eCHTH4Hble JJ.P06H B 06bIKHOBeHHbIe, 3aTeM BbI4HCJIH:
5 1 5 1
I ) 08.- 2 ) 03.1- 3 ) 12.- 4) 2-.16
'8 '3 '6 6'
2 1 4 5
"3. 0 ,06 27. 0 ,5 2,1'7 2,8.17
4 1 5 1
15' 0,9 5""6' 0,08 12' 1,8 3 9 , 1,5
06paTH 06blKHoBeHHble JJ.P06H B JJ.eCHTH4HbIe, 3aTeM BbI4HCJIH:
2 14 3 I
l) ""'5. 0 ,6 2) 25 . 0 ,08 3) 1,4.10 4) 5,2.1""'5
1 49 4 7
0,04' T 0,3. 50 ""'5. 2 ,6 210. 8 ,8
13 3 3 3
0,9. 20 8. 0 ,1 0,72'4 24. 3 ,6
YnpocTH Bblpa>KeHHe:
3 2
2) 4'"3a
6 1
7.""6 Y
307. BbI4HCJIH:
I) 12-_
.2-
3 4 9
1 I
2-.4-5-
2 4
1 2 9
3"3+ 2 9' 20
4 4 3 1
3) 15+15.4-"3
308.
309.
310.
I) 4.3a
1,2x.4
311. PacKpoH cK06KH:
I) 3 (14-2x)
2,4 (a+3)
312. YnpocTH BbIpa>KeHHe:
I) 7a-3a+5a
I ,8x + 2,4x - 2,2x
60
2) ( 2...!...-1...!... ) .22-
3 2 5
( 2..i..+2- )
7 9 3
(
....!... ) .12
12 4
5 3
3) ""6'Tx
3 4
-.-b
8 3
2) 2(+a++)
5 (++
x )
1 2
2) "3a+"3a
5 I
8 x -8 x
4 3
4) "3x,""'5
1 5
w'T b
1
3) "3(6x-9)
(+Y+
)
I I
3) TX-"3x
2 3
""'5 a +W a
6.
Bbl4HCJIH (y C THO), npHMeHHH pacnpeJJ.eJIHTeJIbHbIH 3aKOH YM-
313.
HO>KeHHH:
1 1 1 1
I) 5.3- 2) 9.5- 3) 10.7- 4) 4.1-
5 3 2 4
1 3 5 2
1-.6 100.5- 14.2- 9.3-
3 5 7 9
5 9 1 7
5-.6 4-.22 6-.30 18.-
12 11 15 18
314. BbI4HCJIH KaK MO>KHO npOLLle:
24 7
2) 125 .316
6
.9.!...
8 9
1 9
4 3O '1T
3 5 8
3) 4'12'15
6 3 8
25 . 3 4'9
2...!.... I...!.... 32-
2 3 5
315.
1 4
I) 18"3. 1 1T
7 3
4 20 .1 29
I
' 5.!!.-
58 21
Bbl4HCJIH:
I ) 21-.12.31-.l... 2)
.11-.2
.l...
6 35 7354
3 ) 11-.5.
.71- 4) 2....11-.24.71-
11 5 3 12 5 2
Bbl4HCJIH 3Ha4eHHe Bblpa>KeHHH 1
X -+y, eCJIH:
1 1
I) x=6 H y=3; 2) x=1 26 H y=2 12 "
316.
317.
BbI4HCJIH:
( 13 I ) 3 4 + 5 1 7 .
I) 3 50 - 3 20 . T \2' 15'
2 )
( 1
+ 1.!... ) -7...!.......!...-
;
22 5 10 2 30 8
3) (3+.1 ;4 + 3 .
)' ;1 - 1 +;
4) (2+.-k) (1-k.6+ ). ;4 . :0 "
HaHJJ.H JJ.eCHTH4Hble npH6JIH>KeHHH 06bIKHOBeHHblX
OKpyrJIeHHble JJ.O JJ.eCHTbIX, 3aTeM BbI4HCJIH:
4 25
I) 2,6'9 2) 0,24.1 3 3) 4-0,7'9
.!....O 85 2 55. 4...!... 3,2. 1 7 5 + 1,34
12 ' '3
17 6 1 8 12
1 .5,4 2 24 .1,5 17. 0 ,24+ , '13
JJ.p06eH,
318.
61
319. YnpocTH BbIpa>KeHHe:
I) ..i...I...!...a....!... 2 )
523
3 4 1
4 a .5'"3
1 3
2a.-.-
2 4
320. PacKpoH cK06KH:
I) 8(
+ 38a ) 2) +(
a+f) 3)
(
-
) .18 4...!... ( _...!... )
9 18 2 X 3
321. YnpocTH BbIpa>KeHHe:
I) ...!...a+
a+..i..a 3 1 5
5 5 5 2) """7 m -"""7 m +"""7 m
2 + 3 1 1 I 3
5 a lOa-lOa "3 x -""6 x +4 x
3 ) ...!...b +
b +
b+ ...!... b 1 1
2 8 8 2 4) 2a+Ta+2a-a
2 + 1 + 5 2 4 1 9
gY ""6 Y ""6Y-gY TX+wx+x-wx
322*. PewH ypaBHeHHe, npoBepb ero KopeHb:
I) 5 (x-3)=0; 2) 3 (2x-8)=0; 3) x (x-9)=0.
724
I-.-x.-
8 3 15
4 10 3
2-y.-.3-
5 21 10
3 4
1-.-x.3
4 7
2
3) O,5'"3c.6
5
9.02.- b
, 9
5n.0,2.7
15 (
b -2-c+...!... )
6 18 30
1 ( 5 I
1""6 2x+14y-12)
3.4. 3a.ll.al.JH Ha YMHO>KeHHe .ll.po6eii
n p H M e p I. OiJUH. MeTp TKaH.U CTOUT 15 p, CKOJlbKO CTORl
2 M 3TOU TKaH.u?
3aJJ.a4a pewaeTcH YMHO>KeHHeM: 15.2=30,
llBa MeTpa TKaHH CTOHT 30 p.
n p H Me p 2. OiJUH. KUJl02paMM R6JlOK CTOUT : p, CKOJlbKO
CTORT: I) 3 Kr; 2) 2 Kr; 3)
Kr; 4) 2+ Kr 3TUX R6JlOK?
11 3TY 3aJJ.a4Y pewaeM YMHO>KeHHeM:
I) ..i...3= 4.3 =J!=2 2 (p ) .
5 5 5 5 .,
2) ..i...2=
=
= I
(p ) .
5 5 5 5 .,
4 3 4.3 3
3) 5'4=5:4=5 (p.);
4)
.2
=
..2..-= 4.9 =.2..-= I
(p )
5 4 5 4 5.4 5 5 ,.
62
2
n p H M e p 3. B xope 80 Y'iaU{UXCR, 5 U3 H.UX MaJlb'iUKU, OCTaJlb-
Hbte iJeBO'iKU. CKOJlbKO MaJlb'iUKOB B xope?
HaHJJ.eM CHa4aJIa tOT 80 (80: 5 = 16). TaK KaK
OT 80 B 2 pa3a
1
60JlbWe, l/eM TOT 80, TO 16 YMHO>KHM Ha 2 (16.2=32). TOT >Ke OTBeT
nOJIy4HM, eCJIH HaHJJ.eM npoH3BeJJ.eHHe 80 H
,
80. 2 = 80.2 = 3 2
5 5 .
2
J.1TaK, B xope 32 MaJIbl/HKa. rOBOpHT: Mbl HaWJIH 5 OT 4HCJIa
80.
QT06bl H.aUTU iJP06b OT '1UCAa, H.Y3ICH.O '1UCAO YMH.03ICUTb H.a
STY iJp06b.
ECJIH JJ.P06b MeHbwe I, TO JJ.p06b OT 4HCJIa 6YJJ.eT MeHbwe
3Toro 4HCJIa. ECJIH JJ.P06b 60JIbWe I, TO JJ.P06b OT l/HCJIa 60JIbWe
3Toro 4HCJIa.
n p H Me p 4. HaHJJ.eM : OT 35.
HaHJJ.eM CHa4aJIa *" OT 35: 35:5=7. CJIeJJ.oBaTeJIbHO, HCKO-
Moe 4HCJIO paBHo 7.8 = 56,
8
3anHweM Kop04e: 35'5=56.
A.
323. np04TH 3aJJ.a4H I H 2, 4TO B HHX 06Lllero? qeM OHH OTJIH4a-
IOTCH?
324.
BbI4HCJIH nJIOlll,aJJ.b
1 2
I) 3 2 M H 25 M;
5 3
4) ""6 JJ.M H 5 JJ.M.
npHMoyrOJIbHHKa, eCJIH ero H3MepeHHH:
5 3 1
2) 4 M H ""8 M; 3) 4 JJ.M H 3"3 JJ.M;
325. Bbl4HCJIH nJIOLllaJJ.b KBaJJ.paTa, eCJIH JJ.JIHHa ero CTOpOHbl:
231
I) IT M; 2) "'"4 M; 3) ""2 M.
326. BbI4HCJIH 06"beM npHMoyrOJIbHOrO napaJIJIeJIenHneJJ.a, H3MepeHHH
1 3 4 2
KOToporo: I) 2 JJ.M, 3,5 JJ.M H 1 4 JJ.M; 2) 4 M, 5 M H "3 M.
63
3
5 M,
327. OJJ.HH MeTp TKaHH CTOHT 20 p. CKOJIbKO CTOHT 2 M,
2+ M 3TOH TKaHH?
328. OJJ.HH JIHTp KepOCHHa BeCHT : Kr. CKOJIbKO KHJIOrpaMMOB Be-
10 I 3 5 3
CHT JI, 3 JI, 2 JI, 4 JI. ""6 JI, 3 4 JI KepocHHa?
329. CpeJJ.HHH CKOpOCTb newexoJJ.a 3+
. CKOJIbKO KHJIOMeTpOB
npoHJJ.eT nelllexoJJ. 3a 5 4
4
.. 2 J......;I
, 5 ' 3 -., 6-"
330. TOJIH YBHJJ.eJI MOJIHHIO, a pacKaT
rpoMa YCJIbIWaJI TOJIbKO 4epe3
24 c, Ha KaKOM paccTOHHHH 6bIJIa
rp030BaH TY4a, eCJIH CKOpocTb pac-
npocTpaHeHHH 3BYKa + :M ?
331. OJJ.Ha nJIaCTHHKa HMeeT <POPMY
npHMoyroJIbHHKa, H3MepeHHH KOTO-
1 3
poro 1"5 M H ""4 M. llpyraH nJIa-
CTHHKa HMeeT <POPMY KBaJJ.pa-
T8 co CTOpOHOH I M. Ha CKOJIbKO nJIOll.laJJ.b npHMoyrOJIbHHKa
60JIb
e (HJIH MeHbwe) nJIOlll,aJJ.H KBaJJ.paTa?
332. QT06bI JJ.06paTbcH JJ.O ropoJJ.a, TOJIH npoweJI neWKOM 2
KM
4
333.
1
H npoexaJI Ha aBT06yce -- 4 CO
3
paCCTOHHHe OT JJ.OMa JJ.O ropoJJ.a.
np04TH BHHMaTeJIbHO npHMep 3
OT 4HCJIa.
BbI4HCJIH:
H CKa>KH, KaK HaHTH JJ.P06b
CK 58 1 KM H u
OpOCTblO 2 -;;. aHJJ.H
334.
3
I) 40T 12,
5
""6 OT 18,
3
2) ""'5 OT 25,
I
""6 OT 24,
7
3) "'80T 32, 4)
6
""""7 OT 42,
2
3 0T
5
"'8 0T
1
1 2 ,
3
5""'5'
0,7 OT 42; 0,3 OT 14; 0,85 OT 12; 0,24 OT 200.
335. B WKOJIe 360 Y4alll,HXCH, H3 HHX : COCTaBJIHIOT MaJIb4HKH
CKOJIbKO MaJIb4HKOB B WKOJIe?
64
3
336. B 3epHOXpaHHJIHIlI,e 4800 T 3epHa, 8 KOToporo nweHHLla.
CKOJIbKO TOHH nweHHLlbi B 3epHOXpaHHJIHLlle?
337. Konx03 HMeeT 60000 ra 3eMnH. no.n; rroceBHbIMH KynbTypaMH
2 - 0
3amIT0 3" :3TOH rrnOll.la.n;H. CTanbHa5I rrnOll.la)J.b OTBe.n;eHa rro.n;
rracT6Hll.le. CKonbKO reKTapOB 3aHHMalOT rroceBHbIe KynbTypbI H
rracT6Hll.le oT.n;enbHo?
338. 04epeJJ.Hoe 3aceJJ.aHHe MaTeMaTH4eCKoro KpY>KKa npoJJ.oJI>KaJIOCb
1,5 4, Ha 3HaKOMCTBO C HCTopHeH 06b1KHOBeHHblX .n.p06eH
2
YWJIO 3 3Toro BpeMeHH. OCTaJIbHOe BpeMH peWaJIH 3aJJ.a4H.
CKOJIbKO BpeMeHH peWaJIH 3aJJ.a4H?
3 .
339. 3HHa np04HTaJIa 4 KHHrH, B KOTOpOH 120 cTpaHHLl. CKOJIbKO
cTpaHHLl OCTaJIOCb np04HTaTb?
340.
acca caxapa, nOJIY4eHHaH npH nepepa60TKe caxapHoH CBeK-
1 K u
JIbl, COCTaBJIHeT ""6 MaCCbl CBeKJlbI. OJlX03 OTBe3 Ha caxapHbIH
3aBOJJ. 80 aBTOMaWHH, B cpeJJ.HeM no 3,75 T caxapHoH CBeK-
JIbl Ha Ka>KJJ.oH. CKOJIbKO TOHH caxapa nOJIY4HJIH H3 3TOH
CBeKJIbl?
3
341. llJIHHa npHMoyroJIbHHKa 15 CM, a wHpHHa COCTaBJIHeT ""'5 ero
JJ.JIHHbI. BbI4HCJIH nJIOlll,aJJ.b H nepHMeTp npHMoyrOJIbHHKa.
I 3
342. Bbll/HCJIH: I) "'3 OT 150°; 2) 40T 180°; 2) 0,6 OT 90°,
Ha4epTH 3TH yrJIbl C nOMOLllblO TpaHcnopTHpa.
3
343. OTpe30K AB paBeH 6 10 CM, a oTpe30K CD COCTaBJIHeT
2
3 0T JJ.JIHHbl OTpe3Ka AB. Ha CKOJIbKO oTpe30K CD Kop04e OT-
pe3Ka AB? Ha4epTH 3TH OTpe3KH.
6.
344. BbI4HCJIH:
2
2) I T oT4,5,
7
10 OT 0,4,
5
0,8 OT 2""6;
2
3) "30TO,36,
5
120T 4,8,
10
1,4 OT 21;
3
4) 1 4 oTO,24,
2
""'9 OT 0,81,
2
0,450T 4 3 ,
65
3
I) 40T 15,6,
3
100 OT 32,5,
0,9 OT 6,5;
nOJIOBHHY BCeH npOJJ.YKLlHH 3aBOJJ.a COCTaBJIHIOT C4eTHble MawH-
2
Hbl, H3 HHX 3" - MHKpOKaJIbKYJIHTOpbl. KaKYIO 4aCTb BCe"
npOJJ.YKLlHH 3aBOJJ.8 COCTaBJIHIOT MHKpOKaJIbKYJIHTOpbI?
lllHpHHa KOMH8Tbl 4 M, JJ.JIHHa B 1,5 pa3a 60JIbWe WHpHHbl, a
1
BbiCOTa COCTaBJIHeT 2 4aCTb JJ.JIHHbl. BbI4HCJIH Maccy B03JJ.YXa B
KOMHaTe, eCJIH I M 3 B03JJ.yxa BeCHT npH6JIH3HTeJIbHO I 1 3 0 Kr.
B Mara3HH 3aBe3JIH 600 Kr MYKH. B nepBoH nOJIOBHHe JJ.HSI
1 u u 2
npOJJ.aJIH "'4 BceH MYKH, BO BTOpOH nOJIOBHHe JJ.HH ""'5 OCTaTK8.
CKOJIbKO MYKH OCTaJIOCb HenpOJJ.aHHOH?
u 3 13 9
P>KaHOH xJIe6 COJJ.ep>KHT 250 >KHpOB, 25 6eJIKOB H 20 yrJIe-
4 1
BOJJ.OB. MacJIo >Ke COJJ.ep>KHT ""'5 >KHpOB, 250 6eJIKOB H 200
yrJIeBOJJ.OB. CKOJIbKO 6eJIKOB, >KHpOB H yrJIeBOJJ.OB COJJ.ep>KHTCH B
5 Kr xJIe6a H CKOJIbKO B 500 r MaCJIa?
3a eJJ.HHHLlY JJ.JIHHW B llpeBHeM ErHnTe npHHHMaJIH «LlapcKHH
u 13
JIOKOTb», paBHblH npHMepHO 25 M, H «JlOKOTb npOCTOJIIOJJ.HHa»,
paBHblH ;0 M, <1>apaoHbl H >KpeLlbi 6paJIH JJ.aHb B «LlapcKHx
JIOKTHX», a npOCTOMY HapoJJ.Y npOJJ.aBaJIH MaTepHaJI B «JIOKTHX
npOCTOJIIOJJ.HHOB». npeJJ.nOJIO>KHM, 4TO KyneLl JJ.OJI>KeH OTJl.aTb
JJ.aHb MaTepHeH 75 «LlapcKHx JIOKTeH», a npoJJ.aTb eMY nOCl/aCTJIH-
BHJIOCb 80 «npoCTOJIIOJJ.HHOBbIX JIOKTeH» MaTepHH. CpaBHH KOJIH-
4eCTBO npoJJ.aHHoH MaTepHH C KOJIH4eCTBOM MaTepHH, oTJJ.aHHoH B
JJ.aHb.
CocTaBb H peWH 3aJJ.a4Y Ha CJIO>KeHHe 06blKHOBeHHblX JJ.p06eH.
C u 18 1 KM
KOpOCTb KaTepa B CTOH4eH BOJJ.e 2 -;;-, a CKOpOCTb Te-
2 3 KM 1
4eHHH ""'5
' KaTep npOweJI 1"3 4 B CTOH4eH BOJJ.e H elll,e
1 4 npOTHB Te4eHHH. COCTaBb BbIpa>KeHHe JJ.JIH Bbl4HCJIeHHH
Bcero nYTH, npOHJJ.eHHoro KaTepOM. BbI4HCJIH ero 3Ha4eHHe, ec-
1 1
JIH: I) 1=2; 2) 1=1"3'
352. Y4alll,HeCH np04HTaJIH 0 nJIaBaHHH aTOM Horo JIeJJ.OKOJIa «JIeHHH»
B 1961 r., KOTOpblH BnepBble B HCTOpHH npoweJI CeBepHbIH MOp-
CKOH nYTb 3HMOH. Ha nepeMeHe MaJIb4HKH 06cY>KJJ.aJIH np04HTaH-
Hoe H BblHCHHJIH, 4TO H3 7000 MOpCKHX MHJIb HaH60JIbWee
345.
346.
347.
348.
349.
350.
351.
66
MY>KeCTBO Y MOpHKOB nOTpe60-
5
BaJIO npOXO>KJJ.eHHe 7" 3TOro ny-
TH B naKOBOM JIbJJ.Y (MHOrOJIeT-
HHH JIeJJ. TOJIIlI,HHOH 3-5 M), LITO
CKOpOCTb JJ.BH>KeHHH JIeJJ.OKOJIa
u 9 M
«JleHHH» B cB060JJ.HOH BOJJ.e c'
z' ,
.",. ...q;/"."
<if .
,:: ,//It((
""A/At-,
TOJIIlI,HHOH 60JIee 2 M paBHa
a CKOpOCTb JJ.BH>KeHHH BO JIbJJ.ax
..1...- CKOpOCTH JJ.BH>KeHHH B cBo60JJ.HOH BOJJ.e. YCTaHOBHJIH TaK>Ke,
9
LITO JJ.J1HHa JIeJJ.OKOJIa 134 M, a wHpHHa COCTaBJIHeT npH6JIH3H-
TeJIbHO 0,206 OT ero JJ.JIHHbI, Ha ypoKe MaTeMaTHKH no u 3THM
JJ.aHHbIM Y4alll,HeCH COCTaBHJIH 3aJJ.a4H H peWHJIH HX, CJJ.eJIaH H Tbl
TO >Ke caMoe.
2 u
353*. CYMMa 4eTbIpex l/HCeJI 210, nepBoe 4HCJIO COCTaBJIHeT ""'5 3TOH
1 e 4HCJIO 3
CYMMbI, BTopoe 4HCJIO "'4 nepBoro 4HCJIa, a TpeTb ""'5
CYMMbI OCTaBWHXCH JJ.BYX l/HCeJI. HaHJJ.H 3TH 4HCJIa.
Bblpa3H (y C THO) B npoueHTax:
. 0 84 '.!.-' I 28' ..1...-;
; 0,063.
100' , , 10' , '4 5
Bwpa3H (y C THO) 06bIKHOBeHHblMH H JJ.eCHTH4HbIMH JJ.P06HMH:
19%, 305%, 72%, 28%, 121 %, 2,3%.
BbI4HCJIH:
I) 0,45:0,12
2,002:0,026
J].JlH nOBTopeHHH
354.
355.
356.
2) 0,45:3,75
922.6:65,9
357. CrropTHBHaM Hrpa Ha MeCTHO-
CTH npoxoJJ.HJIa no MapwpYTY
ABCDEFGH (pHC. 3,2). CTapT
H <pHHHW 6blJIH B T04Ke A.
Bbl4HCJlH JJ.J1HHY MapwpYTa,
no KOTOPOMY npoxoJJ.HJIa Hrpa,
3 1
eCJIH CB=r;KM, CD= llOKM
H GH=
KM.
5
3) 1,05:7,5
171.6: 0,78
B
E
c
o
PMC. 3.2
67
358. BbI4HCJIH:
I) 8++ 1 3 7 (3+-2+);
3) 3,2-1,2.2,5 1,2 .
5,6-5,56 0,3'
2) 7
- ( 2
-12- ) ..!-.
3 3 7 19'
4) 125+5.46 + 0,71
89-4.21 0,1'
3.5. HaxO>K.ll.eHHe npou.eHTOB OT .ll.aHHOrO IJHCJ1a
B npeJJ.b1JJ.YllI,eM naparpa<t>e Tbl HaY4HJICH HaXOJJ.HTb JJ.P06b OT 4HC-
JIa. JI.JIH 3Toro HY>KHO 4HCJIO YMHO>KHTb Ha JJ.aHHYIO JJ.P06b. l.JaCTo
3Ta JJ.P06b Bblpa>KaeTCH B npOLleHTax. B 3TOM CJIY4ae rOBopHT, 4TO
HY>KHO HaihH npoueHTbl OT JJ.aHHoro 4HCJIa. B npoCTeHwHx CJIY4aHX Mbl
Y>Ke YMeeM pewaTb TaKHe 3aJJ.a4H. BcnOMHHM 3TO.
n pH M e p l. HaiJ.oeM 18% OT '-LUella 200.
CHa4aJIa HaHJJ.eM 1% OT 4HCJIa 200. JI.JIH 3Toro 4HCJIO 200 pa3JJ.e-
JIHM Ha 100 (200: 100 = 2). nOJIY4eHHblH pe3YJIbTaT YMHO>KHM Ha 4HC-
JIO npoueHTOB, KOTopoe Tpe6yeTcH HaihH (2.18=36). 3Ha4HT, 18%
OT 200 paBHbI 36. 3TOT >Ke pe3YJIbTaT MO>KHO nOJIY4HTb nO-JJ.PyroMY,
YMHO>KaH 4HCJIO 200 Ha JJ.P06b /
o HJIH Ha 0,18, JI.eHcTBHTeJIbHO
18 200.18
200. 100 =100=36, HJIH 200.0,18=36.
Bo061l1,e, 4T06bl HaihH npoueHTbI OT JJ.aHHoro 4HCJla, HY>KHO:
I) Bbtpa3UTb npOlfeHTbt 06btICHOBeHHou UJlU iJecRTU'IHOU iJP06blO;
2) YMH03ICUTb iJaHHoe '1UCllO Ha 3TY iJP06b.
n pH Me p 2. flpu nepepa60TKe caxapHoiJ. CBeKllbt M03/CHV nOllY-
'-LUTb 13% caxapa or BceiJ. MaCCbt CBeKllbt, CKOllbKO caxapa M03/CHO
nOllY'-LUTb U3 50 T cBeKllbt?
JI.JIH peweHHH 3aJJ.a4H HY>KHO HaHTH 13% OT 4HCJIa 50. Bblpa-
3HM 13% 06blKHoBeHHoH HJIH JJ.eCHTH4HOH JJ.P06blO:
13% = /;0 =0,13.
YMHO>KHM JJ.aHHoe 4HCJIO Ha 3TY JJ.P06b:
50.
= 5O.13 =E=6
100 100 2 2 '
HJIH
50.0,13=6,5,
CJIeJJ.oBaTeJIbHO, H3 50 T caxapHoH CBeKJIbl MO>KHO nOJIY4HTb
68
65 T caxapa. npH Bbl4HCJIeHHHX C HCnOJlb30BaHHeM MHKpOKaJIb-
KYJIHTOpa npoueHTbl YJJ.06HO Bblpa>KaTb JJ.ecHTH4HOH JJ.P06blO.
A.
359. np04TH TeKCT naparpa<t>a, npoJJ.YMaH npHMepbl H 06"bHCHH, KaK
HaHTH: a) 15% OT 80; 6) 9% OT 63.
HaHJJ.H (y C THO) :
1
I) W OT 20,
5
100 OT 600.
0,4 OT 50;
2) 30% OT 40, 3) 20% OT 100,
5% OT 300,
80% OT 20;
40% OT 90,
150% OT 200.
360. HaHJJ.H:
1) 11
0 OT 70, 2) 15% OT 420, 3) 9.8% OT 500,
I1g 0 OT 8,5, 38 % OT 95, 11,2 % OT 5,5,
OT 75, 137% OT 0.8, 3,7% OT 1,2,
0,78 OT 135; 94% OT 5,6; 23,5% OT 78.
361. B 6 B KJIacce 35 Y4eHHKoB, B 6 A 80% 3Toro KOJIH4eCTBa.
CKOJIbKO Y4eHHKoB B 6 A KJIacce?
362. B WKOJIe 425 Y4aLllHxcH, 56% H3 HHX JJ.eB04KH, CKOJIbKO JJ.e-
B04eK B WKOJIe?
363. B KJIaCCe 30 Y4alliHxcH, 80% H3 HHX nHOHepbl. CKOJIbKO nHOHe-
pOB B KJlaCce?
364. TeJIo 4eJIOBeKa cOJJ.ep>KHT npHMepHo 64% BOJJ.bl. CKOJIbKO KHJIO-
rpaMMoB BOJJ.bl B 4eJIOBe4eCKOM TeJIe, eCJIH ero Macca 40 Kr?
365. npH cywKe H6JIOKH TepHIOT 91 % CBoeH MaCCbI. CKOJIbKO
KHJIOrpaMMoB cyweHblx H6JIOK nOJIY4HTCH H3 160 Kr CBe>KHx?
366. no nJIaHY TOKapb JJ.OJI>KeH H3rOTOBHTb 400 JJ.eTaJIeH B JJ.eHb.
OH BbInOJIHHJI nJIaH Ha 110%. CKOJIbKO JJ.eTaJIeH H3rOTOBHJI TO-
Kapb?
367. 06LllaH nJIOlll,aJJ.b apXHneJIara CeBepHaH 3eMJIH npHMepHo
37000 KM 2 . nJIOLllaJJ.b oCTpoBa nHoHep COCTaBJIHeT 4,3% OT
nJIOlll,aJJ.H Bcero apXHneJIara. Bbl'{H
JIH nJIOlll,aJJ.b OCTpoBa nHO-
Hep, (OTBeT oKpyrJIH JJ.O cOTeH.)
368. .LJ.,nHHa Y4acTKa npHMoyrOJIbHOH <t>OpMbl 500 M, a wHpHHa co-
CTaBJIHeT 42% ero J],JIHHbI. Bbl4HCJIH nJIOlll,aJJ.b Y4acTKa. (OTBeT
Bblpa3H B reKTapax.)
69
369. CTOpOHbl npHMoyroJlbHHKa 5,4 JJ.M H 3,1 JJ.M. nepHMeTp KBan._
paTa COCTaBJIHeT 80% OT nepHMeTpa npHMoyrOJIbHHKa, BbI4HC_
JIH CTOPOHY KBaJJ.paTa.
370. oaK HMeeT <POp MY npHMoyroJIbHOrO napaJIJIeJIenHneJJ.a, H3Mepe_
HHH KOTOpOro 15 JJ.M, 12 JJ.M H 6 JJ.M. BOJJ.OH 3anOJIHeHO
60% ero 061>eMa. CKOJIbKO JIHTpOB BOJJ.bl B 6aKe?
6.
371. JIY4, npoBeJJ.eHHwH H3 BepwHHbl npHMoro yrJIa, JJ.eJIHT ero Ha
JJ.Ba yrJIa. OJJ.HH H3 3THX yrJIOB COCTaBJIHeT 65% npHMoro
yrJIa. Bbl4HCJIH rpaJJ.ycHble Mepbl OISOHX yr JIOB.
372. JI.,nHHa npHMoyrOJIbHOrO Y4acTKa 120 M, a wHpHHa COCTaBJIHeT
75% JJ.J1HHbI. BcnaxaHo 35% 3Toro Y4aCTKa. CKOJIbKO reKTapOB
He BcnaxaHo?
373. CTopoHa 3eMeJIbHOrO Y4acTKa, HMelOIlI,erO <POPMY KBaJJ.paTa,
140 M. noJJ. H4MeHb 3aHHTO 65% ::noro Y4acTKa. Ha OCTaJIbHOH
nJIOLLlaJJ.H paCTeT OBec. nOCTaBb pa3YMHble Bonpocbl H OTBeTb Ha
HHX.
374. 1-13 750 Y4alll,HXCH WKOJIbl 80% 3aHHMalOTCH B pa3JIH4HblX
KpY>KKaX, H3 HHX 5% - B paJJ.HoKpY>KKe. CKOJIbKO Y4alll,HXCH
3aHHMaeTCH B paJJ.HoKpY>KKe?
375. CocTaBb 3aJJ.a4Y Ha HaXO>KJJ.eHHe npoLleHTa OT 4HCJIa. PewH ee.
376. B OJJ.HOM paHoHe ropoJJ.a 360 KBapTHp, H3 HHX
KBapTHp HMe-
lOT nJIOlll,aJJ.b 55 M 2 Ka>KJJ.aH, 40% KBapTHp HMelOT nJIOLLlaJJ.b 40 M 2
Ka>KJJ.aH, a OCTaJIbHble HMelOT nJIOlll,aJJ.b 28 M 2 Ka>KJJ.aH. Bbl4HCJIH
nJIOlll,aJJ.b Bcex KBapTHp paHoHa,
J],J1H caMonpOBepKH
377.
OlSpaTH B JJ.eCHTH4Hble JJ.P06H:
63 1 13 21 1
100 ' 3 5 , 2 20 ' 50 ' 4 4 ,
06paTH B 06bIKHOBeHHbIe JJ.P06H:
3,6; 0,48; 2, I 05; 0,04.
Bbl4HCJIH JJ.eCHTH4HbIe npH6JIH>KeHHH JJ.p06eH 1 7 7 H
8
I g , oKpyr-
378.
379.
JIeHHble JJ.O COTbIX.
380. PaCCKa>KH, KaK BbmOJIHHeTCH YMHO>KeHHe JJ.BYX H 60JIee 06b1K-
HOBeHHbIX JJ.p06eH.
70
381. Bbl4HCJIH (y C THO):
2 5 1 1 3) 4 I 4) 6 ( ++ I )
I) 5.7 2) 1-.- 7.-.-
3 4 9 7
4 1 1 2 5 4( 2-+)
2.1- -.-.5
9'2 2 3 8
.4 1 I 4 1 8 ( : +2 )
2-.- 9.-.-
12 3 7 9 2
382.
BbI4HCJIH:
15 21
I) 28 ' 40 2)
1 I
28. 1 17
:1 .4+ 4+,0,32 4-
(1++
)
2 7 .
Bbl4HCJIH 3Ha4eHHe BbIpa>KeHHH "3 a + I g , eCJIH.
I
6.2-
9
5
1,2'\8
3 ) ( 2
-1
) .1...!..
3 5 11
1
0,6+6.14
383.
5 3
I) a=O; 2) a =""6; 3) a=1 4 .
384.
2
CKOJIbKO KHJIOMeTpOB npoeJJ.eT BeJIOCHneJJ.HCT 3a l:f 4, eCJIH
13 1 KM;I
ero cpeJJ.HHH CKOpOCTb 2
'
1 2
raJIH KynHJIa 4 Kr KOH<peT no LleHe 25 p. 3a KHJIOrpaMM
H 2- Kr ne4eHbH no LleHe 1,5 p. 3a KHJIOrpaMM. I) KaKYIO
5 6 u
cJJ.a4Y nOJIY4HJIa raJIH C 5 p.? 2) Ha CKOJIbKO py JIeH KOH-
<p eTw JJ.opo>Ke (JJ.eweBJIe), 4eM ne4eHbe?
PacKpoH cK06KH:
I) 3(+X++); 2) (6-14a).+; 3)
(x+IO)
PaCCKa>KH, KaK HaHTH JJ.P06b (npoLleHT) OT 4HCJIa,
BbI4HCJIH:
I)
OT 189, 2) O,40T 150, 3) 68% OT 35, 4) 120% OT 80,
!..... OT 3 2 , ' 0,24 OT 2 9 2 ; 7% oT4,9; 5,4% OT 270.
16 '
Bbl4HCJIH nJIOLLlaJJ.b >KeCTHHOro JIHCTa npHMoyroJIbHOH <P<>PMbl,
4
eCJIH ero JJ.JIHHa 2,5 M H wHpHHa COCTaBJIHeT 5 JJ.JIHHbI.
llJIHHa JJ.HCTaHLlHH TpexJJ.HeBHoH BeJIOrOHKH 6blJIa 480 KM, B nep-
% u
BblH JJ.eHb BeJIOrOHIlI,HKH npOeXaJIH 25 0 Bcero nYTH, a BO BTOpOH
JJ.eHb 55% OCTaBwerOCH nYTH. CKOJIbKO KHJIOMeTpOB npOeXaJIH
BeJIOCHneJJ.HCTbI B TpeTHH JJ.eHb?
385.
386.
387.
388.
389.
390.
71
4. J],EJ1EHHE 06blKHOBEHHbiX J],P06EA.
npOnOPLl.HSI
CaMOCTOHTeJIbHaH pa60Ta 3
T e M a. 83aHMHo o6paTHbie "HCJIa.
1. HaHJJ.H npOH3BeJJ.eHHe:
1 3 4 1 3
I) 2'2; 2) 4'"3; 3) 3"3.10; 4) 0,2.5;
5) 2,5.0,4,
ECJIH He 01lm6cH B BbI4HCJIeHHHX, TO nOJIY4HJI OTBeT: 3Ha4eHHe
Ka}KJJ.oro H3 3THX npOH3BeJJ.eHHH paBHO I,
2. 3anoMHH: iJ8a 'l.UC/la, npou38eiJeHue IWTOpbtX pa8HO eiJuHUl{e,
Ha3bt8CUOTCll 83aUMHO 06paTHblMU 'l.UC/laMU.
3. YKa}KH napbl 4HCeJI, B KOTOpblX 4HCJIa B3aHMHO 06paTHbl:
25 37 32 6
I) ""5 H 2; 2) 7 H "3; 3) ""5 H "3; 4) 2,2 H u;
5) 7 H 7; 6) 0,8 H 1+; 7) 2+ H T'
npH npaBHJIbHblX Bbl4HCJIeHHHX Tbl HarneJI nHTb nap, B KOTOpblX
4HCJIa B3aHMHO 06paTHble.
4. HaHJJ.H 4HCJIO, 06paTHoe JJ.aHHOMY:
3 10 13 1
I) ""'5; 2) u; 3) 7; 4) "3; 5) 10; 6) 9; 7) 100.
nOJJ.YMaH, KaK HaHTH 4HCJIO: I) 06paTHoe 06blKHoBeHHoH JJ.P06H;
2) 06paTHoe HaTypaJIbHoMY 4HCJIY.
5. ECJIH Tbl C 3aJJ.aHHeM CnpaBHJICH H Bce nOHHJI, TO npHrneJI
K CJIeJJ.YlOllI,eMY BbIBOJJ.Y:
I) 4T06bl HaHTH 4HCJIO, 06paTHoe 06blKHOBeHHOH JJ.P06H, HY}KHO
4HCJIHTeJIb H 3HaMeHaTeJIb JJ.P06H nOMeHHTb MeCTaMH;
2) 4HCJIO, 06paTHoe HaTypaJIbHoMY,- 3TO JJ.P06b, 4HCJIHTeJIb KO-
TOpOH I, a 3HaMeHaTeJIb - caMO HaTypaJIbHOe 4HCJIO.
A.
391. KaKoe 4HCJIO 06paTHoe eJJ.HHHLle?
392. n04eMY He CYllI,eCTByeT 4HCJIa, 06paTHoro HYJIIO?
72
393.
HaHJJ.H 4HCJIO, 06paTHoe JJ.aHHOMY:
5 9 11 1 I 1 2
7' 10' ""'5' ""'5' 8' 6, 10, 1000,3 4 ,2""'5'
06paTH JJ.eCHTH4Hble JJ.P06H B 06blKHOBeHHble H HaHJJ.H HM 06paT-
Hble 4HCJIa:
0,1; 0,3; 0,02; 1,2; 2,5; 5,29; 4,017.
Bbl4HCJIH (y C THO):
I)
(
.1,7) 2)(i-.I+).g 3) (12.
) . ;
( 314 ) 2
7 -.-.- -.0 I ( I 5.10 )
4 7 3 3"
394.
395.
396.
2.. (
.2.. )
3 3 7
PeWH ypaBHeHHe:
I) +x=1 2)
y=1
a=1
c=1
14 5
0,8z= I
4) O,5x= I .
3) 8x= I
l
b=1
9
6.
397. BepHo JIH, 4TO:
I) Ka}KJJ.oMY 4HCJIY HaHJJ.eTcH 06paTHoe;
2) cYllI,eCTBYIOT 4HCJIa, Y KOTOpblX HeT 06paTHoro;
3) cYllI,eCTBYIOT 4HCJIa, KOTopble HBJIHIOTCH 06paTHblMH caMH
ce6e;
4) HH OJJ.HO 4HCJIO He HBJIHeTCH 06paTHblM caMOMY ce6e?
398*. HaHJJ.H JJ.P06H, 06paTHble CJIeJJ.YIOIlI,HM:
a 3 3x 2m 3n 7x a I 1
2' -;-, 2' ""'5' 2a ' y' b' m, n, -;-' 3a '
I
I
I
I
399*. YnpOCTH Bblpa}KeHHe:
I)
(
.O,98) 2) : ( : .x)
2a (
.2.. )
3 Y 2a
1
3) 3x (7,9.3x)
4m ( 6 97..2:.. )
n ' 4m
7x ( 2...
)
2a 7x
4.1. J],eJIeHHe 06b1KHOBeHHblX .ll.p06eH
Mbl Y}Ke 3HaeM, 4TO JJ.eJIeHHe - 3TO JJ.eHCTBHe, C nOMOlll,blO KOTO-
poro no npOH3BeJJ.eHHIO H OJJ.HOMY H3 MHO}KHTeJIeH MO}KHO HaHTH
n.PyroH MHO}KHTeJIb. J],eJIeHHe - JJ.eHcTBHe, 06paTHoe YMHO}KeHHIO.
3 5
PaCCMOTpHM, HanpHMep, ypaBHeHHe 4x=""6' nOJIY4HM, 4TO
x=
.2.. M bl nOKa He MO}KeM CKa3aTb, 4eM Y P aBeH x, TaK KaK Mbl
6' 4 '
73
He 3HaeM npaBHJIa JJ.eJIeHHH 06blKHOBeHHblX JJ.p06eH. BblBeJJ.eM ero.
B JJ.aJIbHeHWeM Mbl Y3HaeM, 4TO 06e 4aCTH ypaBHeHHH MO}KHO
YMHO}KaTb Ha OJJ.HO H TO }Ke l/HCJIO, OTJIH4HOe OT HYJIH. YMHO}KHM 06e
3 5
l/aCTH ypaBHeHHH TX=6 Ha 4HCJIO, 06paTHoe JJ.aHHOMY MHO}KH_
3 4
TeJIlO 4' TO eCTb Ha 3:
(
X ) ..i...=
..i...
4 3 6 3'
npHMeHHH 3aKOHbl YMHO}KeHHH K JIeBOH l/aCTH paBeHCTBa, nOJIY4HM:
( .2....i... ) .X=
..i... HJIH I.x=
..i...
43 63' 63'
TO eCT b X =
..i... X = 5.4 =..!.Q.= I ...!...
6 3' HJIH 6.3 9 9 0
1
npOBepKa nOJJ.TBep}KJJ.aeT, 4TO Ig- KopeHb JJ.aHHOro ypaBHeHHH.
.LI.eHcTBHTeJIbHO,
3 1 3 10 5
-.1-=-.-=-
4 9 4 9 6'
npaBaH l/aCTb ypaBHeHHH TO}Ke paBHa
. TaKHM 06pa30M, nOJIY-
5 5
4HJIH BepHoe paBeHCTBO 6=6'
J.1TaK, JJ.JIH HaXO}KJJ.eHHH 4aCTHoro
:
HY}KHO HaHTH npOH3-
5 4
BeJJ.eHHe ""6'3' TO eCTb HY}KHO YMHO}KHTb JJ.eJIHMOe Ha 4HCJIO,
06paTHoe JJ.eJIHTeJIIO,
PaCCY}KJJ.aH TaKHM 06pa30M, nOJIY4HM, HanpHMep:
2-..i...=2-.
= 2.5 _
4 5 4.5 2
3' 5 3 4 3.4- 6 ' npOBepKa: ""'5'""6= 5.6 =3;
...!....
_...!....
_
_...!... 2 1 2.1 1
3' 3 - 3 2 -3.2- 2 ' npoBepKa: 3'""2= 3.2 =3'
B 061l1,eM BHJJ.e 3anHweM:
a. cad
bO{[=b'c
J.1TaK, npaBHJIO JJ.eJIeHHH 06blKHOBeHHblx JJ.p06eH MO}KHO CcpOPMY-
JIHpOBaTb CJIeJJ.YIOIlI,HM 06pa30M: flTOObl paaOellUTb 06bUCHOBeHH.glO
OP06b IUl 06blICHOBeHHg1O OP06b. Hg3ICHO OellUMoe gMH03ICUTb Ha
flUCIlO. 06paTHOe OelluTelllO.
74
nepeJJ. HenOCpeJJ.CTBeHHbIM Bbl4HCJIeHHeM JJ.P06b HY}KHO no 803-
r,lO>KHOCTH COKpaTHTbo
5 2 5 3 5.J -
-1...!.-
n pH M e p l. ""6:"3=""6'""2= #f. 2 - 4 - 4 .
ECJIH cpeJJ.H JJ.aHHblx HMelOTCH CMewaHHble 4HCJIa, TO JJ.O JJ.eJIeHHH
»x HY}KHO 06paTHTb B.HenpaBHJIbHWe JJ.P06H H TOJIbKO nOCJIe 3Toro
BbInOJIHHTb JJ.eJIeHH
. 2 5 . 5 5 3 _ :t..£f l _...!.-
n p H M e p 2. 6: 1 3=""6'"3=""6'""'5- #$. - 2 '
npaBHJIO JJ.eJIeHHH 06blKHOBeHHblX JJ.p06eH npHMeHHIOT H B CJIY-
l/ae, KorJJ.a cpeJJ.H JJ.aHHbIX HMelOTCH HaTypaJIbHble 4HCJIa. 3TH 4HCJIa
3anHCblBalOT B BHJJ.e JJ.P06H CO 3HaMeHaTeJIeM 10
3 3 6 3 1 3.1 1
n p H M e p 3. 4: 6 =-;(1=,'""6="""'4.'6=8'
n p H M e p 4. 8:
=f-: : =f-.
=
o = 13+.
JII060e 4HCJIO, KpoMe HyJIH, HMeeT 06paTHoe 4HCJIO. CJIeJJ.oBa-
reJIbHO Tenepb JJ.eJIeHHe BbInOJIHHMO 6e3 orpaHH4eHHH, 3a HCKJIIO-
, ,
l/eHHeM JJ.eJIeHHH Ha HYJIb, Ha HYJIb AenHTb HeJIb3H.
A.
400 nO BT Op H no TeKcT Y npaBuAo oeAeHuR opo6eu. Pa36epHcb B
. 3 9
npl:'Mepe H 06"bHCHH, KaK pa3JJ.eJIHTb JJ.P06b 5 Ha w.
401. BbI4HCJIH:
9. 3 2 . 1 3) 6 . 4 4) 7 .21
I) s' 4 2) s'w ""'5' 3 16'.32"
6 . 3 1 1 5 . 10 25 . 5
. "3' 9
Y' 14 3' 6 9 27
2 . 4 1. 1 6. 3 4 12
""'5' 5 g' 3 ""'5' 5 15' 35
402. Bbl4HCJIH ( Y C THO), 06"bHCHHH Ka}KJJ.bIH war JJ.eJIeHHH:
5 . 1 7 2 4) 3 . 2
...!.....2.. 2) 3) . IT'
I) 2 . 5 g' 2 JO' 9 7
1 1 . 2 1 1 5 . 1
4 . . s'
y' 3 6. 5 12' ""'5 9
5 1 3 2 8 . 1 1 .1
. g' s'T
IT' 6 W' 3 2
403. Pa36epHcb B npHMepax 2, 3 H 4 H 06"bHCHH, KaK pa3JJ.eJIHTb:
1 I 10
I) 2 4 Ha IT; 2) 15 Ha IT'
75
404. BblllHCJIH:
I ) 9
...!-.
3 . 5
2
3' 4
3 2
5: IT
405.
2) I : I
3)
5
I
3 T : 2
1
25: 11
BblllHCJIH 3Hal/eHHe Bblpa>KeHHH
1
:5
7 3
4
'1
5' 4
7
: I
3 9
9 .
16' x, eCJIH
406.
X=4- 2 1 ; x=18' x= 3
, 8'
Bbl4HCJIH 3Ha lleHHe Bblpa>KeHHH a:
, eCJIH
12 2 5
a= 35 ;a=1 25 .
PeWH ypaBHeHHe, npoBepb ero KopeHb:
I)
x=
2 ) 3 3
2 4 T X = -:;- 3)
3 I 2
5 y =I- 5 -b=l
3 2
I
t=
I _ 1
3 3 T X -"(;
407.
5y=l
3
3x=
7
2x = I
3
408.
06paTH JJ.eCHTH4Hble JJ.P06H
I )
. 0 2.
5' ,I 2) T' 0,4
o 9' 2... 0 24 .
, . 4 ' .. 9
1,2 : 2... 7 .
10 12" 0,75
409. BbI4HCJIH:
18 5
4) 1"9: 1 1"9
7 . 49
36 . 72
24: 3
5
x= 2...' x=
.
4 ' 16 '
a=O; a= I;
I
4) yZ=4
2
T a =6
5
fjy=IO
B 06bIKHOBeHHble, 3aTeM
1
3) I 4: 0,5
I
3 T : 1,25
0,36: 1 :
4) 3,5: 2 +-
4,25: I
2
0,105: I
6
I ) 30- 1 '
. 25
2
'
9
2' 10+10
2 2
5 5 -4:I T
2) ( 6
-5 2... ) .
6 8' 48
21....: ( 1
-
\
4 2 5}
1
: ( II . I I
- 8 )
6.
3) (3,6-1 +:
) '15
(3,5. ; -
): 16
: (6+,3,2-18)
410. BbI4HCJIH:
I) 66f: 40
1 4 2.... 23
8"' 24
22 1
65- : 5 13
2 ) 19:2
' 3
4' 3
51....: 2.!Q.. I Q
11 II' 16
5 1 . 7 . 5
4' . 24
3) 0.85:3
. : 0,75
1 6 . 8. 2 7
"- 9 .' -
10
4,4:3,6:
3
76
BbI4HCJIH:
5 1
411. BblllHCJIH 3Ha4eHHe Bblpa>KeHHH 19: a-b:O,7, eCJIH a=2 T
7
H b=I5'
412. PewH ypaBHeHHe:
5 3
I) 9: x=3 4
. 6 2
s.. T= 3'5
7 I 7
2) sX+T=12
1
5 2+0,6y= 16
1 11
3) 2 6 - I5x=0,15
1 1
o 25z-I-=2-
, 6 3
413.
BbI4HCJIH:
( 1 1 ) 1 2 1
1) 2+ 2 2 :7- 2 '"3+3;
( 3 3 1 ) 1 2 1
2) 4: 100 - 23 2 : I 2' "3+ I 6;
3) (8,75 - 2
) : 7, I + (4 ++ 2
).: 4 1
4.2. 3a)J.alfH Ha .ll.eneHHe .ll.p06eH
n p H M e p I. B Mew.K.e 50 Kr caxapa, HY3/CHO paCljJacoBaTb no
B naKeTbt, no 2 Kr B K.a3/CabtU. CKO/tbKO nO/ty'tuTCR TaK.UX naKeToB?
4T06bl pewHTb 3 aJJ.a4y, HY>KHO Maccy Bcero caxapa pa3JJ.eJIHTb
Ha Maccy caxapa B OJJ.HOM naKeTe. nOJIY4HM 50: 2 = 25. 3Ha4HT,
nOJIY4HTCH 25 naKeTOB.
n p H M e p 2. B RU{UK.e 30 Kr K.OHljJeT. HY3/CHO paCljJacoBaTb UX
3
B naK.eTbt, no "4 Kr B K.a3/CaOM. CK.O/tbKO nO/ty'tuTCR TaK.UX naK.eTOB?
KaK H B nepBOM npHMepe, llT06bl OTBeTHTb Ha Bonpoc 3aJJ.allH,
HY>KHO BbInOJIHHTb JJ.eJIeHHe:
30:
=
30.4
3
-40.
JtfTaK, nOJIY4HTCH 40 naKeTOB KOHcpeT. 3
n p H M e p 3. B K./tacce 18 aeBo'teK, 3TO COCTaB/tReT 5" OT 'tuc/ta
Bcex y'taU{uxcR. CK.O/tbK.O y'taU{uxcR B K./tacce?
B " 3
3TOH 3aJJ.a4e H3BeCTHO, 4TO "5 OT llHCJIa Bcex Y4alllHxCH
KJlaCCa paBHHeTcH 18. TorJJ.a MO>KHO HaHTH
OT 061l1,ero l/HCJIa Y4a-
l1I.HXCH. 3TO 6YJJ.eT B 3 pa3a MeHbwe, 4eM 18, TO eCTb 18:3=6.
5
061l1,ee 4HCJIO Y4alll,HXCH COCTaBJIHeT 5' TO eCTb B 5 pa3 60JIbWe, 4eM
1
5' 3Ha4HT, 4HCJIO Bcex Yllalll,HXCH KJIaCCa eCTb 6.5=30.
11TaK, B KJIacce 30 Y4aIll,HXcH.
77
TaKOH }Ke pe3YJIbTaT MO}KHO nOJIY4HTb TOJIbKO OJJ.HHM JJ.eHCTBHeM _
JJ.eJIeHHeM Ha JJ.P06b. LleHCTBHTeJIbHO,
18:
= 18.5 - 30.
5 3
B nOCJIeJJ.HeM npHMepe Mbl HaWJIH 4HCJIO no ero JJ.P06H. Bo061l.{e
'1To6bl HaUTU fl.UC/W no ezo opo6u, M03ICHO pa30ellUTb Ha STY op06
COOT8eTCT8YIOUlee eu fl.UCIlO
OJJ.HaKO B npOCTblX CJIY4aHX, B 4aCTHOCTH npH YCTHblX Bbll/HCJIe-
HHHX, LleJIeco06pa3HO TaKHe 3aJJ.a4H pewaTb JJ.BYMH JJ.eHCTBHHMH,
CHa4aJIa JJ.eJIeHHeM JJ.aHHoro l/HCJIa Ha 4HCJIHTeJIb JJ.P06H HaHJJ.eM
OJJ.HY 4aCTb HCKOMoro 4HCJIa. 3aTeM nOJIY4eHHoe 4HCJIO YMHO}KHM Ha
3HaMeHaTeJIb JJ.P06H. Pe3YJIbTaT JJ.BYX JJ.eHCTBHH H eCTb HCKOMoe
4HCJIO,
n p H M e p 4. HaiJ.iJeM. I.f.UCItO,
K.OTOpOi!O paBHbt 24.
C - I
HallaJIa HaHJJ.eM 3 OT HCKOMoro 4HCJIa:
24:2= 12.
3Ha4HT, HCKOMoe 4HCJIO paBHo 12.3 = 36.
A.
414.
415.
np04TH 3aJJ.a4H I H 2, 4TO B HHX 06ll1,ero? 4eM OHH OTJIH4aIOTCH?
nJIOlll,aJJ.b npHMoyrOJIbHHKa paBHa I
M 2 . Bbl4HCJIH ero JI.JIHHY,
eCJIH wHpHHa
M J.... M
M I J..- M
3 ' 2 ' 5 ' 3 .
416.
KaKOH nYTb npOHJJ.eT rpynna TYPHCTOB 3a I 4, eCJIH 2J.... KM
4
OHa npOXOJJ.HT 3a
ll 3a
4 3a J......;I
4' 5' 2 ...,
CKOJIbKO CTOHT I M TKaHH eCJIH
M CTOHT 5
p ;I
'4 5 ..
C onblTHoro Y4aCTKa, nJIOlll,aJJ.b KOToporo 3
ra, C06paJIH 225 Ll
nWeHHLlbI. CKOJIbKO LleHTHepoB nweHHLlbl B cpeJJ.HeM c06pa-
JIH C Ira?
06'beM OJJ.HOH KOMHaTbl 60 M 3 , a JJ.pyroH - B 1 + pa3a MeHbwe.
Bbl4HCJIH 06'beM JJ.pyroH KOMHaTbl.
CeMa Ha4epTHJI JJ.Ba yrJIa. OKa3aJIOCb, 4TO OJJ.HH yrOJI paBeH 130 0 ,
- 2 1
a JJ.pyroH - B 2 pa3a MeHbwe. Ha4epTH H Tbl 3TH yrJIbl.
PaCCTOHHHe Me}KJJ.Y JJ.BYMH ropOJJ.aMH 210 KM, noe3JJ. npoweJI 3TO
2
paCCToHHHe 3a 4 3 4. 06paTHO OH weJI co CKOpOCTblO
417.
418.
419.
420.
421.
78
50
K: . Bo CKOJIbKO pa3 CKOpOCTb noe3JJ.a Ha 06paTHoM nYTH
60JlbWe (MeHbwe)?
422. Ha CKOJIbKO KyCKOB MO}KHO pa3pe3aTb JIHCT 4>aHepbl, nJIOlll,aJJ.b
KOToporo 2
M 2 , eCJIH nJIOlll,aJJ.b OJJ.Horo KycKa JJ.OJI}KHa 6blTb
paBHoH 0,3 M 2 ?
423. ABTOMaT paccpacoBblBaeT npHnpaBY JJ.JIH MHCHblX 6JIIOJJ. no
1
0,03 Kr B naKeT. CKOJIbKO naKeTOB nOTpe6yeTcH JJ.JIH 22 2 Kr
npHnpaBbl?
7
424. 3a n KHJIOrpaMMoB KOHcpeT ynJIaTHJIH 2 lOP. COCTaBb Bblpa}Ke-
HHe JJ.J1H BblllHCJIeHHH CTOHMOCTH I Kr KOHcpeT, Bbl4HCJIH ero 3Ha-
3 1
4eHHe, eCJIH: I) n=""'5; 2) n= IT; 3) n= 1,2.
425. PaccMoTpH peweHHe 3aJJ.a4H 3. nOBTOpH npaBHJIO HaXO}KJJ.eHHH
O - 3
4HCJIa no ero JJ.P06H. 6'bHCHH, KaK HaHTH 4HCJIO, 4 KOToporo
paBHbl 12.
HaHJJ.H 4HCJIO (y C THO):
1
1) 3 KOToporo paBHbl 2.
3
4 KOToporo paBHbl 15,
426.
4
2) ""'5 KOToporo paBHbI 16,
2
3"" KOToporo paBHbl 1,8,
427.
0,6 KOToporo paBHbl 12;
HaHJJ.H 4HCJIO:
I) : KOToporo paBHbl 7,
0,25 KOToporo paBHbl 12,
7 3
"8 KOToporo paBHbl 2 4 ;
30H np04HTaJIa 150 cTpaHHLl.
CKOJIbKO CTpaHHLl B KHHre?
0,7 KOToporo paBHbI 1,4.
428.
2) 0,45 KOToporo paBHbl 3,6,
0,04 KOToporo paBHbl 0,5,
3
1""'5 KOToporo paBHbl 0,8.
4-
JTO COCTaBJIHeT :f BceH KHHrH.
429.
3
B KJlaCCe 12 MaJIb4HKOB. 3TO COCTaBJIHeT "8 Bcex Y4alll,HXCH
KJIacca. CKOJIbKO Y4alll,HXCH B KJlaCce?
WHpHHa naCT6HllI,a, HMelOIlI,erO cpOPMY npHMoyrOJIbHHKa, 240 M.
3TO COCTaBJIHeT
ero JJ.JIHHbI. KaKoBa nJIOlll,aJJ.b naCT6HllI,a?
8
(OTBeT Bblpa3H B reKTapax.)
nJIOlll,aJJ.b 03epa oaJIxaw BO BpeMH HH3KOH BOJJ.bl paBHa
79
430.
431.
35
17 500 KM 2 , LITO COCTaBJIHeT 44 nJIOlll,aJJ.H 03epa BO BpeMH Bbl-
COKOH BOJJ.bl. Bbl4HCJIH nJIOlll,aJJ.b 03epa OaJIXaW BO BpeMSj
BblCOKOH BOJJ.bI.
432. Macca TOBapa, ynaKoBaHHoro B HIlI,HK, 20,8 Kr, 4TO COCTaBJIHer
8
9 MaCCbl HIlI,HKa C TOBapOM. Bbl4HCJIH Maccy nYCToro HIlI,HKa.
6.
5 2
433. nepBaH aBTOMaWHHa 3a (; II npoexaJIa 60 KM, BTopaH 3a 3" l/
npoexaJIa 54 KM. CKOpOCTb KaKoH MaWHHbl 60JIbwe? Bo CKOJIbKO
pa3?
434 3 u u 3
. ""'5 BceH KOJIX03HOH 3eMJIH OTBeJJ.eHO noJJ. naWHIO. epHOBblMH
3 u C
3aHHTO ""4 3TOH nJIOlll,aJJ.H. KOJIbKO 3eMJIH HMeeT KOJIX03, eCJIH
3epHOBblMH 3aceHHO 1080 ra?
3
435. KorJJ.a LlHMa ynJIaTHJI 3a nOKynKH ""'5 Bcex CBOHX JJ.eHer, y He-
ro OCTaJIOCb elll,e 90 K. CKOJIbKO JJ.eHer 6blJIO y LlHMbl?
436. reoJIorH npOWJIH JJ.OJIHHY, pacnOJIO>KeHHYIO Me>KJJ.Y ropaMH, 3a
B u 2 u I
TPH JJ.HH. nepBblH JJ.eHb OHH npOWJIH 9' BO BTOpOH 3" Bcero
nYTH H B TpeTHH OCTaBWHeCH 28 KM, Bbl4HCJIH JJ.JIHHY nYTH, npo-
XOJJ.HllI,erO no JJ.OJIHHe.
437 B u I u 3
. nepBblH JJ.eHb TypHCTbl npOWJIH "3' a BO BTOpOH JJ.eHb T HaMe-
4eHHoro nYTH, Ha TpeTHH JJ.eHb OCTaBaJIOCb npOHTH nOCJIeJJ.HHe
10 KM, nOCTaBb pa3YMHble BonpOCbl H OTBeTb Ha HHX,
438*. (C Tap H H H a H 3 a JJ. a 4 a
J1.<1>.MarHH LlKoro l ,) Cnpo-
CHJI HeKTO Y4HTeJIH: «CKOJlbKO
y Te6H B KJlaCCe Y4eHHKOB, TaK
KaK X04Y OTJJ.aTb K Te6e B Y4eHHe
CBoero CblHa?» Y4HTeJlb OTBe-
THJI: «ECJIH npHJJ.eT Y4eHHKOB elll,e
CTOJIbKO >Ke, CKOJIbKO HMelO, H
nOJICTOJIbKO, H 4eTBepTaH 4aCTb,
H TBOH CbIH, TorJJ.a 6YJJ.eT y MeHH
100 Y4eHHKOB». CnpaWHBaeTCH:
CKOJIbKO Y4eHHKOB 6WJIO y Y4H-
TeJI H ?
I J1eoHTHii <1>HJlHnnOBHlJ MarHHLlKHii (1669-1739) - H3BecTHblii PYCCKHH npeno-
JlaBaTeJJb, aBTop 3HaMeHHToro YlJe6HHKa cApHIjJMeTHKa» (1703 r.).
80
9 COCTaBb 3aJJ.a4Y Ha HaXO>KJJ.eHHe 4HCJIa no 3aJJ.aHHoH ero JJ.po-
43 .
6H PewH ee. n r py n-
. Ul UXCH 04HIlI,aJIH KaTOK OT CHera, epBaH
40*. TpH rpynnbl Y4a n
4 7 BTo p aH .!..Toro 4TO OCTaJIOCb, a TpeTbH OCTaB-
na 04HCTHJIa \2' 3'
250 M 2 B bl4HCJIH nJlOlll,aJJ.b KaTKa.
WHeCH .
AnA nOBTOpeHHA
441. Bbl4HCJIH (y C THO):
1
4) 0,9 : 2
2-. 02
5' ,
3) 7. 2
W'
1
4:""'5
2. 9
T'T!
1,5:0,5
1
I) 0,4+2
7 2
9-3
1
0,1+3
2) 36.
4
0,5.""'5
4 3
7 4
442. Bbl4HCJIH (y C THO) :
I) + OT 4HCJIa 6,
0,8 OT 4HCJIa 5,
20% OT 4HCJIa 40;
2
2) "30T 9 p.,
0,3 OT 5 M,
10% OT 2 T
3) + OT 90 0 ,
OT 180 0 ,
6
0,5 OT 60 0 ,
443.
BbInOJIHH JJ.eHcTBHH:
I) (1,25 + I {-) : 29 +
2 ; 2)
(3++
: 15-+
H3Mepb yrJIbl TpeyrOJIbHHKa ABD
H TpeyroJIbHHKa BCD (pHC. 4.1)
CpaBHH HX rpaJJ.ycHble Mepbl H
cJJ.eJIaH BbIBOJJ.bl.
Kopnyc 3aBOJJ.a HMeeT 4>OPMY
npHMoyrOJIbHOrO napaJIJIeJIenH-
neJJ.a. .llJIHHa Kopnyca 80 M, WH- A
5
pHHa COCTaBJlHeT s: ero JJ.JIHHW
H BblCOTa 30% ero WHpHHbI. nOCTaBb
OTBeTb Ha HHX.
PMC. 4.1
pa3YMHwe BonpocW H
( 2 3 02) .2,
0,75+ 1""'5.\4-' '""'5'
( ...!-. I.!..+.!..\ .0,7 + 0,3.
2 3 3/
444
c
445.
81
4.3. HaxO>K.ll.eHHe QHCJ1a no ero npOl.leHTaM
B npeJJ.blJJ.YllI,eM naparpacpe Mbl HaXOJI.HJIH llHCJIO no ero JI.P06H.
ECJIH 3Ta JI.P06b Bblpa}KeHa B npOLleHTax, TO rOBopHT, 4TO HY>KHQ
HaHTH 4HCJIO no ero npOLleHTaM.
n p H M e p I. 3a KOHTpO/tbHYIO pa60TY no M.aTeM.aTUKe OT MeTKy
«4» nO/tY'tu/tu 9 Y'teHUKOB, 3m COCTaB/tJleT 36% OT Bcex Y'taw,UXCR
K/tacca, CKO/tbKO Y'taw,uXCJl B K/tacce?
npH peweHHH 3aJJ.allH HY}KHO HaHTH TaKoe 4HCJIO, 36% KOTO-
poro paBHbl 9, 11.JIH 3Toro Bblpa3HM npOLleHTbl 06blKHoBeHHoH HJlH
JJ.eCHTHllHOH JJ.P06blO:
36% = 1
=0,36.
LlaJIee nOCTynHM no npaBHJIY HaXO>KJJ.eHHH 4HCJIa no ero JJ.P06H,
TO eCTb pa3JJ.eJIHM Ha 3TY JJ.P06b COOTBeTCTBYlOllI,ee eH llHCJIO, nOJIY-
4HM:
. 36 9.100
9. 100 = ""36 =25, HJIH 9:0,36=25.
B KJIaCCe 6blJIO 25 Y4aIll,HxCH.
11TaK, 4T06bl HaHTH llHCJIO no ero npoLleHTaM, MO>KHO:
I) 8blpaauTb npoqeHTbl ooblICHo8eHHoii lUlU iJecRTU'l.HOii iJP06blO;
2) paaiJeJtuTb iJaHHoe 'l.uc/to Ha STY iJPOOb.
B npOCTblX CJIYllaHX, B llaCTHOCTH npH YCTHblX BbI4HCJIeHHHX,
npOLleHTbl Bblpa}KalOT 06blKHOBeHHOH JJ.p06blO, H TorJJ.a HCKOMoe llHC-
JIO MO}KHO HaiiTH JJ.BYMH JJ.eHCTBHHMH (CM. npHMep 3,
4.2) : JJ.eJIeHHeM
HaHJI.eM 1%, HJIH OJJ.HY COTYIO 4HCJIa, 3aTeM pe3YJIbTaT YMHO>KHM Ha
100 (HaHJJ.eM 100%, HJIH LleJIOe).
n p H M e p 2. HaiioeM. 'tUC/to, 5% KOmpoi!o paBHbt 15.
nOJIY4HM 15:5=3, 3.100=300. HCKoMoe llHCJIO 300.
oe3YCJIOBHO, TaKOH >Ke pe3YJIbTaT MO}KHO nOJIYllHTb H CJIeJJ.YIOIlI,HM
06pa30M:
15: I
=300, HJIH 15:0,05=300.
npH Bbl4HCJIeHHHX C HCnOJIb30BaHHeM MHKpOKaJIbKYJIHTOpa
npOLleHTbl YJJ.06HO Bblpa}KaTb JJ.eCHTH4HOH JJ.P06blO.
A.
446. npOl/TH TeKCT naparpacpa, npoJJ.YMaH npHMepbl H 061>HCHH, KaK
HaHTH 4HCJIO, eCJIH 20% ero paBHbl 60,
82
447. HaHJJ.H 4HCJIO (y C THO):
I) 2.. KOToporo paBHbl 6,
10
8 KOToporo paBHbl 24,
10
KOToporo paBHbl 12;
100
2) 2% KOToporo paBHbl 4,
8 % KOTOpOro paBHbl 32,
20% KOToporo paBHw 80,
120% KOToporo paBHbl 240.
448. HaHJJ.H 4HCJIO, eCJIH:
I) 24 % ero paBHbl 72,
42% ero paBHbl 6,3,
375% ero paBHbl 525;
3) 99% ero paBHbl 990,
210% ero paBHbl 6,3,
12,5% ero paBHbl 7,5;
2) 3,8% ero paBHbl 28,5,
52, 1% ero paBHbl 10,42,
100,7% ero paBHbl 4,028;
4) 123% ero paBHbl 2,46,
0,3 % ero paBHbl 0,27,
2500% ero paBHbl 500.
449. Macca MyKH, 3aTpa4eHHoH Ha Bbme4KY xJIe6a, COCTaBJIHeT 75%
OT 061l1,eH MaCCbl Bbme4eHHoro xJIe6a. CKOJIbKO KHJIOrpaMMoB
xJIe6a MO}KHO nOJIYllHTb H3 6 Kr, 18 Kr, 324 Kr, 3 T MYKH?
450. 8 YlleHHKoB, 4TO COCTaBJIHeT 25% Y4alll,HXCH KJlaCCa s 3a C KOHT-
pOJIbHYIO pa60TY no MaTeMaTHKe nOJIY4HJIH oTMeTKY« ». KOJIb-
KO Y4alll,HXCH B KJlaCce?
451. npH nOMOJIe nweHHLlbl nOJIY4aeTcH 80% O MYKH. CKOJIbKO unweHH;
Llbl HY}KHO CMOJIOTb, 4T06bl nOJIY4HTb 48 Kr nweHH4HOH MYKH,
452. KOJIX03 npoJJ.aJI rocYJJ.apcTBY 264,5 T MHca, 3TO COCTaBJIHeT
115% nJIaHa nOCTaBOK MHca. KaKoB nJIaH nOCTaBKH MHca Y 3TO-
ro KOJIX03a?
453. B nOCeJIKe 1650 KBapTHp HMelOT LleHTpaJIbHOe OTonJIeHHe. 3TO
COCTaBJIHeT 55% Bcex KBapTHp B nOCeJIKe. CKOJIbKO KBapTHp B
nOCeJIKe? %
454. Macca KpOBH B3pOCJIOro 4eJIOBeKa COCTaBJIHeT B cpeJJ.HeM 7,5 0
OT ero 061l1,eH MaCCbl. CKOJIbKO KpOBH Y 4eJIOBeKa, eCJIH ero
Macca 72 Kr? 100 Kr?
455. .llJIHHa npHMoyroJIbHHKa 82 CM, a wHpHHa COCTaBJIHeT 65% ero
JJ.J1HHbI. Bbl4HCJIH nJIOlll,aJJ.b H nepHMeTp npHMoyrOJlbHHKa.
456. OJJ.Ha CTopoHa TpeyrOJIbHHKa 12 JJ.M, a JJ.Be JJ.Pyrwe COCTaBJIHIOT
80% 3TOH CTOpOHbI. BblllHCJIH nepHMeTp TpeyrOJIbHHKa,
457. ABT
M06HJIb BbleXaJI H3 OJJ.Horo ropoJJ.a B JJ.pyroH. B nepBblH 4ac
OH npoexaJI 68 KM, HJIH 40% Bcero paccToHHHH Me}KJJ.Y ropoJJ.a-
MH, a BO BTOpOH 4ac 72 KM. CKOJIbKO KHJIOMeTpOB OCTaJIOCb
npoexaTb?
83
6.
458. KJIaJJ.oBIlI,HK BblJJ.aJI MaJIHpy 18% KOJIH4eCTBa BceH KpaCKH, nOCJIe
4ero Ha CKJIaJJ.e OCTaJIOCb elll,e 492 Kr KpacKH. CKOJIbKO
KHJIOrpaMMoB KpacKH BblJJ.aHO MaJIHpy?
459. 36 Y4acTHHKoB copeBHoBaHHH no JIbl}KaM CTapTOBaJIH Ha 1I.H-
CTaHLlHH 3 KM, a OCTaJIbHble 55% Y4acTHHKoB cTapTOBaJIH Ha
JJ.HCTaHLlHH 2 KM. nOCTaBb pa3YMHble Bonpocbl H OTBeTb Ha
HHX,
460. MY}K4HHbl Ha 3aBOJJ.e COCTaBJIHIOT 75% Bcero KOJIH4eCTBa
pa60 ' lHx. )KeHIlI,HH Ha 3aBOJJ.e 216. Ha CKOJIbKO }KeHIlI,HH MeHb-
we, 4eM MY}KllHH?
461. TOBap BMeCTe C ynaKoBKoH CTOHT 40,8 p, CTOHMOCTb yna-
KOBKH COCTaBJIHeT 2% CTOHMOCTH TOBapa. CKOJIbKO CTOHT
TOBap 6e3 ynaKoBKH?
462. COCTaBb 3aJJ.a4Y Ha HaXO>KJJ.eHHe 4HCJIa no ero npoLleHTaM.
PewH ee.
463*. J10CH COCTaBJIHIOT 30% OT 061l1,ero 4HCJIa KOCYJIb H JIOCeH,
>KHBYIll,HX B 3anOBeJJ.HHKe. CKOJIbKO KOCyJIb }KHBeT B 3anOBeJJ.HH-
Ke, eCJIH 4HCJIO JIOCeH Ha 144 MeHbwe 4HCJIa KOCYJIb?
464*. TynoH yroJI pa3JJ.eJIHJIH Ha TPH 4aCTH. OJJ.HH H3 06pa30BaBwHxcH
yrJIOB COCTaBJIHeT 40% Tynoro yrJIa, BTOpOH COCTaBJIHeT 20%
nepBoro yrJIa, a TpeTHH paBeH 78 0 . Bbl'lHCJIHTe BeJIH4HHY Tynoro
yrJIa.
4.4. I.JTO nOKa3b1BaeT l.faCTHOe .ll.BYX I.fHCen
np04TH BHHMaTeJIbHO cJIeJJ.YIOIlI,HH TeKCT H OTBeTb Ha BonpOCbl
K 3aJJ.aHHIO 465.
OTHOCHTeJIbHO 3Ha4eHHH 4aCTHoro JJ.BYX 4HCeJI HMelOTCH TPH
B03MO}KHOCTH:
l. Lf aCTHoe 6o/tbUle I. HanpHMep, 8: 5 = : = I
= 1,6> 1.
B 3TOM CJIY4ae 4aCTHoe nOKa3bIBaeT, BO CKOJIbKO pa3 JJ.eJIHMOe
3
60JIbWe JJ.eJIHTeJIH, 4HCJIO 8 60JIbWe 4HCJIa 5 B IT' HJIH 1,6, pa3a.
2. LfacTHoe paBHo I. HanpHMep, 10,7: 10,7= 1. B 3TOM CJIY4ae
JJ.eJIHMOe H JJ.eJIHTeJIb - paBHble 4HCJIa.
3 " IH .63 I
. ...,aCTHoe MeHbUle . anpHMep, 6.8=8=4=0,75< '
84
I
I.
3
""4
-\
PHC. 4.2
B 3TOM CJIYllae 4aCTHoe nOKa3bIBaeT, KaKYIO 4aCTb COCTaBJlHeT JJ.eJIH-
Moe OT JJ.eJIHTeJlH, HTaK, 4HCJIO 6 COCTaBJlHeT
OT 4HCJla 8 (pHC, 4.2).
4aCTHoe JJ.BYX llHCeJI 4aCTO Bblpa}KalOT B npOLleHTax, TO eCTb
B COTblX JJ.OJIHX. B paCCMOTpeHHblX npHMepax nOJIy4HM:
8:5= 1,6= :
= 160%,
10,7: 10,7= I = :: = 100%,
6:8=O,75= 1
0 =75%.
TaKHM }Ke 06pa30M Bblpa3HM B npoLleHTax llaCTHble:
5 ' 8 - 0 625 = 62,5 =625 %
. - , 100 ,0,
4,56: 120=0,038= 130
=3,8% H T. JJ..
HTaK, 4T06bl Bblpa3HTb 4aCTHoe B npoLleHTax, HY}KHO: 'I.aCTHOe
YM.H03ICUTb Ha 100 U IC no/tY'l.eHHOM.Y npou38eoeHuIO npunucaTb 3HaIC
npol{eHTa.
4acTHoe, Bblpa}KeHHOe B npoLleHTax, nOKa3bIBaeT, CKOJIbKO npoueH-
TOB COCTaBJIHeT JJ.eJIHMOe OT JJ.eJlHTeJIH, n03TOMY, 4T06bl Y3HaTb, CKOJlb-
KO npoLleHToB COCTaBJIHeT OJJ.HO llHCJIO OT JJ.pyroro, HY>KHO: 1) nep-
Boe 4HCJIO pa3JJ.eJIHTb Ha BTopoe; 2) nOJIY4eHHoe 4aCTHoe Bwpa3HTb
B npoLleHTax.
n p H M e p 1. B KJlacce 25 Y'ta14uxCJl, U3 HUX 10 iJe60'teK. CKO/tbKO
npOljeHmo6 6cex Y'ta14uxCJl KJlacca COCma6Jl5llOm iJe60'tKU?
10
Ami perneHH5I 3a,D,a'IH HY)I{HO 'IaCTHOe 25 BbIpa3HTb B npou;eHTax,
TeM caMblM Y3HaeM, CKOJIbKO npoueHTOB COCTaBJI5IeT 'IHCJIO 10 OT 'IRCJIa
25:
10 = 0,4 = 40%.
25
3Ha'IHT, ,D,eBO'IKH COCTaBJIHIOT 40% Bcex Y'IallI,HXcH.
85
n p H M e p 2. BM.eCTO nAaHOBblX 75 aeraAeii pa6ol.f.uii
TOBUA 80 aeTaJl.eii. Ha CK.OAbK.O npOl-{eHToB BbtnoAHeH nAaH?
Bblpa3HM llaCTHOe 80 B n p OLleHTax:
75
80
75
1,067= 106,7%.
3Hal/HT, pa6011HH BbmOJIHHJI nJIaH Ha 106 7%
, 0, TO eCTb nepe-
BbmOJIHHJI ero Ha 6,7% (106,7%-100% =6.7%),
U320_
A.
465. I, 4TO nOKa3bIBaeT 4acTHoe, eCJIH OHO: I) 60JIbWe I; 2) paB-
HO I; 3) MeHbwe I? npHBeJJ.H npHMepbl.
2. KaK Bblpa3HTb llaCTHoe JJ.BYX 4HCeJI B npoLleHTax? npH-
BeJJ.H npHMepbl.
3. 4TO nOKa3blBaeT llacTHoe, Bblpa}KeHHoe B npoLleHTax?
4. KaK Y3HaTb, CKOJIbKO npoLleHToB COCT8BJIHeT OJJ.HO llHCJIO
OT JJ.pyroro?
5. CKOJIbKO npoLleHToB COCTaBJIHeT 4HCJIO nHoHepoB OT l/HCJIa
Bcex Y4alll,HXCH TBoero KJIacca (npoJJ.YMaH npHMepbl I H 2)?
KaK 6YJJ.ewb BbI4HCJIHTb?
466. 061>HCHH CJIeJJ.YIOIlI,He Bbll/HCJIeHHH:
I) 6: 10=0,6=60%
9:4=2,25=225%
7:8=0,875=87,5%
3,6: 3,6= 1 = 100%
4TO nOKa3bIBaeT OTBeT?
467, CKOJIbKO npoLleHTOB COCTaBJIHeT:
I) 3 OT 5, 2) 75 OT 18,75, 3) 76 OT 47,5, 4 ) J..-OTJ..-
4 2 '
12 OT 8, 39 OT 195, 621 OT 180, I 5 13
SOT "i5'
100 OT 50; 14,7 OT 19,6; 1,6 OT 2,5: 3
""'50T 1,5?
CKOJIbKO npoLleHToB COCTaBJIHeT: I) I ra OT 5 ra' 2 ) 3
6 Kr' 3 ) 2 I ' Kr OT
, JJ.M OT M; 4) 18 0 OT pa3BepHYToro yr JIa?
CKOJIbKO npOLleHTOB COCTaBJIHeT (oTBeT oKpyrJIH JJ.O .u.ecHTblx):
2) 2 I I
3: 1 3=2"=0,5=50%
0,15:0,03=5=500%
54:72=0,75=75%
I. 1 2
6. 2 12= 25 =0,08=8%
468.
469.
86
1 2
1) 5 OT 6, 2) 32 OT 290, 3) 3,3 OT 29, 4) TOT 3'
4
3 OT 7, 19 OT 157, 5,3 OT 495, 6 OT 1""'5'
5
2 OT 9; 89 OT 645; 0,8 OT 7,4; "'80T 1,2?
470. 1-13 25 KOHTpOJIbHblX pa60T 6 OLleHHJIH Ha «5». CKOJIbKO npoLleH-
TOB COCTaBJIHIOT 3TH pa60Tbl OT 06rnero 4HCJIa pa60T?
471. 1-13 32 Y4alll,HXCSl KJIaCCa no 60JIe3HH OTCYTcTBOBaJIH 4 YllaIll,HX-
CH. CKOJIbKO npOLleHTOB Bcex Y4alll,HXCH KJIaCCa OTCYTCTBOBaJIH
H CKOJIbKO npOLleHTOB npHcYTcTBOBaJIH?
472. B JIHTeHHOM Llexe H3rOTOBHJIH 320 JJ.eTaJIeH, 8 H3 HHX OKa3aJIHCb
C JJ.ecpeKTaMH CKOJIbKO npOLleHTOB JJ.eTaJIeH C JJ.
cpeKTaMH?
473. 1-13 40 wTpacpHblx 6pOCKOB, CJJ.eJIaHHbIX 6aCKeT60JIbHOH KOMaH-
JJ.OH, 6blJIO 36 nonaJJ.aHHH. Bbll/HCJIH npOLleHT nonaJJ.aHHH.
474. Mara3HH 3a JJ.eHb npoJJ.aJI 28 4epHO-6eJIb1X H 42 LlBeTHblX
TeJIeBH30pa. CKOJIbKO npoLleHToB COCTaBJIHeT l/HCJIO npoJJ.aHHblx
LlBeTHblX TeJIeBH30poB OT 061l1,ero KOJIH4eCTBa npoJJ.aHHblx
TeJIeBH30poB?
475. nJIOrnaJJ.b npHMoyrOJIbHHKa 14 JJ.M 2 , ero JJ.JIHHa 5 JJ.M. CKOJIb-
KO npoLleHToB COCTaBJIHeT wHpHHa npHMoyrOJIbHHKa OT ero
JJ.J1HHbl?
476. H3 215 TbIC. py6JIeH, BblJJ.eJIeHHblX JJ.JIH KanHTaJIbHOrO peMoHTa
3JJ.aHHH, B nepBoM KBapTaJIe H3pacxoJJ.oBaHo 42 TWC. py6JIeH.
CKOJIbKO npoLleHToB COCTaBJIHeT 3Ta cYMMa OT cpeJJ.cTB, Bbl-
JJ.eJIeHHbIX JJ.JIH peMoHTa? (OTBeT oKpyrJIH JJ.O JJ.ecHTblx.)
6.
477. no pe3YJIbTaTaM nepenHcH HaCeJIeHHH B 1979 r. B fpY3HH-
CKOH CCP npO}KHBaJIO B ropOJJ.ax 2 60 I 000 4eJIOBeK, a B CeJIb-
CKOH MeCTHOCTH 2 414 000. B TO }Ke BpeMH B PC<1>CP npo-
}KHBaJIO B ropoJJ.ax 95 374 000 4eJIOBeK, a B CeJIbCKOH MeCTHOCTH
42 177 000. B J1HTOBCKOH CCP B ropoJJ.ax npO}KHBaJIO 2 062 000
H B CeJIbCKOH MeCTHOCTH I 336 000 4eJIOBeK. C nOMOlll,blO npo-
LleHTOB cpaBHH 4HCJIO ropoJJ.cKHX H CeJIbCKHX }KHTeJIeH B JJ.aHHbIX
pecny6JIHKax.
478. COCTaBb 3aJJ.a4Y, B KOTOpOH HaJJ.O HaHTH, CKOJIbKO npoueHToB
COCTaBJIfieT OJJ.HO l/HCJIO OT JJ.pyroro, PewH ee.
479*. nepBoe 4HCJIO COCTaBJIHeT 80% OT BToporo. CKOJIbKO npOLleHTOB
COCTaBJIHeT BTopoe 4HCJIO OT nepBoro?
87
480. B TypHHpe no 6aCKeT60JIY npH-
HHMaJIH Y4acTHe KOMaHJJ.bl nHTblX
H weCTblX KJIaCCOB, Jtb 45
wTpa4>Hblx 6pOCKOB KOMaHJJ.bl nH-
TblX KJlaCCOB H H3 56 wTpa4>Hblx
6POCKOB KOMaHJJ.bl weCTblX KJIaC-
COB 6blJIO COOTBeTCTBeHHO 34 H
40 nonaJJ.aHHH. Pe3YJIbTaT wTpa4>-
HblX 6pOCKOB KaKoH KOMaHJJ.bl
6blJI J1Y4we?
4.5. H3MeHeHHe BeJ1HlfHHbI B npOl.leHTax
MHome BeJIH4HHbl nOCTOHHHO MeHHIOTCH, HanpHMep TeMnepa-
Typa B03JJ.yxa, ypOBeHb BOJJ.bl B Mope, e>KeMeCH4Hoe KOJIH4eCTBO
ocaJJ.KOB, 061l1,aH CYMMa JJ.eHer B c6epKacce, pOCT H Macca 4eJIOBeKa
H T, JJ.. BpeMH OT BpeMeHH TaK>Ke MeHHIOTCH LleHbl Ha 'rOBapbl, 3apnJIa-
Ta pa604HX H JJ.P,
H3MeHeHHH BeJIH4HH 4aCTO xapaKTepH3YlOTCH C nOMOlll,blO npoueH-
TOB.
n pH Me p I. 0.0 CHUJlCeHUR l{eH XOAOiJUAbHUK. CTOUA 250 p.,
nOCAe CHUJlCeHUR 230 p. Ha CK.OAbK.O npOl{eHTOB CHU3UAaCb
CTOUMOCTb xOAoduAbHUK.a?
D.JIH peweHHH 3aJJ.al/H Y3HaeM CHa4aJIa, Ha CKOJIbKO py6JIeH
H3MeHHJIaCb LleHa XOJIOJJ.HJIbHHKa. nOJIYl/HM 250 - 230 = 20 (p,). Te-
nepb HaHJJ.eM. CKOJIbKO npoLleHToB COCTaBJIHeT nOJIY4eHHaH pa3HOCTb
OT nepBOHa4aJIbHOH CTOHMOCTH XOJIOJJ.HJIbHHKa:
- 008 - 8%
250-' - 0'
3Ha4HT, CTOHMOCTb XOJIOJJ.HJIbHHKa nOHH3HJIaCb Ha 8%.
n p H Me p 2. KOAX03 nAaHupoBaA nOAYI.f.UTb C 1 ra B cpeiJHeM
29 Ll 3epHOBblX. a nOAY'iUA 32 Ll. Ha CK.OAbK.O npOl{eHTOB K.OA-
X03 nepeBbLnOAHUA nAaH? Ha CK.OAbK.O npOl{eHTOB BbLnOAHUA nAaH?
CHa4aJIa onpeJJ.eJIHM, Ha CKOJlbKO ueHTHepOB KOJIX03 nepeBbInOJI-
HHJI nJIaH: 32-29=3 (u). naJIee Bbll/HCJIHM, CKOJIbKO npOLleHTOB
3
COCTaBJIHeT 3Ta pa3HOCTb OT nJIaHa: 29
0, I 03 = 10,3 %. 3Ha-
4HT, nJIaH nepeBbInOJIHeH Ha 10.3%, a nJIaH BbInOJIHeH Ha 110,3%
000% + 10,3% = 110,3%).
88
HTaK, 4T06bl Y3HaTb, Ha CKOJIbKO npoLleHToB YBeJIH4HJIaCb HJIH
yMeHbWHJIaCb JJ.aHHaH BeJIH4HHa, He06xoJJ.HMO HaHTH:
I) Ha CICOAbICO eiJuHuq g8eAU'IU/laCb UAU gMeHbUJ.UAacb 3Ta
8eAU'l.UHa;
2) CICOAbICO npoqeHT08 COCTa811Jl.eT nOAg'l.eHHOJl pasHocTb OT
nep80Ha'l.aAbHOZo SHa'l.eHUR 8eAU'l.UHN.
A.
481. npOJJ.YMaH npHMepbl I H 2 H HTor naparpa4>a. 06"bHCHH, KaK
HaHTH U3MeHeHue BeAUI.f.UHbl B npol{eHTax. eCJIH OHa YBeJIH4H-
JIaCb C 16 JJ.O 24.
482. Bblpa3H (y C THO) B npOLleHTax H3MeHeHHe BeJIH4HHbl:
3) OT 30 JJ.O 36,
OT 70 JJ.O 77,
OT 50 JJ.O 150.
483.
I) OT 4 JJ.O 6, 2) OT 100 JI.O 120,
OT 10 JJ.O 20, OT 200 JJ.O 160,
OT 10 JJ.O 8; OT 40 JJ.O 48;
Bblpa3H B npoLleHTax H3MeHeHHe:
I) OT I p. JJ.O 90 K" 2) OT 25 Ll JJ.O 3 T,
OT 800 CM 3 JJ.O 1,6 JJ.M 3 , OT 4 M 2 JJ.O 450 JJ.M 2 ,
OT 4000 Kr JJ.O 5 T; OT I 4 JJ.O 30 MHH.
Bblpa3H B npoLleHTax H3MeHeHHe BeJIH4HHbl:
1) OT 4500 JJ.O 5400, 2) OT 4,8 JJ.O 3,6,
OT 240 JJ.O 180, OT 6 JJ.O 4,5,
OT 3400 JJ.O 1700; OT 15,4 JJ.O 38,5,
npH 06pa60TKe JJ.eTaJIH Macca ee YMeHbWHJIaCb C
216 Kr. Ha CKOJIbKO npoLleHToB YMeHbWHJIaCb Macca
UeHa Ha KOBep CHH3HJIaCb C 450 JJ.O 369 p. Ha
npoLleHToB CHH3HJIaCb LleHa Ha KOBep?
nJIOlll,aJJ.b napHHKoB B COBX03e YBeJIH4HJIH 3a roJJ. C 2,5 JJ.O
3,6 ra. Ha CKOJIbKO npOLleHTOB YBeJIH4HJIH nJIOlll,aJJ.b napHHKOB?
B CBH3H C nOCTpOHKOH HOBoro MHKpopaHOHa MapwpYT aBTO-
6yca yJJ.JIHHHJICH Ha 18%. KaKaH JJ.JIHHa HOBoro Mapwpy-
Ta, eCJIH JJ.J1HHa npe>KHerO MapwpYTa 20 KM? v
no nJIaHY pa604HH JJ.OJl>KeH 6blJI H3rOTOBHTb 128 JJ.eTaJIeH B
JJ.eHb, OH H3rOTOBHJI 160 JJ.eTaJIeH. Ha CKOJIbKO npoueHToB
pa604HH BWnOJIHHJI nJIaH?
484.
485.
270 JJ.O
JJ.eTaJIH?
CKOJIbKO
486.
487.
488.
489.
6.
490.
no nJIaHY OJJ.HH pa604HH JJ.OJI>KeH 6blJI H3rOTOBHTb 60 JJ.eTaJIeH, HO
OH nepeBbInOJIHHJI nJIaH Ha 5%. D.pyroH pa604HH JJ.OJI>KeH 6blJI
89
H3rOTOBHTb 40 JJ.eTaJIeH, a H3rOTOBHJI 43 JJ.eTaJIH. nOCTaBb
pa3YMHble Bonpocbl H OTBeTb Ha HHX.
491. 110 CHH>KeHHH LleH TOBap CTOHJI 120 p. BblllHCJIH LleHY TOBapa
nOCJIe JJ.BYX nOCJIeJJ.OBaTeJIbHbIX CHH>KeHHH, eCJIH nepBoe CHH-
>KeHHe 6blJIO Ha 10%, a BTopoe Ha 5%.
492*. BblcoTa COCHbl B Ha4aJIe rOJJ.a 1,8 M, a K KOHLlY rOJI.a
OHa YBeJIH4HJIaCb Ha 15%. KaKoBa 6YJJ.eT JJ.J1HHa COCHbl 4epe3
TPH rOJJ.a, eCJIH ee npHpoCT 3a Ka>KJJ.bIH CJIeJJ.YIOIlI,HH rOl{
COCTaBJIHeT 90% npHpOCTa npeJJ.blJJ.YllI,erO rOJJ.a?
493*. CTOPOHY KBaJJ.paTa YBeJIH4HJIH Ha 20%. Ha CKOJlbKO npo-
LleHTOB YBeJIHl/HJIaCb nJIOlll,aJJ.b KBaJJ.paTa? nepHMeTp KB3JI.-
paTa?
4.6. COBMecTHbie .ll.eHCTBHH C .ll.eCHTHlfHblMH
H 06b1KHOBeHHblMH .ll.P06HMH
B HeKOTopblX 3aJJ.a4ax l/HCJIOBble JJ.aHHble MorYT 6blTb Bblpa>KeHbl
H 06bIKHoBeHHbIMH, H JJ.eCHTH4HbIMH JJ.P06HMH, H HaTypaJIbHblMH 4HC-
JIaMH. B 3TOM CJIY4ae BCTaeT Bonpoc, KaK nOCTynHTb: 06paTHTb 06b1K-
HOBeHHble JJ.P06H B JJ.eCHTH4Hble HJIH, Ha060poT, JJ.eCHTH4Hble B
06bIKHOBeHHble? CJIeJJ.yeT HMeTb B BHJJ.Y, 4TO B >KH3HH JJ.eCHTHll-
Hble JJ.P06H HCnOJIb3YIOTCH llalll,e, 4eM 06bIKHOBeHHble. Ho He Bcem3
06blKHOBeHHYIO JJ.P06b MO>KHO 3aMeHHTb KOHe4HOH JI.eCHTH4HOH
(CM.
3.1-3.2), a 3aJJ.a4a Tpe6yeT T04Horo OTBeTa, TorJJ.a 3aJJ.a4Y
npHXOJJ.HTCH pewaTb B 06blKHoBeHHblx JJ.P06HX,
n p H M e p I, Bbl4HCJIHM 3Ha4eHHe Bblpa>KeHHH
( : ++ ).2-3,8,0,35,
4 1
TaK KaK ""'5=0,8 H "4=0,25, TO nOJIY4HM:
(0,8+0,25).2-3,8'0,35= 1,05.2-3,8.0,35=2,1-1,33=0,77.
n p H M e p 2. Bbl4HCJIHM 3Ha4eHHe Bblpa>KeHHH
8,5-(
+
).3,8.
2
TaK KaK JJ.P06b 3 HeJIb3H 3aMeHHTb KOHe4HOH JJ.ecHTHllHOH
JJ.P06blO, TO JJ.eCHTH4Hble JJ.P06H 3aMeHHM 06bIKHOBeHHbIMH:
85 17 38 19
8,5=10="'2 H 3,8=10=5"'
90
11aJlee nOJIYllHM:
17 16+9 19 _ 17
2- 24 '5"- 2
25.19 _ 17 95 _ 204-95 _ 109 - 4
24.5 "'2- 24 - 24 24 24'
A.
494.
BHHMaTeJIbHO np04HTaH TeKCT, paCCMOTpH npHMepbI. KorJJ.a
LleJIeco06pa3HO pewaTb 3aJJ.a4H B JJ.eCHTH4HbIX JJ.P06HX. a KorJJ.a
B 06bIKHOBeHHbIX?
Bbl4HCJlH (y C THO):
I) 1,8+ 12,2 2) 2100-95 3)
709+ 131 5,19-4,09
17 5 19 7
24 + 24 36 - 36
495.
4) 3500:5
1,8:0,03
1
2: 6
40. 102
0,01.6,5
2 5
5"'s
1
10+0,5
...!....-03
2 '
1
-.001
3 '
1.
6. 0 ,2
496. Bbl4HCJIH 3Ha4eHHe Bblpa>KeHHH:
I) 16112:53+26928:33 2) (10 1
-8
.
1 ( 11 5
12
304.16-43.19 36 : 12-97 +13
( 7 12 ) 1
2,88: 15+0,11.894 18. 1 7+3 .""'5
2:
+ 1,8.
( 0,26 - do ).3 ;
497. Bbl4HCJIH 3Ha4eHHe Bblpa>KeHHH 0,2x-+, eCJIH x=4; x=2+;
x = 6,25
1 1
498. 4HCJIO 4,8 YBeJIH4b: I) Ha 3 6 ; 2) B 2"4 pa3a.
499. 4HCJIO 5 1 7 2 YMeHbwH: 1) Ha 4,5; 2) B 2,4 pa3a.
500.
BbmOJIHH JJ.eHCTBHH: 1
2- 3 4 1
I) 0,08 + 3,93 . 2) ; 3) .-0,12'12
0,02 0,3' 4
.0 6 1-
6 ' 3
KynHJIH 2+ Kr BHHorpaJJ.a no LleHe 0,8 p. 3a KHJIOrpaMM
H ap6Y3, KOTOpblH CTOHJI 1+ p, CKOJIbKO 3anJIaTHJIH 3a no-
501.
KynKY?
502. OJIH KynHJIa 2,5 M WeJIKOBOH JIeHTbl KpacHoro LlBeTa, 3anJIa-
THB 3a Hee 3 p" H 3+ M JIeHTbl 3eJIeHOrO LlBeTa, 3anJIaTHB
91
3a Hee 2,56 p. Ha CKOJIbKO py6JIeH I M 3eJIeHOH JIeHTb)
JJ.eweBJIe I M KpacHoH?
503. CKOpOCTb aBTOM06HJIH 75 K'I M . KaKoe paccToHHHe npoeJJ.eT
aBTOM06HJIb: I) 3a 2,5 4; 2) 3a : 4; 3) 3a 50 MHH?
504. Bbl4HCJIH CKOpOCTb aBTOM06HJIH, eCJIH 24 KM OH npoexaJJ:
2
I) 3a 3'" 4; 2) 3a 0,75 4; 3) 3a 36 MHH.
505. BepTOJIeT cJJ.eJIaJI nepBYIO nocaJJ.KY Ha paccToHHHH 57,5 KM
OT B3JIeTHOH nJIO lllaJJ.KH , npOJIeTeB 1
Bcero nYTH. Bbll/HCJJH
AnHHY Bcero nOJIeTa.
506. CTOpOHbl npHMoyroJIbHHKa + M H 0,3 M, HaHJJ.H CTOPOHY
KBaJJ.paTa, nepHMeTp KOToporo paBeH nepHMeTpy JJ.aHHoro
npHMoyrOJIbHHKa. Bbl4HCJIH ero nJIOlllaJJ.b.
6.
507. Bbl4HCJIH 3Ha4eHHe BbIpa}KeHHH:
I) 27 (1828-999)+(987 + 16 353):85;
2) 4,3-3,5+ 1,44:3,6+3,6: 1,44.(0,1-0,02);
( 7 5 ) 1 1 1 1
3) 8-12 .11T-T+ 3 6":33'";
3. 4 3 5 I
4 ) -.12 + 35.- + -.-- 0 2 5 '-.
5' '5 46 ' '2'
5) 51
-22,68:0,45+(
+
) '1 2
;
6) (13,75.2-rr--7,75):( 4
: 1++ 14 : -8
-) ;
7) (5 :: -(1
+2.6 )).(4,2-2,76).2 1 1 2 ;
8) (15 1
-38,902:( 10 2 1 5 -2,7 )):( 1 7 2 + :6 :2*).
508. BbI4HCJIH 3Ha4eHHe Bblpa}KeHHH 47;-a+8:b, eCJIH a=O,75
H b=
7 .
509. Bbl4HCJIH 3Ha4eHHe Bbl p a}KeHu,-, (3a-0,74).a I ) 0 5
nn 3 2 ,eCJIH: a=,;
a+ 5 .1 3
2
2 ) a=-
3 .
92
510. KaToK npHMoyrOJIbHOH <t>OPMbl, JJ.JIHHa KOToporo 82 M H WHPH-
Ha 55 M, HY}KHO 04HCTHTb OT CHera. CKOJIbKO TOHH CHera
1
HY}KHO BbIBe3TH, eCJIH TOJIIlI,HHa CHe}KHOrO nOKpOBa ""'5 M H
I M 3 CHera BeCHT 125 Kr?
511. Tpoe pa604Hx 3apa60TaJIH BMeCTe 500 p. nepBblH nOJIy4HJI
Bcex JJ.eHer, BTOpOH - B 1,75 pa3a 60JIbWe nepBoro.
4
nOCTaBb pa3YMHble BonpOCbl H OTBeTb Ha HHX,
512. PaccToHHHe Me}KJJ.Y JJ.ByMH npHCTaHHMH 45 KM. KaTep npoweJI
5 I 1 0 u
- 3Toro pacCTOHHHH 3a _ 3 4. CTaJIbHOH nYTb KaTep npo-
12
KM n
llIeJI co CKOpOCTblO 21 -. OCTaBb pa3YMHble Bonpocbl H
'I
OTBeTb Ha HHX.
513*. KOTOpbIH ceH4ac 4ac, eCJIH:
1) npoweJJ.llIaH 4aCTb CYTOK Ha 3,5 l/ JJ.JIHHHee oCTaBweHcH;
2) npoweJJ.waH 4aCTb CYTOK Ha 6+ 4 Kop04e OCTaBweHCH?
AJIH nOBTOpeHHH
514. Ha30BH (pHC. 4.3): I) Bce JIY4H; 2) Bce oCTpble yrJIbl; 3) Bce
Tynble yr JIb!.
515. H3Mepb yrJIbl (pHC. 4,4) AOD, DOC, BOC H AOB, CpaB-
HH HX rpaJJ.ycHble Mepb!. HaHJJ.H CYMMY Bcex 3THX yrJIOB.
516. Bbl4HCJIH (y C THO):
I) 40% OT 180, 2)
1
3'" OT 33,
0,7 OT 5;
90% OT 6,
3
T OT 24,
0,2 OT 0,8;
3) 15% OT 200,
5
"9 OT 90,
0,24 OT 10.
c
A
PHC. 4.3
PHC. 4.4
93
HaHJJ.H 4HCJIO, eCJIH:
I) 32% ero paBHbl 256, 2) 7,4% ero paBHbl 0,111,
785
12 ero paBHbl 63, 9"" ero paBHbl 'T;
0,19 ero paBHbl 0,76; 0,45 ero paBHbl 2,7.
CKOJIbKO npoLleHToB COCTaBJIHeT OJJ.HO 4HCJIO OT JJ.
yroro:
4
2) 1""'5 OT 9,
5 2
45 OT 40; "'60T IT; 8,1 OT 5,4?
Bbl4HCJIH nJIOlll,aJJ.b 3eMeJIbHOrO Y4acTKa, eCJIH BcnaxaHHble 45 ra
COCTaBJIHIOT 90% Bcero 3eMeJIbHOrO Y4aCTKa.
B KOJIX03e noJJ. MOpKOBb 3aHHJIH 1,8 ra. ;:ho Ha
we, 4eM npeJJ.ycMaTpHBaJIOCb no nJIaHY. CKOJIbKO
noJJ. MOpKOBb no nJIaHY?
BbI4HCJIH:
517.
518.
I) 7 OT 28,
519.
520.
521.
3) 0,15 OT 3,
15% 60JIb-
OTBOJJ.HJIOCb
( 2 3 ) 2 ;
8 2 ; 0,72;
34 2 ; 5,82;
(1ft
4.7. OKPY>KHOCTb H Kpyr
Ha pHcYHKe 4.5 H306pa}KeH H3BeCTHblH Te6e 4epTe}KHblH HHCT-
PYMeHT - LlHPKYJIb. Ha KOHLle OJJ.HOH ero HO}KKH oCTpHe, a Ha KOHLle
JJ.PyroH - rpH<t>eJIb. ECJIH YCTaHoBHTb HO}KKY C oCTpHeM Ha 6YMary,
TO JJ.PyraH HO}KKa npH Bpalll,eHHH 6YJJ.eT onHCbl-
BaTb OKPY>KHOCTb (pHC. 4.6). T04Ka, B KOTOpOH
HaXOJJ.HTCH oCTpHe, Ha3bIBaeTCH LleHTpOM OKPY}K-
HOCTH. UeHTp OKPY}KHOCTH 06bl4HO 0603Ha4alOT
6YKBOH O. Bce T04KH OKPY>KHOCTH Jle>KaT B 0.lI.-
HOM nJlOCKOCTH H HaXO.lI.HTCH Ha O.lI.HHaKOBOM
paCCTOHHHH OT LleHTpa.
PHC. 4.5
94
PHC. 4.6
PHc.4.7
CoeJJ.HHHM OTpe3KOM LleHTp OKPY}K-
IIOCTH C npOH3BOJIbHOH T04KOH A 3TOH
OKPY}KHOCTH (pHC. 4.7). nOJIY4eHHblH
OTpe30K AO, a TaK}Ke ero JJ.JIHHa Ha3bl-
BalO TCH pa.lI.HycoM OKPY}KHOCTH. 0603Ha- A
qHM paJJ.HYC OKPY}KHOCTH 6YKBOH r. Bce
paJJ.HYCbl OJJ.HOH OKPY}KHOCTH paBHbl
Me}KJJ.Y C060H. npOBeJJ.eM 4epe3 LleHTp
OKPY}KHOCTH OTpe30K, coeJJ.HHHIOIlI,HH
jXBe T04KH OKPY}KHOCTH (pHC. 4.8). no-
J1Yl/eHHbIH oTpe30K AB, a TaK}Ke JJ.JIHHa 3Toro oTpe3Ka Ha3bIBaeTCH
.lI.HaMeTpoM OKPY}KHOCTH. 0603Ha4HM JJ.HaMeTp OKPy}KHOCTH 6YK-
BOH d. lI.HaMeTp B JJ.Ba pa3a JJ.JIHHHee paJJ.Hyca. C nOMOlll,blO <t>OPMYJIbl
3TO MO}KHO 3anHcaTb TaK:
B
PMC. 4.8
d=2r.
OKPY}KHOCTb JJ.eJIHT nJIOCKOCTb Ha JJ.Be 4aCTH, 4aCTb nJIOCKOCTH
BHYTPH OKPY}KHOCTH BMeCTe C OKPY}KHOCTblO 06pa3yeT Kpyr (pHC. 4.9).
lleHTp, paJJ.HYc, JJ.HaMeTp OKPY}KHOCTH HBJIHIOTCH oJJ.HoBpeMeHHo
QeHTpoM, paJJ.HycoM, JJ.HaMeTpoM KPyra. PaccToHHHe OT LleHTpa JJ.O
J11060H T04KH Kpyra He npeBblwaeT paJJ.Hyca.
A.
522. npoBepb no TeKcTY, nOMHHWb JIH Tbl:
I) KaK pacnOJIaralOTCH T04KH Ha OKPY}KHOCTH OTHOCHTeJIbHO
LleHTpa;
2) 4TO TaKoe paiJuyc H iJuaM.eTp OKPY}KHOCTH, KaKHM COOT-
HoweHHeM OHH CBH3aHbl Me}KJJ.Y C060H;
3) 4eM OTJIH4aeTCH OKPY}KHOCTb OT Kpyra.
0"" \J 'i)K H ° c 1"6
H
-B
PHC. 4.9
PMC. 4.10
95
523. Ha4epTH JII06yIO OKPY}KHOCTb. H3Mepb ee paJJ.HYc H JJ.HaMeTp,
524. HaHJJ.H JJ.HaMeTp OKPY}KHOCTH, eCJIH ee paJJ.HYc 4 CM; 6,8 CM;
1 2
4,2 JJ.M; 7,1 JJ.M; 0,6 M; 30,84 M; ""4 M; ""'5 M.
525. HaHJJ.H paJJ.HYc OKPY}KHOCTH, eCJIH ee JJ.HaMeTp 6,8 CM; 18 CM;
3 7 1
5,3 JJ.M; ""4 JJ.M; I M; 210 M; 42,6 M; 6 2 M.
526. Ha4epTH OKPY>KHOCTb, paJJ.HYc KOTOpOH 5,2 CM. 0603Ha4b LleHTp
OKPY}KHOCTH H npOBeJJ.H JJ.Ba ee JJ.HaMeTpa.
527. Ha30BH T04KH, KOTopble npHHaJJ.Jle}KaT: I) OKPY}KHOCTH; 2) KPY-
ry; 3) He npHHaJJ.JIe}KaT Kpyry (pHC. 4.10),
6.
528. Ha4epTH OKPY}KHOCTb, paJJ.HYc KOTOpOH 3,4 CM. npOBeJJ.H JJ.Ba ee
paJJ.Hyca, KOTopble 06pa3YIOT npHMOH yroJl.
529. Ha4epTH OKPY}KHOCTb. OTMeTb Ha OKPY}KHOCTH T04KH A, B H C
TaK, 4T06bl OTpe30K AB OKa3aJICH JJ.HaMeTpOM OKPY}KHOCTH,
Ha4epTH OTpe3KH AC H BC. H3Mepb yrOJI ACE.
530. UeHTp OKPY}KHOCTH - TOl/Ka 0, JJ.JIHHa ee paJJ.Hyca r. rJJ.e
pacnOJIO}KeHa T04Ka P, eCJIH OP<r, OP=r, OP>r?
531 *. Ha4epTH OKPY}KHOCTb H npHMoyrOJIbHHK TaK, 4T06bl Y OKPY}K-
HOCTH H Y CTOpOH npHMoyrOJIbHHKa 6blJIa: I) OJJ.Ha 061llaH
T04Ka; 2) JJ.Be 061l1,He T04KH; 3) TPH 061llHe T04KH; 4) 4eTblpe
061llHe T04KH.
4.8. Jl.J1HHa OKPY>KHOCTH
3 a JJ. a 4 a. LlBeTO'iHaR K/lYM.6a UM.eeT l/JopM.Y Kpyza, auaM.eTp
KOTOpOZO 5 M, 3TY K/lYM.6y HYJ/CHO o6HecTu aepHOM.. KaKou a/lUHbl
nO/locy aepHa HYJ/CHO nOaZOTOBUTb, eC/lU a/lUHY nO/lOCKU C'iUTaTb no
BHYTpeHHeM.Y KpUlO?
l..lT06bI pewHTb 3TY 3aJJ.a4Y, He06xoJJ.HMO HaHTH JJ.JIHHY OKPY}K-
HOCTH,
MO}KHO nonp060BaTb CJJ.eJIaTb 3TO HenocpeJJ.cTBeHHbIM H3Mepe-
HHeM, HanpHMep, C nOMOlll,blO «waralOllI,ero» LlHPKYJIH (pHC, 4.11).
Ho nOJIY4eHHblH pe3Y.'1bTaT, KOHe4HO, HeT04HbIH, nOToMY 4TO JJ.JIHHY
OKPY}KHOCTH Mbl 3aMeHHJIH JJ.JIHHOH JIOMaHOH JIHHHH.
npoBeJJ.eM TaKoH onblT. B03bMeM KOHcepBHYIO 6aHKY. ECJIH «onOH-
caTb» 6aHKY HHTKOH, a 3aTeM pacnpHMHTb ee, TO JJ.JIHHa HHT-
96
I
PHC. 4.11
PMC. 4.12
KH 6YJJ.eT npH6JIH3HTeJIbHO paBHa JJ.JIHHe OKPY}KHOCTH 6aHKH, l..lT06bl
nOJIY4HTb 60JIee T04HblH pe3YJIbTaT, HY}KHO HHTKOH «onOHcaTb» 6aHKY
HeCKOJlbKO pa3, a 3aTeM JJ.JlHHY BceH HHTH pa3JJ.eJIHTb Ha KOJIH-
4eCTBO «onOHCblBaIOIlI,HX» KpyrOB. Tenepb H3MepHM JJ.HaMeTp 6aH-
KH JIHHeHKOH HJIH JJ.BYMH yrOJIbHHKaMH (pHC, 4.12). nOCMOTpHM,
BO CKOJIbKO pa3 JJ.JIHHa OKPY}KHOCTH 6aHKH 60JIbWe ee JJ.HaMeTpa.
LlJIH 3Toro pa3JJ.eJIHM JJ.JIHHY OKPY}KHOCTH Ha JJ.HaMeTp. ECJIH H3MepeHHH
CJJ.eJIaHbl TllI,aTeJIbHO, TO 4aCTHoe OT JJ.eJIeHHH 6YJJ.eT B npeJJ.eJIax
OT 3,1 ... JJ.O 3,2. T04Hble MaTeMaTH4eCKHe paccY}KJJ.eHHH (c HHMH
Tbl n03HaKOMHWbCH B CTapWHX KJlaCcax) nOKa3bIBalOT, 4TO JJ.JIHHa
JlI060H OKPY}KHOCTH 60JlbWe CBoero JJ.HaMeTpa B OJJ.HO H TO }Ke 4HCJIO
pa3. 3TO 4HCJIO 0603Ha4aeTCH rpe4eCKOH 6YKBOH 31 (nH). Ee T04Hoe
3Ha4eHHe Bblpa}KaeTCH 6eCKOHe4HOH JJ.eCHTH4HOH JJ.P06blO:
31=3,14159265... ,
B JJ.aJIbHeHWeM MbI 6YJJ.eM nOJIb30BaTbCH npH6JIH}KeHHbIM 3Ha4e-
HHer,j 31
3, 14.
ECJlH 0603Ha4HTb llJIHHy OKPy}KHOCTH 6YKBOH C, a ee JJ.HaMeTp d, TO
nOJIY4HM <t>OPMYJIY JJ.JIH Bbl4HCJIeHHH JJ.JlHHbl OKPY}KHOCTH:
C=31d.
/lUHa OKpgJ/CHOCTU pa8Ha npOU38eaeHUW
UC/la 31 Ha aU
Tp
OKpgJ/CHOCTU.
Mbl 3HaeM, 4TO d:= 2r, n03TOMY <t>OPMYJIY JJ.JIH Bbll/HCJIeHHH
JJ.JlHHbI OKPY}KHOCTH MO}KHO 3anHcaTb B BHJJ.e:
C=231r.
97
Tenepb MO}KHO peWHTb 3aJJ.a4Y, npHBeJJ.eHHYIO B Ha4aJIe naparpa-
<t>a.
TaK KaK JJ.HaMeTp KJlYM6bl paBeH 5 M, TO 3anHWeM:
C=11.5
3,14.5= 15,7 (M).
3Ha4HT, HY}KHO nOJJ.rOTOBHTb nOJIOcKY JJ.epHa JJ.JIHHOH npHMepHQ
15,7 M.
Ha MHorHX MHKpOKaJIbKYJIHTOpaX JJ.JIH 4HCJIa 11 eCTb OTJJ.e
bHaH
KJlaBHwa. ECJIH Ha}KaTb Ha Hee, TO Ha 3KpaHe nOHBHTCH npH6JIH-
}KeHHOe 3Ha4eHHe 11, 06bl4HO C CeMblO .lI.eCHTH4HblMH 3HaKaMH:
11
3,1415926,
n p H M e p. ECJIH JJ.HaMeTp OKPY}KHOCTH 3,8 M, TO Ha M'1KpO-
KaJIbKYJIHTOpe JJ.J1HHy OKPY}KHOCTH HaHJJ.eM no CJIeJJ.YlOllI,eH cxeMe:
00 3 [J8 [;]
Ha 3KpaHe BblCBeTHTCH OTBeT 11,938051, KOTUpbIH MO}KHO oKpyr-
JIHTb, HanpHMep, JJ.O JJ.eCHTbIX (11,9 M), JJ.O COTblX (11,94 M) H T. JJ..
A.
532. nepeCKa}KH TeKCT naparpa<t>a C nOMOlllblO BonpocoB:
I) n04eMY JJ.JIHHY OKPY}KHOCTH HeJIb3R H3MepHTb JIHHeHKoH",)
2) KaK MO}KHO H3MepHTb JJ.J1HHy OKPY}KHOCTH KaKOrO-JIH60 npeJJ.-
MeTa, HanpHMep CTaKaHa?
3) 4TO 0603Ha4aeTCH 6YKBOH 11?
4) KaK Bbl4HCJIHeTCH JJ.JIHHa OKPY}KHOCTH?
533. H3Mepb JJ.OMa JJ.HaMeTpbI H JJ.JIHHbl OKPY}KHOCTeH JJ.BYX pa3-
JIH4HblX CTaKaHOB. Bbl4HCJIH 4aCTHble OT JJ.eJIeHHH JJ.JIHH OKPY}K-
HocTeH Ha cooTBeTcTBYlOlllHe JJ.Haw.eTpbI. (OTBeTbI oKpyrJIH JJ.O
JJ.eCHTbIX. )
534. Bbl4HCJIH JJ.JIHHY OKPY}KHOCTH, eCJIH JJ.HaMeTp ee 5,8 CM; 42 CM;
9,7 JJ.M; 0,54 M; 43 M.
535. Bbll/HCJIH JJ.JIHHY OKPY>KHOCTH, ec-
JIH paJJ.HYc ee 6 CM; 8,5 CM;
0,8 JJ.M; 4,6 JJ.M; 0,35 M; 24 M.
536. H3Mepb JJ.HaMeTp OKPY}KHOCTH,
H306pa}KeHHOH Ha pHcYHKe 4.12
H Bbl4HCJIH ee JJ.JIHHY.
537. YqeHHKH opraHH30BarrH copeB-
HOBaHHe no qmrypHoMY KaTaHHIO
Ha BeJIOCHnep;ax. B 3THX copeBHO-
98
BaHHHX HY}KHO 6blJIO npoexaTb 4 Kpyra no OKPY}KHOCTH pa-
JJ.HyCOM 3 M. KaKoe paccToHHHe npOeXaJIH BeJIOCHneJJ.HCTbl B
3TOM BHJJ.e <t>HrYPHoro KaTaHHH? (OTBeT oKpyrJIH JJ.O eJJ.HHHIl.)
538. lI.JIH o<t>opMJIeHHH 3aJIa K HOBOMY roJJ.Y AHH Bblpe3aJIa H3
LlBeTHoH 6YMarH KPY}KKH paJJ.HycoM 8 CM. KaKOH «nYTb»
HY}KHO 6bIJIO npOJJ.eJIaTb HO}KHHLlaMH, 4T06bI BbIpe3aTb 30 TaKHX
KPY}KKOB? (OTBeT oKpyrJIH JJ.O eJJ.HHHLl.)
6.
539. 4eMY paBeH JJ.HaMeTp OKPY}KHOCTH, eCJIH JJ.JIHHa ee 13,4 CM?
540. 4eMY paBeH paJJ.HYc OKPY}KHOCTH, eCJIH JJ.JIHHa ee 56,32 M?
54 1, CpaBHH nepHMeTp KBaJJ.paTa, CTopoHa KOToporo 9 CM, C JJ.JIH-
HOH OKPY}KHOCTH, JJ.HaMeTp KOTOpOH 9 CM.
542. H3 npHMoyrOJIbHHKa BbIpe3aH
nOJIYKpyr (pHC. 4,13), BbInOJIHH
He06xoJJ.HMble H3MepeHHH H Bbl-
l/HCJIH nepHMeTp oCTaBweHcH
<t>HrYPbI.
543. B TeXHH4eCKOM KpY}KKe AHJJ.peH
CKOHCTpYHpOBaJI MOJJ.eJIb caMO-
JIeTa, Bo BpeMH HCnblTaHHH MO-
JJ.eJIb npOJIeTeJIa 3a 1,8 C nOJIHbIH Kpyr paJJ.Hyca 20 M, KaKoBa
CKOpOCTb nOJIeTa ero MOJJ.eJIH?
544. npH Bbl4HCJIeHHH JJ.JIHHbI OKPY}KHOCTH B lI.peBHeM BaBHJIOHe
3a 11 4aCTO npHHHMaJIH l/HCJIO, paBHoe TpeM. npeJJ.nOJIO}KHM,
4TO BaBHJIOHHHHH BbI4HCJIHJI JJ.JIHHY OKPY}KHOCTH, paJJ.HYc KOTO-
pOH 40 M. Ha CKOJIbKO OTJIH4aeTCH ero OTBeT OT OTBeTa
Yl/eHHKa 6 KJIaCCa (11
3,14)?
545*. LlHaMeTp OKPY}KHOCTH YBeJIH4HJIH Ha CM, Ha CKOJIbKO YBeJIH-
4HJIaCb JJ.JIHHa OKPY}KHOCTH?
PHC. 4.13
4.9. nJ1oUJ,a.ll.b Kpyra
3 a JJ. a 4 a. Ha nOAy U2pOBOU KOM.HaTbt HYJ/CHO 3aKpaCUTb KPY2,
paiJuyc KOTOp020 2 M, CKOAbKO nOTpe6yeTcR iJAR 3T020 KpacKu,
eCAU Ha KaJ/CiJbtU KBaiJpaTHbtU M.eTp pacxoiJyeTcR 0,2 Kr KpacKu?
4T06bl pewHTb 3TY 3aJJ.a4Y, HY}KHO HaHTH nJIOmaJJ.b Kpyra. KaK
3TO cJJ.eJIaTb?
CTpOfHMH paccY}KJJ.eHHHMH JJ.OKa3aHO, LITO nJIOlllaJJ.b Kpyra B 11
99
= tr .
.'
PHC. 4.14
pa3 60JlbWe nJIOlll,aJJ.H KBaJJ.paTa, CTOpOHa KOTOpOrO paBHa paJJ.HYCY
Kpyra. ECJIH 0603Ha4HM paJJ.HYC Kpyra 6YKBOH " TO nJIO].UaJJ.b KBaJJ.-
paTa, CTOpOHa KOTOpOrO paBHa paJJ.HYCy Kpyra, paBHa ,2. CJIeJJ.OBa-
TeJIbHO, nJIOlll,aJJ.b Kpyra Bbl4HCJIHeTCH no <popMYJIe (pHC. 4.14)
S =1t,2,
'boobt 8bt'lUC/lUTb n/lOUfaiJb ICpyza, HY3ICHO 1C8aiJpaT paiJuyca
YMH03ICUTb Ha 1t,
C nOMOlll,blO 3Toro npaBHJIa MO}KeM pewHTb 3aJJ.a4Y, C<POPMYJIH-
pOBaHHylO B Ha4aJIe naparpa<pa. HaHJJ.eM no <popMYJIe nJIOlll,aJJ.b
Kpyra, KOTOpblH HY}KHO 3aKpacHTb:
S=3,14.4=12,56 (M 2 ).
TaK KaK Ha Ka}KJJ.bIH KBaJJ.paTHbIH MeTp pacxoJJ.yeTcH 0,2 Kr
KpaCKH, TO Bcero HY}KHO KpacKH 0,2. 12,56
2,5 (Kr).
l..h06bl nOJIb30BaTbCH MHKpOKaJIbKYJIHTOpOM JJ.JIH Bbl4HCJIeHHH
nJIOwaJJ.H Kpyra, He06xoJJ.HMO YMeTb HaXOJJ.HTb KBaJJ.paT 4HCJIa,
TO eCTb npOH3BeJJ.eHHe l/HCJIa caMO Ha ce6H. 3TO 04eHb npoCTO:
BBOJJ.HM 4HCJIO, Ha}KHMaeM Ha KJIaBHWY yMHO}KeHHH H Ha KJIaBHWY
OTBeTa. Ha 3KpaHe nOHBHTCH KBaJJ.paT 3Toro 4HCJIa.
n pH Me p l. Bbl4HCJIHM 4 2 .
CxeMa: 4
I I
n p H M e p 2. Bbl4HCJIHM nJIOlll,aJJ.b Kpyra, paJJ.HYC KOToporo
3,7 CM.
CxeMa: 3 D 7
G
G G
Ha 3KpaHe nOJIY4HM 4HCJIO 43,008402, KOTopoe npH He06xo-
JJ.HMOCTH MO}KHO oKpyrJIHTb, HanpHMep, JJ.O JJ.eCHTbIX, (43,0 CM 2 ), JJ.O
COTblX (43,01 CM 2 ) H T. JJ..
100
HCTOpHlJeCKHe CBe.ll.eHHH
EllI,e B JJ.aJIeKOM npOWJIOM JIIOJJ.HM 6bIJIH H3BeCTHbI MHorHe
reoMeTpH4ecKHe <PHrYPbl, B TOM 4HCJIe OKPY}KHOCTb H Kpyr. 06
3TOM cBHJJ.eTeJIbCTBYIOT pacKonKH, rJJ.e 6blJIH HaHJJ.eHbI pa3JIH4HbIe
YKpaweHHH, nocyJJ.a, OCTaTKH JJ.peBHHx coopY}KeHHH. 3Ha4HT, elll,e
TorJJ.a npHxoJJ.HJIOCb pewaTb 3aJJ.al/H Ha Bbl4HCJIeHHe JJ.JIHHbI OKPY}K-
HOCTH H nJIOwaJJ.H Kpyra, CeHl/aC H3BeCTHO, LITO 3Ha4eHHeM l/HCJIa 1t
B pa3HbIe BpeMeHa CJlY}KHJIH pa3J1H4HbIe 4HCJIa. TaK, B lI.peBHeM
ErHnTe (OKOJIO 3500 JIeT Ha3aJJ.) 3a 3Ha4eHHe 4HCJIa 1t npHHHMa-
JIOCb 3,16, a JJ.peBHHe pHMJIHHe C4HTaJIH, LITO 1t = 3,12, Bce 3TH
3Ha4eHHH 1t 6blJIH onpeJJ.eJIeHbl onbITHblM nYTeM. BeJIHKHH Y4eHblH
lI.peBHeH rpeLlHH ApxHMeJJ. (287-212 rr. JJ.O H, 3.) on peJJ.eJIHJI , 4TO
3Hal/eHHe l/HCJIa 1t HaXOJJ.HTCH B n p eJJ.eJIax 3
< 1t < 3 ..!- HJlH
71 7 '
3,1408... < 1t < 3,1428,.. , npH nOMOIll,H COBpeMeHHbIX 3JIeKTpOHHO-
Bbl4HCJIHTeJIbHblX MaWHH JJ.JIH 1t Bbl4HCJIeHO KOJIH4eCTBO JJ.eCHTHl/HbIX
3HaKOB CBblwe MHJIJIHOHa.
lI.JlH 0603Ha4eHHH l/aCTHOro OT JJ.eJIeHHH JJ.JIHHbI OKPY}KHOCTH
Ha JJ.HaMeTp BnepBble HCnOJIb30BaJl 6YKBY 1t aHrJIHikKHH MaTeMa-
THK lI.}KOHC B 1706 r., HO 06wenpHHHTblM 3TO 0603Ha4eHHe CTaJIO
6JIaroJJ.apH pa60TaM BeJIHKOrO MaTeMaTHKa J1. 3HJIepa (1707-1783),
l/JIeHa neTep6yprcKoH aKaJJ.eMHH HaYK. OH Bbl4HCJIHJI JJ.JIH 4HCJIa 1t
153 JJ.eCHTHl/HbIX 3HaKa.
A.
546. np04HTaH TeKCT naparpa<pa, BblJJ.eJIH OCHOBHoe. KaK CBH3aHbl
Me}KJJ.Y C060H nJIOlll,aJJ.b Kpyra H nJIOwaJJ.b KBaJJ.paTa?
547. Bbl4HCJIH nJlOlll,aJJ.b Kpyra, eCJIH ero paJJ.HYC 6 CM; 4,2 CM;
12 JJ.M; 3.4 JJ.M; 0,5 M; 23 M.
548. Bbll/HCJIH nJIOwaJJ.b Kpyra, eCJIH
ero JJ.HaMeTp 18 CM; 3,6 JJ.M;
15 JJ.M; 2 M; 8,5 M.
549. H3Mepb paJJ.HYc Kpyra, H306pa-
}KeHHoro Ha pHcYHKe 4.10, BbI-
4HCJIH ero nJIOlll,aJJ.b H JJ.JIHHY
OKPY}KHOCTH.
550. BOKpyr <pJIarWTOKa pa36HJIH nJIO-
llI,aJJ.KY, HMelOllI,YIO <POp MY Kpyra,
paJJ.HYC KOTOpOro 3,5 M, Bbll/HCJIH
101
PHC. 4.15
PHC. 4.16
PHC. 4.17
nJIOlll,aJJ.b 3TOH nJIOlllaJJ.KH. (OTBeT oKpyrJIH JJ.O JJ.ecHTblx.)
551. npH peCTaBpaLlHH TpeX3Ta}KHOH Kpyr JIOH 6awHH Bce nOJIbI
nOKpblJIH KaMeHHblMH nJIHTKaMH. CKOJIbKO KBaJJ.paTHblx MeTpoB
nJIHTKH nOTpe60BaJIOCb, eCJIH BHYTpeHHHH JJ.HaMeTp 6awHH
6,4 M? (OTBeT oKpyrJIH JJ.O eJJ.HHHLl.)
6.
552. PaJJ.HYc OJJ.HOH OKPY}KHOCTH, H306pa}KeHHOH Ha pHcYHKe 4.15,
12 CM, a paJJ.HYc JJ.pyroH - Ha 3 CM 60JIbwe. BbI4HCJIH nJIOlll,aJJ.b
KOJIbLla, 06pa30BaHHoro 3THMH OKPY}KHOCTHMH.
553. B napKe pa36HJIH 5 LlBeT04HbIX KJIYM6, HMelOllI,HX <t>OPMY Kpyra
JJ.HaMeTpoM 1,5 M. Ha Ka}KJJ.bIH KBaJJ.paTHbIH MeTp npeJJ.noJIa-
raeTCH BblcaJJ.HTb no 9 LlBeTKOB, C<t>OPMYJIHPYH Bonpoc K
3aJJ.a4e. PewH ee.
554*, Ha Ka}KJJ.oH CTopoHe KBaJJ.paTa nOCTpoeH nOJIYKpyr (pHC, 4.16).
CJJ.eJIaH He06xoJJ.HMble H3MepeHHH H Bbl4HCJIH nJIOlll,aJJ.b nOJIY-
4HBweHCH <t>HrYPbI.
555*. CTopoHa KBaJJ.paTa, H306pa}KeHHOrO Ha pHcYHKe 4.17, paB-
Ha 4 CM. 4eMY paBeH paJJ.HYC Kpyra? Bo CKOJIbKO pa3 nJIO-
maJJ.b KBaJJ.paTa 60JIbWe nJIOlll,aJJ.H Kpyra? (OTBeT oKpyr JIH JJ.O
COTbIX.)
J].JIH nOBTOpeHHH
556. BbI4HCJIH H nOCTpoH nOJIY4eHHble yrJIbl:
I) 30% OT 1800; 2) 60% OT 700; 3) 45% OT 160 0 .
102
557.
BbIl/HCJIH (oTBeT oKpyrJIH JJ.O eJJ.HHHLl):
I) 74% OT 130°, 2) 92% OT 90°,
6% OT 78 0 , 8,4% OT 180 0
-.!. OT 1600; 0,24 OT 880;
9
Ha4epTH yrJIbl, paBHble 35° H 80°, TaK, 4T06bI BepwHHbI
yrJIOB COBnaJJ.aJIH H OJJ.Ha cTopoHa 3THX yrJIOB 6
JIa 6w 06-
llI,eH. (PaccMoTpH JJ.Ba CJIY4aH,)
B 6 KJIaCCe 3a KOHTpOJIbHYIO pa60TY no MaTeMaTHKe oLleHKY
«5» nOJIY4HJIH 25% Y4alll,HXCH KJIaCCa, oLleHKY «4» - 35%,
oLleHKY «3» - 30%, a OLleHKY «2» - 10% Bcex Y4aIll,HXcH.
nOCTpOH cToJI64aTYlO JJ.HarpaMMY pacnpeJJ.eJIeHHH OLleHOK.
BbI4HCJIH:
521
I) 0,75: 6 +2,5'""'5- 1 : 1 9 ;
2 ) (( 4
- 3 .!.! ) .
+ 1
( 3..!....- 22- )) : 3..!....,
12 24 7 17 18 12 3
3HaMeHHTbIH JJ.peBHerpe4eCKHH MaTeMaTHK ApXHMeJJ. onpeJJ.eJIHJI,
4TO 3
<Jt<3..!...., CpaBHH JJ.JIHHbl OKPY}KHOCTH, 3aMeHHH n
71 7
4HCJIaMH 3
H 3 T' eCJIH paJJ.HYc OKPY}KHOCTH 497 CM-.
3) 54% OT 165°,
2,3% OT 136°.
OT 36 0 .
558.
559.
560.
561.
4.10. KpyroBOH ceKTOp
npOBeJJ.eM JJ.Ba paJJ.Hyca OA H OB Kpyra, H306pa>KeHHOrO
Ha pHcYHKe 4.18. TaKHM 06pa30M Mbl pa3JJ.eJIHM Kpyr Ha JJ.Be 4aCTH,
Ka}KJJ.aH H3 KOTOpblX Ha3bIBaeTCH KpyroBbiM ceKTopoM. Ha pHcYHKe
ceKTopbI 3awTpHxoBaHbI no-pa3HoMY. YroJI AOB 3awTpHxoBaHHoro
ceKTopa - 3TO OCTpblH yroJI. YroJI JJ.pyroro ceKTopa 60JIbWe pa3-
BepHYToro yrJIa. 3Hal/HT, rpaJJ.ycHaH Mepa 3TOro yrJIa 60JIbWe 180°.
B
PHC. 4.18
PHC. 4.19
103
Ha pHcYHKe 4.19 Kpyr pa3JJ.eJIeH Ha l/eTbl_
pe ceKTopa, yroJI Ka>KJJ.oro H3 HHX paBel-I
B 90°. CYMMa 3THX yrJIOB 06pa3yeT nOJlHblii
yroJl. TaK KaK 90°+90°+90°+90°=
= 360°, TO MO}KeM 3aKJlIOl/HTb, 4TO rpa-
AYCHaH Mepa (BeJlHl/HHa) nOJlHOrO yrJla
paBHa 360°.
ECJIH Kpyr pa3JJ.eJIeH Ha JJ.Ba ceKTOpa
H yroJI OJJ.Horo ceKTopa H3BeCTeH, TO MO)K-
HO Bbl4HCJIHTb yrOJI JJ.pyroro ceKTopa, lI.JIH
3Toro HY}KHO H3 BeJIH4HHbl nOJIHOrO yrJIa
BbI4eCTb BeJIH4HHY H3BecTHoro yrJIa. Ha-
npHMep, eCJIH OCTpbIH yrOJI AGB paBeH 60° (pHC. 4.18), TO yrOJl
JJ.pyroro ceKTopa pa BeH 360° - 60° = 300°.
C
PHC. 4.20
A.
562.
np04TH TeKCT H:
1) nOKa}KH Ha pHcYHKe 4.18 KPYZOBOii ceKTOp Hero yroJI'
2) CKa)KH, 4eMY paBHa rpaJJ.ycHaH Mepa nOJIHOrO yrJIa:
H3Mepb yrJIbl AGB H BGe (pHC. 4.20) H Bbll/HCJIH B
yrJIa AGe. eJIH4HHY
Pa3JJ.eJIH Kpyr Ha ceKTopbI TaK, 4T06bI yrJIbl Tpex ceKTopoB
6blJIH paBHbl 80 0 , 50 0 H 90 0 . 3aKpaCb 3TH TPH ceKTopa pa3-
HblMH LlBeTaMH. 4eMY paBeH yrOJI He3aKpaweHHoro ceKTopa?
Kpyr pa36HJIH Ha TPH ceKTopa. YrOJI OJJ.Horo ceKTopa paBeH
132 0 , BToporo 59 0 . Bbl4HCJIH yrOJI TpeTbero ceKTopa.
Bbl4HCJIH yroJI ceKTopa, eCJIH OH COCTaBJlHeT OT nOJIHOrO
yrJIa: I) ; ; 2) f-; 3) [52 ; 4) 70%; 5) 40,5%; 6) 27,3%.
(OTBeTbl, Bblpa}KeHHble LI.p06HbIMH 4HCJIaMH, oKpyrJIH JJ.O ueJIoro
4HCJIa rpaLI.ycoB,)
Pa3JJ.eJIH Kpyr Ha ceKTopbl, yrJIbl KOTOpblX COCTaBJIHIOT 20%
30% H 42,5% nOJIHOrO yrJIa. 3aKpacb Ka}KJJ.bIH H3 3THX ceKT
:
pOB pa3HblMH LlBeTaMH. 4eMY paBeH yroJI He3aKpaweHHoro
ceKTopa?
563.
564.
565.
566.
567.
6.
568.
Kpyr pa3JJ.eJIeH Ha TPH ceKTopa, YroJl OJJ.Horo ceKTopa paBeH
48 0 , a yrJIbI JJ.BYX JJ.pymx ceKTOpoB paBHbI Me)KJJ.Y C060H. Bbl-
4HCJIH 3TH yrJIbl.
104
569. Kpyr pa3JJ.eJIeH Ha TPH ceKTopa. YroJI OJJ.Horo ceKTopa COCTaB-
1
JIHeT 3' a BToporo 40% nOJIHorO yrJIa. BbI4HCJIH yrJIbl Bcex
3THX ceKTopoB.
570. Kpyr pa3JJ.eJIeH Ha TPH ceKTopa, YroJI OJJ.Horo ceKTopa COCTaB-
2
JIHeT 30% nOJIHOrO yrJIa, a BToporo ""'5 yrJIa nepBoro ceKTopa.
Bbl4HCJIH yrJIbl 3THX ceKTopoB.
4.1 1. KpyroBaH .ll.HarpaMMa
B 4eTBepToM KJlaCCe Mbl HaY4HJIHCb CTpOHTb JIHHeHHble H CTOJI6-
4aTbIe JJ.HarpaMMbl. Pa3JIH4Hble 4HCJIOBble JJ.aHHble 6bIBaeT YJJ.06HO
H306pa}KaTb C nOMOlllblO KpyroBblx ceKTopOB.
n p H M e p. H 3 27 y'-taU{uxcR. K/laCCa 9 y'-taU{uxcR. 3a KOH.T-
pO/lbH.YIO pa60TY nO/ly'-tu/lU Ol.{eH.KY «5», 15 y'-taU{uxcR. - «4» u 3
y'-taU{uxcR. - «3».
Ope}KJJ.e Bcero HaHJJ.eM, KaKYIO 4aCTb Bcex Y4alll,HXCH COCTaBJIHIOT
Te, KOTopble nOJIY4HJIH oLleHKY «5», «4» H «3» cOOTBeTcTBeHHo.
9 1 15 5
OLleHKY «5» nOJIY4HJIH '27=3' oueHKY «4» 27 =9 H OLleHKY «3»
2..=
Y4alll,HXCH, CYMMa nOJIY4eHHblx JJ.p06eH JJ.OJI>KHa 6bITb paB-
27 9
1 5 1
Ha l. npoBepHM: 3+9+ 9 = l.
OYCTb Kpyr H306pa}KaeT 4HCJIO Bcex Y4alll,HXCH KJIaCca. Pa3JJ.e-
1 5 1
JIHM ero Ha ceKTopbl, COOTBeTCTBYlOlllHe JJ.P06HM 3' 9 H 9. TOrJJ.a
KOJIH4eCTBO Yl/alllHXCH, nOJIY4HBWHX oLleHKY «5», 6YJJ.eT H306pa-
1
}KaTb ceKTOp, yrOJI KOToporo paBeH 120° (360°. 3= 120°), oueH-
KY «4» _ yroJI, paBHbIH 200 0 (360 0 .
= 200 0 ), H oLleHKY «3»-
1
yroJI, paBHblH 40 0 (360 0 . 9= 40 0 ), CYMMa rpaJJ.ycHblX Mep 3THX
yrJIOB JJ.OJI}KHa paBHHTbCH nOJIHoMY yrJIY. OpoBepHM: 120 0 +2000 +
+ 400 = 360 0 . Tenepb C nOMOlll,blO TpaHcnopTHpa nOCTpOHM JJ.Ba
ceKTopa, yr JIbl KOTOpblX 120 0 H 40 0 . Y rOJI TpeTbero ceKTopa JJ.OJI}KeH
paBHHTbCH 200 0 . CeKTopbl 3aKpacHM pa3HblMH UBeTaMH HJIH 3a-
wTpHxyeM no-pa3HoMY, TaKHM 06pa30M Mbl nbJIY4HM KpyroBYIO
AHarpaMMY pacnpeJJ.eJIeHHH oueHOK KOHTpOJIbHOH pa60Tbl (pHC. 4,21).
105
..4"
PHC. 4.21
PHC. 4.22
A.
571.
np04TH BHHMaTeJIbHO TeKCT naparpa<t>a. OnHWH BCe 3Tanbl no-
CTpOeHHH KPYZOBOii iJuazpaMMbl (Ha4HH C YCJIOBHH 3aJJ.a4H),
KpyroB
H JJ.HarpaMMa (pHC. 4,22) nOKa3bIBaeT pacnpeJJ.eJIeHHe
JJ.HeBHOH HOpMbl nHTaHHH 4eJIOBeKa. KaK pacnpeJJ.eJIHeTCH
JJ.HeBHaH HopMa nHTaHHH 4eJIOBeKa?
nOCTpOH KpyroBYIO JJ.HarpaMMY, H306pa}Ka101lI,YIO KOJIHl/eCTBO
Manb4HKOB H JJ.eBOl/eK TBoero KJIaCCa,
Cywa 3aHHMaeT 29 % a M u
0, HpOBOH OKeaH 71 % nOBepxHocTH
3eMJIH, B ceBepHoM nOJIywapHH cywa 3aHHMaeT 39 % a MH-
po u 61 % 0,
BOH OKeaH 0; B IO}KHOM }Ke nOJIywapHH cywa - 19%
MHpOBOH OKeaH 81 %. nOCTpOH KpyroBYIO JJ.HarpaMMY pacnp
JJ.eJIeHHH CYWH H MHpoBoro OKeaHa Ha nOBepxHoCTH 3eMJIH:
I) Ha BceM 3eMHOM wape; 2) B ceBepHoM nOJI y wa p HH" 3 ) B IO}K-
HOM '
nonywapHH. (BeJIHl/HHbI yrJIOB nOJIY4eHHbIx ceKTopoB
oKpyrJIH JJ.O LleJIOrO l/HCJIa rpaJJ.ycoB,)
572.
573.
574.
6.
575. B KpY}KKe IOHblX TeXHHKOB 50 Y4aIllHXCH: 5 H3 HHX KOHCTPYH-
pOBaJIH MOJJ.eJIH paKeTbI, 10 - MOJJ.eJIH CaMOJIeTa, 20 CTpOHJIH
KapTHHrH, 15 - KaTepa. nOCTpOH KpyroBYIO JJ.HarpaMMY pac-
npeJJ.eJIeHHH l/JIeHOB KpY}KKa IOHblX TeXHHKOB.
576. MHpOBOH OKeaH COCTOHT H3 4eTbIpex OKeaHOB: THxoro, nJIO-
llI,aJJ..b KOTo
oro 179,7; ATJIaHTH4ecKoro, nJIOlllaLI.b KOToporo
93,4, HHJJ.HHcKoro, nJIOlll,aJJ.b KOToporo 749 H C
106 ' , eBepHoro
JIeJJ.OBHToro, nJIOlll,aJJ.b KOToporo 13,1 MJIH. KM 2 . nOCTpOH
KpyroBYIO JJ.HarpaMMY pacnpeJJ.eJIeHHH nJIOlll,aJJ.H MHpoBoro
OKeaHa,
4.12. UJap
<1>YT60JIbHbIH MH4, TeHHHCHblH MHl/, HJJ.PO, TOJIKaeMOe Ha copeB-
HOBaHHHX no JIerKOH aTJIeTHKe, H JJ.pyrwe npeJJ.MeTbl (pHC. 4.23) JJ.alOT
HaM npeJJ.CTaBJIeHHe 0 reOMeTpHl/eCKOH <t>Hrype, KOTopaH Ha3bIBaeTCH
wapOM. nOBepXHOCTb wapa - KpHBaH nOBepXHOCTb, OHa Ha3bIBaeTCH
c4IepOH. Bce T04KH c<t>epbI yJJ.aJIeHbl OT OJJ.HOH onpeJJ.eJIeHHOH T04KH,
KOTopaH Ha3bIBaeTCH LleHTpoM mapa HJIH LleHTpoM c4Iepbl, Ha OJJ.HHa-
KOBoe paccTOHHHe. OTpe30K (a TaK}Ke ero JJ.JIHHa), KOTOpbIH coeJJ.H-
HHeT JII06ylO T04KY c<t>epbI C LleHTpoM wapa, Ha3bIBaeTCH pa,IJ.HycoM
mapa (c<t>epbI). 06bIl/HO paJJ.HYc wapa (c<t>epbI) 0603Ha4alOT 6YKBOH
R (pHC. 4.24),
nYCTb wap CJJ.eJIaH H3 JJ.epeBa. ECJIH MbI paCnHJIHM 3TOT wap
Ha JJ.Be 4aCTH (pHC. 4.25), TO YBHJJ.HM, 4TO ce4eHHe wapa HMeeT
<t>OPMY Kpyra. Jho6oe CeqeHHe mapa nnOCKOCTblO eCTb KPyr. PaJJ.HYc
3Toro Kpyra TeM MeHbwe, l/eM JJ.aJIbWe OT LleHTpa wapa HaXOJJ.HTCH
ceKYlllaH nJIOCKOCTb (pHC. 4.26). Kpyr C HaH60JIbWHM paJJ.HycoM
I \
PHC. 4.23
107
PHC 4.24
PHC. 4.25
nOJIYl/aeTCH, eCJIH paCCel/b wap nJIOCKOCTblO, npOXOJJ.HllI,eH 4epe3
ero LleHTp, B 3TOM CJIY4ae paJJ.HYC Kpyra 6YJJ.eT paBeH paJJ.HYCY wapa
1I.HaMeTp :noro Kpyra Ha3bIBaeTCH H JJ.HaMeTpOM wapa (ccpepbI). P.u
aMeTp mapa !a8eH yiJ8oeHHOMY paiJuycy. ECJIH 0603Ha4HTb JJ.HaMeTp
wapa 6YKBOH D H paJJ.HYC 6YKBOH R, TO nOJIYl/HM CPOPMYJIY
D=2R.
3eMJIH, Ha KOTOpOH MbI mHBeM, npH6JIHmeHHO HMeeT CPOPMY
wapa. n03TOMY ee Ha3blBalOT H 3eMHbIM wapOM. MOJJ.eJIblO 3eMHoro
wapa HBJIHeTCH rJI06yc (pHC. 4.27),
A.
577. np04TH TeKCT H:
I) Ha30BH HeKoTopbIe npeltMeTbl, HMelOlllHe CPOPMY mapa'
2) CKamH, LITO TaKoe cl/Jepa, l.{eHTp wapa H paouyc i oua-
MeTp) wapa.
PHC. 4.26
108
PHC. 4.27
578. BbIl/HCJIH JJ.HaMeTp wapa, eCJIH ero paJJ.HYc paBeH:
I) 42 CM; 2) 5,8 JJ.M; 3) 0.25 M; 4) 6378 KM.
579. Bbl4HCJIH paJJ.HYc wapa, eCJIH ero JJ.HaMeTp paBeH:
I) 2,6 CM; 2) 3,04 JJ.M; 3) 1,7 M; 4) 1540 KM.
580. B Llexe H3rOTOBHJIH JJ.Be JJ.eTaJIH, HMelOllI,He CPOPMY wapa:
OJJ.Ha paJJ.HycoM 4,5 CM, a JJ.PyraH JJ.HaMeTpoM 1,2 JJ.M. 06e
JJ.eTaJIH nOKpbIJIH OJJ.HHaKOBbIM CJIOeM KpacKH. .LJ.,nH KaKOH
JJ.eTaJIH nOTpe60BaJIOCb 60JIbWe KpacKH? n04eMY?
581, B Mara3HH npHBe3JIH JJ.Ba OJJ.HHaKOBbIX HIllHKa C pa3HblMH MH-
4aMH, B OJJ.HOM HIllHKe 6blJIH TeHHHCHble MHl/H, a B JJ.pyroM -
MH4H JJ.JIH HaCTOJIbHOrO TeHHHca. B KaKOM HIllHKe 6blJIO 60JIbWe
MHl/eH? nOl/eMY?
582. 1I.HaMeTp 3eMJIH npH6JIH3HTeJIbHO paBeH 12 800 KM, a JJ.HaMeTp
J1YHbl 3480 KM. Bo CKOJIbKO pa3 JJ.HaMeTp J1YHbI MeHbwe' JJ.Ha-
MeTpa 3eMJIH? (OTBeT oKpyrJIH JJ.O JJ.ecHTblx.)
6.
583. KaKYIO CPOPMY HMeeT JII060e Cel/eHHe wapa nJIOCKOcTblO?
584. TpH ce4eHHH wapa nJIOCKOCTHMH HMelOT nJIOlll,aJJ.H 7 CM 2 ,
15 CM 2 H 1,2 JJ.M 2 . OJJ.HO H3 JJ.aHHbIX Cel/eHHH npoxoJJ.HT 4epe3
LleHTp wapa. KaKoe? n04eMY?
585. .LJ.,nHHa oKpymHocTH caMoro 60JIbWOrO Cel/CHHH wapa nJ1OC-
KOCTblO paBHa 25,12 CM. BbIl/HCJIH paJJ.HYc wapa.
4. t 3. n ponopu.HH
n p H M e p. MaMa ynAaTuAa 39 K. 3a 1.5 Kr MYKU, a 6a6ywKa
52 K. 3a 2 Kr MYKU. BbtR.CHU, no OOUHaKOBOU AU l.{eH.e 6bt./ta
KynAena MYKa.
l..h06bl onpeJJ.eJIHTb CTOHMOCTb 1 Kr KynJIeHHOH MaMOH H 6a6ywKOH
MYKH, HymHo HaHTH l/aCTHble COOTBeTCTBeHHO:
39: 1,5 H 52:2.
Bbl4HCJIeHHH nOKa3bIBalOT, LITO B 060HX CJIY4aHX CTOHMOCTb
Kr MYKH 26 K. 3Ha4HT, 3TH 4aCTHble paBHbl H MomHO 3anHcaTb:
39 ' 1 _ 52 ' 2 39 _ 52
. ,5- . HJIH T.5-2'
TaKHe paBeHCTBa Ha3bIBalOT npOnOPQHHMH.
BepHoe pa8eHCT80 iJ8YX 'lacTHblX Ha3bl8aeTCll nponop'4ueu.
109
nponOPLlHHMH HBJIHeTCH ellle, HanpHMep, TaKHe paBeHCTBa:
5 2,5 2 7
""4="'2 H 3= 10,5 '
TaK KaK
= 1,25 H 2 2 5 = 1,25;
=0,666... H 1
5 =0,666...
H I 3 1
o paBeHCTBO "'2=""'5 He HBJIHeTCH nponopLlHeH, TaK KaK "'2=0,5,
3
a ""'5=0,6.
B 6YKBeHHoM BHJJ.e nponopLlHH 3anHCblBaeTCH CJIeJJ.YIOIlI,HM 06-
pa30M:
a:b=c:d HJIH
=..E....
b d'
4aCTHoe l/HCeJI a H b Ha3bIBaeTCH HHOr.lla OTHoweHHeM a K b. 003-
TOMY 3TH nponOPLlHH MO>KHO np04HTaTb: ca, ae/leHHoe Ha b, paBHO
c, ae.AeHHOM.Y Ha d» HJIH: «OTHoweHue a K b paBHO OTHoweHUIO c K d».
EYKBbl a H d 0603Ha4alOT KpaHHHe IfJleHbI, b H C - Cpe,IJ.HHe 4J1eHbI
nponopLlHH.
cpeJJ.HHe 4JIeHW KpaHHHH 4JIeH cpeJJ.HHH 4JIeH
1\ I 590.
I
a : b == c :d a c
\ I b - d
\ "
KpaHHHe 4JIeHW cpeJJ.HHH 4JIeH KpaHHHH 4JIeH
KaK BHJJ.HM, 3HaKOM JJ.eJIeHHH B nponopIlHH MO>KeT CJIY>KHTb JJ.p06HaH
4epTa HJIH JJ.BoeT04He. B JJ.aJIbHeHWeM Mbl 6YJJ.eM nOJIb30BaTbCH B
OCHOBHOM JJ.P06HOH 4epTOH.
A.
586. HCnOJIb3YH TeKcT, 06"bHCHH:
I) n04eMY paBeHCTBa += : H
:
=
2 HBJIHIOTCH nponop14 u -
RM.U (Ha30BH KpaHHHe (cpeJJ.HHe) 4JIeHbi 3THX nponoPLlHH);
2) n04eMY paBeHcTBo 4;} =
He HBJIHeTCH nponopIlHeH
110
587.
,
\
npOBepb (y C THO), KaKHe H3 paBeHCTB HBJIHIOTCH nponopLlHHMH
(Ha30BH KpaHHHe H cpeJJ.HHe l/JIeHW nponopLlHH):
I
3) 56:7= 1:""8
36 _ 0,48
""6- 0,08
I) 9:3=0,3: I 2)
J..-._ O,2
2 -0,4
3:0,1 =60:2
2.._ 0 ,7
10 - 0,01
18
2 20
8,4 21
4= 100
2,4 _ 2
6 5
6.
588.
npOBepb, HBJIHIOTCH JIH JJ.aHHble paBeHCTBa nponopLlHHMH:
47,4_3,16. 2 ) 4-.!.'2-2'2... 3)
= 2,02 .
I) 12- 0,8 ' 7' - . 8 ' 8,5 0,1'
4) 43,4:3,1 =0,7:0,28; 5) 1 9 0 :
= 1,2:0,8.
J].JlH nOBTOpeHHH
589.
PeWH (y C THO)
I) x+30= 120
y-4,9= 1,1
4 1
""'5- z =""'5
PeUIH ypaBHeHHe:
3 5
I) ""4x=""8
t=
2. m =6
6 8 3
u=.l... 2.. 5 =9 5 ( z- /2 ) =5
7 14 4
noe3JJ. OmpaBJIHeTCH B 10 l/ 40 MHH H npH6blBaeT Ha CTaHIlHIO
Ha3Hal/eHHH B 13 4 20 MHH. Bbl4HCJIH paccToHHHe Me>KJJ.Y CTaH-
KM
LlHHMH, eCJIH cpeJJ.HHH CKOpOCTb noe3JJ.a 60
.
UeHY Ha nblJIeCOC CHH3HJIH Ha 10%, B pe3YJIbTaTe 4ero OH
CTOHT Tenepb 38,7 p. CKOJIbKO CTOHJI nblJIeCOC JJ.O CHH>KeHHH
LleHbl?
Bo BpeMH BeJIHKOH OTe4ecTBeHHoH BOHHbl norH6JIo npHMepHo
20 MJIH. COBeTCKHX rpa>KJJ.aH. 3TO COCTaBJIHeT 40% OT 06-
llI,ero KOJIH4eCTBa nom6wHx BO BpeMH BTOpOH MHpOBOH BOHHbl.
CKOJIbKO 4eJIOBeK nom6JIo 80 BpeMH BTOpOH MHpOBOH BOHHbl?
ypaBHeHHe:
2) 4x= 1,2
0,5t = 3
I
-- y=9
4
3) y:20=30
8:5=0,2
1
x:T=6
2) 5/= 1,25
3) 2 (x-0,5)=7
3...!... ( + I...!... ) = 6.!.!..
3 y 8 12
591.
592.
593.
III
4.14. OCHoBHoe CBOHCTBO npOnOp1l.HH
4T06bl npOBepHTb HBJIHeTCH JIH JJ.aHHoe paBeHCTBO nponopLlHeH,
Mbl Bbll/HCJIHeM 3Hal/eHHe Ka>KJJ.oro H3 JJ.aHHbIX l/aCTHbIX. npoBepKa
ynpolllaeTCH, eCJIH Mbl BOCnOJIb3yeMcH OCHOBHblM CBOHCTBOM npo-
nOpu.HH. B l/eM OHO 3aKJIIOl/aeTCH?
pac fA u
aCCMOTpHM nponOpu.HIO b=d' nCnOJIb3yeM OCHOBHoe CBOHCTBO
4aCTHoro (CM.
2.2): YMHO>KHM JJ.eJIHMOe H JJ.eJIHTeJIb l/aCTHOro
a d c n
b Ha ,a JJ.eJIHMOe H JJ.eJIHTeJIb l/aCTHOro d Ha b. OJIYl/HM:
a.d c.b
b.d - d.b '
113 paBeHCTBa 3THX 4aCTHblX BbITeKaeT paBeHCTBO JJ.eJIHMblX (nOl/eMY?):
a.d=c.b.
06paTHO: H3 paBeHCTBa a.d=c.b MO>KHO BblBeCTH npOnOpLlHIO
a c 3 d u
b=d' aMeTHM, LITO a. - 3TO npOH3BeJJ.eHHe KpaHHHX 4JIeHOB, a
c. b - npOH3BeJJ.eHHe cpeJJ.HHX 4JIeHOB nponopLlHH. 3Hal/HT, np0U38e-
iJeHue ICpaUHUX 'lAeH08 pa8HO npoU38eiJeHulO cpeiJHux VleH08 npo-
nopquu.
3TO H eCTb OCHOBHoe CBOHCTBO nponopLlHH, no OCHOBHOMY
CBOHCTBY nponopu.HH MO>KeM CKa3aTb, 4TO, HanpHMep, nponopLlHH
1,6 2,4
T="3 COCTaBJIeHa BepHO, TaK KaK
1,6.3=4,8 H 2,4.2=4,8, TO eCTb 1,6.3=2,4.2.
5 11
Ho paBeHCTBO ""6=12 HeBepHO, TaK 5.12* 11.6, n03TOMY OHO He
HBJIHeTCH npOnOpu.HeH. OCHOBHoe CBOHCTBO nponopLlHH n03BOJIHeT
pewaTb ypaBHeHHH HOBoro THna.
n 2 5
P H Me p I. PeWHM ypaBHeHHe "3=x'
I1cnOJIb3YH OCHOBHoe CBOHCTBO nponopLlHH, nOJIYl/HM:
2x=3.5,
2x=15,
x=15:2,
x=7,5.
OCHOBHoe CBOHCTBO nponopLlHH n03BOJIHeT TaK>Ke H3 JJ.aHHOH
nponopLlHH BbIBOJJ.HTb HOBbie.
112
\ 5 10
n p H Me p 2, PaccMoTpHM nponopLlHIO ""6=12' oTKYJJ.a
5.12=6.10. nOCJIeJJ.Hee paBeHcTBo MO>KeM nOJIY4HTb, BHJJ.HMO,
H H3 CJleJJ.YIOIlI,HX nponopLlHH:
5 6 12 10
10=12 H ""6=""'5'
OTCIOJJ.a CJleJJ.yeT, 4TO cpeJJ.HHe (KpaHHHe) 4J1eHbI npOnOpu.H H MO]KHO
nOMeHHTb MeCTaMH.
A.
594. npOl/TH TeKCT naparpa4>a H CKa>KH:
I) B 4eM 3aKJII04aeTCH OCHOBHoe CBOUCTBO nponop/.{uu;
4 100 u
2) n04eMY paBeHcTBo -=5 HBJIHeTCH nponopLlHeH;
0,2
3) n04eMY paBeHCTBO 3:,2 = 25 3 He HBJIHeTCH nponopLlHeH.
595. npoBepb (y C THO). HCnOJIb3YH OCHOBHoe CBOHCTBO nponopLlHH,
KaKHe H3 CJIeJJ.YIOIlI,HX paBeHCTB HBJIHIOTCH nponopLlHHMH.
2 4
I) ""'5=10
5:6= 10: 12
2 ) 1,2_
"3-4
3 ) 0,4_ 0.01
32 -8
...!.... 3 =2-: 12
6' 3
..!.&..._ 360
0,01 2
1,2:0,4=3: 10
6,6 0,3
D- 0,05
7_...!...
0,7 - 10
596. npoBepb, HCnOJIb3YH OCHOBHoe CBOHCTBO nponopLlHH, HBJIH-
IOTCH JIH JJ.aHHble paBeHcTBa nponopLlHHMH:
597.
598.
I ) 0,25 _ 7,5 . 2 2 75 0,44 3) 245 7
TI"-"42' ) 1:25 = 0,02 ; 100 ="3;
4 ) ...!.....1..= I...!...: 1.1..; 5) _ 4 1 :0,5=- 6 1 :- 3 1 ; 6) 3+:25=2: 15.
2 . 5 6 4
x 1
npOJJ.YMaH npHMep I H 06'bHCHH, KaK peWHTb ypaBHeHHe ""6=3"'
PeWH ypaBHeHHe:
I ) x -
2) 5:4=25:y
7-2
5 t
3"=""6
I_x
3"-18
x:8= 1,5:2
3) x _ 9
0,6 - 0,5
4,5_ 12,5
-Y-T
2,5 _!
1,25 - Y
1 1
2:a=2 T : 14"
113
6.
599. PewH ypaBHeHHe:
I ) 3x _1,5 2) x+2 5 2x-1 _ 1
8 -"2 _ 4 =- 2 3)
3 -"2
_ I y-5 4 6 2
5x -3"
=""'5 3x+2 =""'5
=
4 2 5 10
5 15 3+z =3" 6 - 2x-3
600. npOJJ.YMaH npHMep 2 H JJ.J1H Ka>KJJ.oro H3 CJl eJJ.YIOIlI,HX
paBeHcTB
HanHWH JJ.Be nponopLlHH:
1) 2.20=5.8; 2) 12.a=6.7; 3) x.l=u.y,
601. HanHwH JJ.Be nponopLlHH, npOH3BeJJ.eHHe K pa '
H u x
rI n l/JIeHOB KO-
TOpblX paBHo 12.
602. HanHwH JJ.Be nponopLlHH, npoH3BeJJ.eHHe C p eJJ.HHx
pblX paBHO 8. l/JIeHOB KOTO-
603*. HaHJJ.H 3Hal/eHHH x H 6
y, TaKHe, LITO bl Ka>KJJ.oe H3 JJ.BYX paBeHCTB
_ 2 y 3
y -"3 H 24 =""4 6blJIO BepHblM.
4.15. npHMaH npOnOp1l.HoHaJJbHOCTb BeJ1WIHH
2 n Y 3 CTb I Kr TOBapa CTOHT 2 p, nOJJ.Cl/HTaeM CTOHMOCTb HanpHMep
Kr, Kr, 0,5 Kr, 5 Kr 3Toro TOBapa: "
KOJIHl/eCTBO TOBapa, Kr I 2 3 0,5 5
CTOHMOCTb TOBapa, p. 2 4 6 I 10
BHJJ.HO, LITO CTOHMOCTb TOBapa pa3JIHl/Ha H OHa 3aBHCHT OT
KOJIH4eCTBa KynJIeHHOro TOBapa, a OTHoweHHe CTOHMOCTH TOBapa
K ero KOJIHl/eCTBY eCTb l/ H C JI 0 n 0 C T 0 H H H B
o e. JJ.aHHOM CJIYl/ae
OH
paBHO CTOHMOCTH I Kr 3Toro TOBapa (B Py6JIHX) 2
D.eHcTBHTeJIbHO ' TO eCTb .
,
246
1="2="3=...=2.
ECJIH JJ.Be BeJIHl/HHbl H3MeHHIOTCH TaKHM 06pa30M, LITO OTHowe-
HHe COOTBeTCTBYlOlllHX 3Ha4eHHH 3THX BeJIH4HH OCTaeTCH l/HCJIOM
nOCTOHHHblM, 10 TaKHe BeJlHl/HHbl Ha3blBalOTCH npHMO nponOPLlHOHaJIb-
HhlMH. 113 COOTBeTCTBYlOllI,HX 3Hal/eHHH npHMO nponopLlHOHaJIbHblX Be-
JIH4HH MO>KHO COCTaBJIHTb nponopLlHH.
114
I1TaK, KOJIHl/eCTBO TOBapa Hero CTOHMOCTb - npHMO npo-
nopLlHOHaJIbHWe BeJIHl/HHW. npHMO nponopLlHOHaJIbHWMH BeJIHl/HHa-
MH, HanpHMep, 6YJJ.YT nYTb, npOHJJ.eHHbIH npH nOCTOHHHOH CKOpOCTH,
H BpeMH; JJ.J1HHa H Macca npOBOJIOKH nOCTOHHHOH TOJIIlI,HHbl; JJ.J1HHa
OKPY>KHOCTH H JJ.HaMeTp H JJ.p.
npHMO nponopLlHOHaJIbHble BeJIHl/HHbl MO>KHO oxapaKTepH30BaTb
elll,e H TaK: C YBeAUlieHueM (YMeHbweHueM) OiJHOU. BeAUliUHbl B
HeCKOAbKO pa3 iJpyzafl BeAUliUHa YBeAUliUBaeTCfl (YMeHbwaeTCfl) BO
CTOAbKO 3/Ce pa3. HanpHMep, JJ.JIH nOKpaCKH Ka>KJJ.oro KBaJJ.paTHoro
MeTpa nOJIa paCXOJJ.yeTcH OJJ.HHaKOBoe KOJIHl/eCTBO KpaCKH. TorJJ.a C
YBeJIHl/eHHeM nJIOlllaJJ.H OKpaWHBaeMoro nOJIa B 2, 3, ... pa3a YBe-
JIHl/HBaeTCH H KOJIH4eCTBO Tpe6yeMoH KpacKH B 2, 3, ... pa3a.
3Hal/HT, nJIOlll,aJJ.b nOJIa H KOJIH4eCTBO Tpe6yeMoH KpacKH - npHMo
nponopLlHOHaJIbHWe BeJIH4HHW.
n p H Me p. 'h06bl nOKpacuTb nOA nAow,aiJblO 16 M 2 , nOTpe60Ba-
AOCb 3,2 Kr KpacKu. CKOAbKO nOTpe6yeTcfl TaKOu. KpacKu, liT06bl
nOKpacuTb nOA nAow,aiJblO 12 M 2 ?
PewHM 3TY 3aJJ.al/Y JJ.BYMH cnoc06aMH.
n e p B bl H C n 0 C 0 6, OnpeJJ.eJIHM CHa4aJIa, CKOJIbKO KpacKH no-
Tpe6yeTcH, l/T06bl nOKpacHTb I M 2 nOJla. nOJIYl/HM 3,2: 16=0;2 (Kr),
D.JIH nOKpacKH nOJIa nJIOlll,aJJ.blO 12 M 2 nOTpe6yeTcH B 12 pa3 60J1bWe
KpaCKH, TO eCTb 0,2.12=2,4 (Kr).
B TOp 0 H C n 0 C 0 6 (c nOMOlll,blO n ponopLlHH). 0603Ha l/H M HC-
KOMoe KOJIH4eCTBO KpaCKH l/epe3 x, TaK KaK nJIOlll,aJJ.b nOJIa H KqJIH-
l/eCTBO KpaCKH - npHMO nponopLlHOHaJIbHble BeJIHl/HHbl, TO COCTaBHM
nponopLlHIO:
x _3,2
\2-16'
OTK YJJ. a 16x= 12.3 , 2 H X= 12.3,2 =24
16 ' .
o T BeT. He06xoJJ.HMO 2,4 Kr KpacKH.
A.
604. npOl/TH TeKCT naparpa<l>"a H CKa>KH, KaK xapaKTepH3YIOTCH
npflMo nponop/{uoHaAbHble BeAUliUHbl. 3aTeM npoBepb (y C T-
H 0), B KaKHX H3 CJIeJJ.YIOIlI,HX Ta6JIHLl JJ.aHbI npHMo nponopLlHo-
HaJIbHble BeJIH4HHbl x H y:
115
I)
x I 2 3 4
Y 5 10 15 20
2)
x 2 4 6 8
y 16 32 48 64
3)
x I 0,5 0,1 p,OI
y 10 5 I I
4)
x 27 18 9 0,9
y 3 2 0,1 0,01
605.
610.
Bbll/HCJIH nepHMeTp KBaJJ.paTa, eCJIH ero CTopoHa paBHa 2 JJ.M;
6 .lI.M; 4,5 JJ.M; 7,2 JJ.M. OYJJ.YT JIH nepHMeTp KBaJJ.paTa Hero
CTopOHa npHMO nponOPLlHOHaJIbH
MH BeJIHl/HHaMH? (06nHCHH
OTBeT. )
ECJIH CTOPOHY KBaJJ.paTa YBeJIHl/HTb (YMeHbWHTb) B 2, 3, 7, 10
pa3, TO BO CKOJIbKO pa3 YBeJIHl/HTCH (YMeHbWHTCH) nepHMeTp
KBaJJ.paTa?
B
l/HCJIH nJIOUlaJJ.b KBaJJ.paTa CO CTOpOHOH a, eCJIH a = 2 CM;
a=4 CM; a=8 CM. OYJJ.YT JIH nJIOlll,aJJ.b KBaJJ.paTa Hero CTOpOHa
npHMO nponopLlHOHaJIbHblMH BeJIHl/HHaMH? nOl/eMY?
3a 3 l/ Ha MeJIbHHLle CMOJIOJIH 27 T nWeHHl/HOH MYKH. CKOJIbKO
TOHH nWeHHl/HOH MYKH MO>KHO CMOJIOTb 3a 8 l/, eCJIH TeMn
pa60Tw He H3MeHHTCH?
CKOJIbKO CTOHT 3,25 M TKaHH, eCJIH 0,8 M 3TOH TKaHH CTO-
HT 5,4 p.?
3 I _
3a 4 q TypHCTbI npOWflH 2 4" KM. KaKoe paCCTOHHHe npOHJJ:YT TY-
pHCTbI 3a 2 q; 3a 3 q; 3a 3,5 4, eCJIH OHH HAYT C nOCTOHIIHOH CKO-
pOCTbIO?
1J.Boe pa60l/HX H3rOTOBHJIH 3a onpeJJ.eJIeHHOe BpeMH 19 JJ.eTaJIeH.
CKOJIbKO pa60l/HX nOTpe6yeTcH, l/T06bl 3a TO >Ke BpeMH H3ro-
TOBHTb 38, 76, 133 JJ.eTaJIH?
1J.Ba KOM6aHHa «1J.OH» y6HpalOT 3a onpeJJ.eJIeHHOe BpeMH YPO>KaH
C nOJIH nJIOUlaJJ.blO 32 ra. C nOJIH KaKOH nJIOUlaJJ.H y6ePYT 3a
TO >Ke BpeMH YPO>KaH 3, 4, 6, 7 KOM6aHHoB «1J.oH»;I
npH nepeJIeTe cepaH BopoHa MO-
>KeT JIeTaTb co CKOpOCTblO 50
lJ '
a cKBopeLl - co CKOpOCTblO 74
.
'I
KaKoH nYTb npOJIeTHT cKBopeLl 3a
BpeMH, KOTopoe nOTpet;JyeTcH BO-
pOHe, ljT06
npOJIeTeTb 75 KM?
606.
607.
608.
609.
611.
612.
613.
.1\ .
- £,
'-
'("1.;-.
.
:;..
-::-
116
614.
3a pa60TY, BbInOJIHeHHYIO 3a 15 l/, pa60l/HH nOJIYl/HT 19,5 p,
CKOJIbKO Py6JIeH OH nOJIYl/HT 3a 8 l/, eCJIH TeMn pa60T
Ta-
KOH >Ke?
4 M J 3epHa p>KH BeCHT 2,8 T. CKOJIbKO TOHH BeCHT 6,5 M 3 3ep-
Ha p>KH?
615.
6.
616. KaKHe H3 CJIeJJ.YIOIlI,HX BeJIHl/HH HBJIHIOTCH npHMo nponopLlH-
OHaJIbHblMH (npoBepb):
I) nJIOrnaJJ.b npHMoyrOJIbHHKa H JJ.JIHHa, eCJIH wHpHHa ero 6 CM;
2) 06neM npHMoyrOJIbHOrO na p aJIJI 2 eJIenHneJJ.a H BbICOTa, eCJIH
nJIOUlaJJ.b ero OCHOBaHHH 15 JJ.M ;
3) nJIOlll,aJJ.b Kpyra H pa.ll.HYC;
u u BpeM H eCJIH CKO p OCTb JJ.BH>KeHHH no-
4) npoHJJ.eHHwH nYTb H ,
50 KM
CTOHHHa: -;;- ; ;I
5) CKOpOCTb JJ.BH>KeHHH H BpeMH, eCJIH JJ.JIHHa nYTH 120 KM.
617. 10 M J B03JJ.yxa BeCHT 13 Kr. CKOJIbKO BeCHT B03JJ.YX 6 B
OMHaTe,
JI.,TIHHa KOTOpOH 4,2 M, WHpHHa 3,5 M H BblCOTa 2, M,
(OTBeT oKpyrJIH JJ.O JJ.ecHTblx,)
618. 30H KynHJIa B Mara3HHe 18 H6JIOK. 3TH H6JIOK
xa3JJ.eJIHJIH
Me>KJJ.Y MaMOH, nanOH H 30eH B OTHoweHHH 2: : , TO eCTb
MaMa nOJIYl/HJIa 2 l/aCTH, nana I l/aCTb, a 30H 3 l/aCTH Bcex
U;I
H6JIOK. CKOJIbKO H6JIOK nOJIY4HJI Ka>KJJ.bIH.
npOCJIe.ll.H 3a peweHHeM 3TOH 3aJJ.al/H: u
1) Bce KynJIeHH
e H6JIOKH COCTaBJIHIOT 2+ I +3=6 (l/aCTeH).
2) TaK KaK 6 l/aCTHM cooTBeTcTBYIOT 18 H6JIOK, TO Ha OJJ.HY
l/aCTb npHXOJJ.HTCH 18:6=3 (H6JIOKa).
3) MaMa nOJIYl/HJIa 2.3=6 (H6JIOK), nana 1.3=3 (H6JIOKa)
H 30H 3.3=9 (H6JIOK),
619. J1aTYHb - 3TO CnJIaB MeJJ.H H LlHHKa, MaCCbl KOTOP 6 blX OTHOC I H 2 T O CH
KaK 3:2.
JIH H3rOTOBJIeHHH KYCKa JIaTYHH Tpe yeTcH r
MeJJ.H. CKOJIbKO Tpe6yeTcH UHHKa JJ.JIH H3rOTOBJIeHHH 3Toro KYC-
Ka JIaTYHH?
620. Pa3JJ.eJIH l/HCJIO 480 Ha JJ.Be l/aCTH B OTHoweHHH 3: 5.
621. Pa3JJ.eJIH l/HCJIO 60 Ha l/aCTH B OTHoweHHH 2:3:5.
622. HOBoe cepe6po (aJIbnaKa) - 3TO CnJIaB HHKeJIH, UHHKa H MeJJ.H
B OTHoweHHH 3:4: 13. CKOJlbKO KHJIorpaMMOB Ka>KJJ.oro MeTaJIJIa
HY>KHO B3HTb, l/T06bl nOJIYl/HTb 4 Kr HOBoro cepe6pa?
117
623*. 4acbl co CTpeJIKaMH l/epe3 Ka>KJJ.ble 3 l/ OTCTalOT Ha 2 MH
" H,
,epe3 KaKOH HaHMeHbWHH npOMe>KYToK BpeMeHH OHH nOKa>KYT
npaBHJIbHOe BpeMH?
624*. (C Tap H H H a H 3 a JJ. a l/ a cpeJJ.Hea3HaTCKoro Yl/eHOrO Be-
PYHH.) ECJIH 10 JJ.HpxeMOB (JJ.eHe>KHaH eJJ.HHHLla) npHHOCHT JJ.oxo.n.
5 JJ.HpxeMoB 3a JJ.Ba MeCHLla, KaKOH JJ.OXOJJ. npHHecYT 8 JJ.HpxeMOB
3a TPH MecHLla?
4.16. MacwTa6
.lI.onYCTHM, LITO Hac HHTepeCYeT, KaK 6YJJ.YT pacnOJIO>KeHbl KOMHaTbl
KBapTHpbl B CTpOHllI,eMCH JJ.OMe. .lLnH 3Toro MO>KHO nOCMOTpeTb
nJiaH KBapTHpbl (pHC. 4.28). Ha nJIaHe Bce pa3Mepbl YMeHb-
weuHbl B OJJ.HO H TO >Ke l/HCJIO pa3. Bo CKOJIbKO pa3 pa3Mepbl B
JJ.eHCTBHTeJIbHOCTH 60JIbWe pa3MepOB Ha nJIaHe, nOKa3bIBaeT Macw-
Ta6 nJiaHa. Ha pHcYHKe 4.28 CJJ.eJIaH nJIaH B MacwTa6e 1: 100,
3TO 03Hal/aeT, LITO Bce pa3Mepbl B JJ.eHCTBHTeJIbHOCTH B 100 pa3
60JIbWe, l/eM COOTBeTCTBYIOUlHe pa3Mepbl Ha nJIaHe. HanpHMep, eCJlH
1 :100
.
KVXHR
CrOIlOBaR
T l-
I a:
<II
m
a: BaHHaR 0
c:t
<II <II
* <:;
0
x
:s
0.
c:::
)f( "lIlaR
KOMHara
CnaIlbHR
.
PHC. 4.28
118
Ha nJIaHe JJ.JIHHa npHXO>KeH paBHa 8 CM, TO B JJ.eHCTBHTeJIbHOCTH
3Ta )J.JIHHa paBHa 8.100=800 (CM), TO eCTb 8 M. MacwTa6 1 :5000
03Hal/aeT, LITO JJ.J1HHa JII060ro oTpe3Ka B JJ.eHcTBHTeJIbHOCTH B 5000 pa3
60JIbWe, l/eM JJ.J1HHa cooTBeTcTBYIOUlero oTpe3Ka Ha nJlaHe. .lI.JIHHa Ha
nJIaHe H cooTBeTcTBYIOUlaH eH JJ.J1HHa B JJ.eHcTBHTeJIbHOCTH - 3TO
nponopLlHOHaJIbHble BeJIH4HHbl: OTHoweHHe JJ.JIHHbl Ha nJIaHe K COOT-
BeTcTBYIOUleH )J.JIHHe B JJ.eHcTBHTeJIbHOCTH - l/HCJIO nOCTOHHHoe.
BOJIbWHe TeppHTopHH, HanpHMep rocYJJ.apcTBa HJIH l/aCTH CBeTa,
306pa>Ka1OTCH Ha reorpa4>Hl/eCKHX KapTax. Ha KapTax Bce pa3Mepbl
TaK>Ke YMeHbweHbI B OJJ.HO H TO >Ke l/HCJIO pa3. 3TO nOKa3blBaeT
MaCWTa6 KapTbl, HanpHMep I: 100 000 H JJ.p.
A.
625. npOl/TH TeKCT naparpa4>a H 06'bHCHH, 4TO 03Hal/aeT Macm-
m6 I: 2000, YKa3aHHblH Ha nJlaHe 3eMeJIbHOrO Yl/aCTKa.
626. Ha nJIaHe 3eMeJIbHOrO Yl/aCTKa YKa3aH MacwTa6 I: 1000,
H3BecTHo, LITO paCCTOHHHe Me>KJJ.Y JJ.BYMH TOl/KaMH Ha nJIaHe:
I) I CM; 2) 2,5 CM; 3) 4,9 CM; 4) 10 CM, Bbl4HCJlH COOTBeT-
CTBYIOUlHe paccToHHHH Ha Yl/aCTKe.
627. Ha pHcYHKe 4.29 H306pa>KeH nJIaH 3eMeJIbHOrO Yl/aCTKa npH-
MoyrOJIbHOH 4>OpMW. CJJ.eJIaH He06xoJJ.HMwe H3MepeHHH H BW-
l/HCJlH nepHMeTp H nJIOlll,aJJ.b 3eMeJIbHOrO Yl/aCTKa,
628. CJJ.eJIaH He06xoJJ.HMble H3MepeHHH (CM, pHC. 4.28) H BbI4HCJIH:
1) nJIOlll,aJJ.b H nepHMeTp KYXHH;
2) nJIOlll,aJJ.b H nepHMeTp CTOJIOBOH;
3) nJIOrnaJJ.b H nepHMeTp >KHJIOH KOMHaTbl;
4) nJIOUlaJJ.b H nepHMeTp CnaJIbHH.
629. Hal/epTH nJIaH Ka6HHeTa MaTeMaTHKH B MacwTa6e I: 100.
630. PaccToHHHe Ha KapTe Me>KJJ.Y ropOJJ.aMH 18 CM. KaKoe JJ.eHcT-
BHTeJIbHOe paccToHHHe Me>KJJ.Y ropoJJ.aMH, eCJIH MacuJTa6 Kap-
Tbl I: 500 OOO?
631. no KapTe onpe,JJ;eJIH, Ha KaKOM 1: 2000
paCCT05lHHH (npH6JIH3HTeJIbHO)
HaXO,JJ;HTCH: 1) lleTep6ypr OT
MOCKBbI; 2) MHHCK OT KHeBa; 3)
KHlllHHeB OT MOCKBbI; 4) EaKY OT
T6HJIHCH.
632. .lI.JIHHa peKH HeBbl 74 KM. 4eMY
paBHHeTcH ee JJ.JIHHa Ha KapTe,
MacwTa6 KOTOpOH I: 2 000 OOO? PHC. 4.29
119
6.
1 :10
-
633. Hal/epTH nJIaH OJJ.HOH KOMHaTbl
cBoeH KBapTHpbl.
634. 125 M Ha MeCTHOCTH COOTBeT-
CTBYIOT 2,5 CM Ha nJIaHe. KaKo-
BO paccToHHHe Me>KJJ.Y JJ.BYMH TOl/-
KaMH Ha MecTHocTH, eCJIH Ha
nJIaHe OHO 4,2 CM?
635. Pa3Mepbl nOJIa KOMHaTbl Ha nJIa-
He 4 CM H 6 CM. B JJ.eHCTBH-
TeJIbHOCTH MeHbwaH CTopOHa no-
JIa 5 M. HaHJJ.H JJ.J1HHY 60JIb-
weH CTOpOHbl.
636. Pa60l/HH nOJIYl/HJI 3aJJ.aHHe Bbl- PHC. 4.30
pe3aTb H3 >KeCTH 80 JJ.eTaJIeH
(pHC. 4,30). BbmOJIHH He06xoJJ.H-
Mble H3MepeHHH H Bbll/HCJIH nJIOlll,aJJ.b OJJ.HOH JJ.eTaJIH. CKOJIbKO
>KeCTH nOTpe6yeTcH
JIH H3rOTOBJIeHHH 80 JJ.eTaJIeH?
637*. 4TO 03Hal/aeT MacwTa6: I) 10: I; 2) 100: I? B KaKOM CJlY4ae
npHMeHHeTcH TaKOH MaCWTa6?
638*. .lI.JIHHa KpblJIa HaceKoMoro, HapHcoBaHHoro B MacwTa6e 50: I,
paBHa 15 CM, KaKoBa ero JJ.eHcTBHTeJIbHaH JJ.JIHHa?
AJiR nOBTopeHHR
639. HaHJJ.H (y C THO):
I) 50% OT 700, 2) : OT 240,
4% OT 9,
OT 36,
120% OT 20; ..!. OT 28'
7 '
640. HaHJJ.H (y C THO) l/HCJIO:
I) 20% KOToporo8O, 2) + KOToporo9, 3) 10% KOToporo 15,
5% KOTOporo 100. : KOToporo 21,
KOToporo 3,2,
80% KOTOpOro 16;
KOToporo 40; 50% KOToporo 7.
641. CKOJIbKO npOLleHTOB COCTaBJIHeT (y C THO):
I) 7 OT 10, 2) 10 OT 50, 3) 18 OT 9,
240T 100; 7 OT 35; 60 OT 20?
3) 80% OT 50,
7
lOOT 6,
20%
OT 300.
120
j
1
642.
nOCJIe YBeJIHl/eHHH 3apnJIaTbl pa60l/ero Ha 10% OH CTaJI nOJIY-
l/aTb 195 p. B MeCHLl. KaKoBa 6blJIa 3apnJIaTa pa60l/ero?
B 6H6JIHOTeKe 20400 KHHr, H3 HHX 8% Ha HHOCTpaHHblX
H3blKax. CKOJIbKO KHHr Ha HHOCTpaHHblX H3blKax B 6H6JIHOTeKe?
UeHa I M TKaHH CHH3HJIaCb C 4,5 JJ.O 3,6 p. Ha CKOJIbKO
npoLleHToB CHH3HJIaCb CTOHMOCTb TKaHH?
B OJJ.HH roJJ. KOJIX03 c06paJI C nOJIH nJIOIuaJJ.blO 100 ra 2500 Ll
3epHa, a B CJIeJJ.YIOIlI,HH roJJ. C nOJIH nJIOlll,aJJ.blO 110 ra 2800 Ll
3epHa. KaKaH BeJIHl/HHa B03pOCJIa 60JIbwe: nJIOlll,aJJ.b nOJIH HJIH
YPO>KaH? Bo CKOJIbKO pa3? LITO eUle MO>KHO Bbll/HCJIHTb?
Bbll/HCJIH:
5 2 3 4 9 25.10
I) 3""6+ 2 ""3 2) 5- 2 8 3) 27'\6 4) 36 ' 63
16 7 2
_ I
1
. I ..!.2. 22.4
25 +10 15 3 7 30 ""'5.15
+ ...!... 3..!.2.-1
6.1...!... 6:3- 3 1
27 18 24 16 15
06paTH 06blKHOBeHHble JJ.P06H B JJ.eCHTHl/Hbie H Bbll/HCJIH:
I) 13,75-1
2) :
-O,097 3) 1,8.+ 4) 0,32: :
I,OI+
I
+2,025
'0,44 3+:0,14
06paTH JJ.eCHTHl/Hble JJ.P06H B 06blKHoBeHHble H Bbll/HCJIH:
4...!... + 03 2 ) 28-1..!. 3) .!..!...14 4) 1 7 2 : 0 ,21
I) 6' , 3 21'
...!... + I 5
-O 25 2,7.1- 9 1 1,8: 1- 3 1
12' 9'
BbmOJIHH JJ.eHCTBHH:
I) (4+-3
),0,6; 2) 22:(1
-0,3);
1 2-.
5:
2+0,4 18 14 7
3) 1 ; 4) 1 1 ; 5) 1
1-- -+- 6 0,25+- 3
3 18
Bbll/HCJIH 3Hal/eHHe Bblpa>KeHHH:
643.
644.
645.
646.
647.
648.
649.
650.
I) (147.29-22800: 75+ 19): 17;
2) 1,35:2,7 +6,02- 5,9+0,4:2,5'(4,2-1,075);
3) (1 110 +
). 1 3 7 + /0 :+;
4) 18:( : - : )-6.+.
121
4.17. 06paTHaSi npOnOpLl.HOHaJ1bHOCTb BeJ1H..HH
nYCTb nJIOUlaJJ.b npHMoyrOJIbHHKa paBHa 60 CM 2 , TorJJ.a JJ.J]HHbl ero
CTOpOH a H b MO>KHO Bblpa3HTb, HanpHMep, TaKHMH HaTypaJIbHblMH
l/HCJIaMH:
a, CM I 2 4 6 12 20
b, CM 60 30 15 10 5 3
1-13 Ta6JIHLlbl BHJJ.HO, LITO a H b pa3J1Hl/HbI, HO npOH3BeJJ.eHHe
HX COOTBeTCTBYlOllI,HX 3Hal/eHHH nOCTOHHHO H paBHO 60.
ECJIH JJ.Be BeJIHl/HHbl H3MeHHIOTCH TaK, 4TO npOH3BeJJ.eHHe COOT-
BeTcTBYIOUlHX 3Hal/eHHH 3THX BeJIHl/HH OCTaeTCH l/HCJIOM nOCTOHHHbIM,
TO TaKHe BeJIHl/HHbl Ha3blBalOTCH 06paTHO nponopu.HOHaJlbHblMH. 3Ha-
l/HT, npH nOCTOHHHOH nJIOlll,aJJ.H CTOpOHbl npHMoyrOJIbHHKa 06paTHO
nponopLlHOHaJIbHbI.
06paTHO nponopLlHOHaJIbHble BeJIHl/HHbl MO>KHO oxapaKTepH30BaTb
TaK: C YBeAUlieHueM (YMeHbweHueM) OaHOU. BeAUliUHbl B HeCIWAbKO
pa3 apyzaR BeAUliUHa YMeHbwaeTCR (YBeAUliUBaeTCR) BO CTOAbKO
e pa3, HanpHMep, eCJIH JJ.J]HHa nYTH nOCTOHHHa, TO C YBeJIHl/e-
HHeM CKOpOCTH B 2, 3, ... pa3a BpeMH npe6blBaHHH B nYTH Ol/e-
BHJJ.HO YMeHbwHTcH B 2, 3, ... pa3a, TaKHM >Ke 06pa30M, eCJIH l/HCJIO
pa60l/HX, KOTopoe nOTpe6yeTcH JJ.JIH BWnOJIHeHHH JJ.aHHoH pa60Tw,
YMeHbwHTcH B HeCKOJIbKO pa3, TO BpeMH BbmOJIHeHHH pa60Tbl YBe-
JIHl/HTCH BO CTOJIbKO >Ke pa3 (eCJIH TeMn pa60Tbl Bcex pa604HX
OJJ.HHaKOBbIH) .
n p H Me p, Tpoe pa60liux BbtnOAHRIOT pa60TY 3a 6 l/, CKOAbKO
BpeMeHU nOTpe6yeTcR, eCAU aaHHYIO pa60TY 6yaYT BbtnOAHRTb aBoe
pa60liux?
PewHM 3Ty 3aJJ.al/Y JJ.BYMH cnoc06aMH.
n e p B bl H C n 0 C 0 6. ECJIH Tpoe pa60l/HX BbmOJIHHIOT pa60TY 3a
1
6 4, TO 3a I l/ OHH BbmOJIHHIOT "'6 l/aCTb pa60TbI. TorJJ.a, OJJ.HH
pa604HH BbmOJIHHeT 3a I l/ +:3= 1
(l/aCTb pa60TbI). CJIeJJ.o-
2 1
BaTeJIbHO, JJ.Boe pa60l/HX BbmOJIHHIOT 3a I l/ 18=9 (l/aCTb pa-
60Tbl) If BCIO pa60TY OHH JJ.eJIalOT 3a 9 l/.
B TOp 0 H C n 0 C 0 6. 4HCJIO pa604Hx H BpeMH JJ.J]H BbmOJIHeHHfI
JJ.aHHoH pa60Tbl - 06paTHo nponopLlHOHaJIbHble BeJIHl/HHbI. 3Ha4HT,
122
pa3a (
= 1,5) , TO
BO CTOJIbKO >Ke pa3.
eCJIH l/HCJIO pa60l/HX Y M e H b W H T C H B 1,5
BpeMH BbmOJIHeHHH pa60Tbl Y Be JI H 4 H T C H
CJIeJJ.OBaTeJIbHO, nOJIY4HM 6.1,5=9 (l/).
o T BeT. D.Boe pa60l/HX BbmOJIHHT JJ.aHHYIO pa60TY 3a
651.
652.
9 4.
A.
npOl/TH TeKCT naparpa4>a H CKa>KH, KaK xapaKTepH3YIOTCH
06paTHO nponop/{uOHaAbHbte BeAuliuHbt. B l/eM OTJIHl/He npHMo
nponopLlHOHaJIbHWX BeJIHl/HH OT 06paTHO nponOPLlHOHaJIbHWX?
B KaKHX H3 CJIeJJ.YIOUlHx Ta6JIHLl JJ.aHbl 06paTHO nponopLlHO-
HaJIbHble BeJIH4HHbl x H Y (YCTHO):
I)
x I 2 3 4
Y 6 3 2 1,5
x 2 3 5 7
Y 7,5 5 3 2
2)
3)
x I 3 4 24
Y 24 8 6 I
x 0,1 0,3 25 100
y 200 40 0,8 0,2
4)
653.
B Mara3HHe HMelOTCH KOH4>eTbl no LleHe I p., 1,5 p., 2 p,
H 3 p. 3a KHJIOrpaMM. CKOJIbKO KHJIOrpaMMoB KOH4>eT Ka>KJJ.oro
BHJJ.a MO>KHO KynHTb Ha 3 p.? UeHa I Kr H Macca KOH4>eT, KynJIeH-
3 P - 06 p aTHo n p onopLlHOHaJIbHble BeJIHl/HHbl. (06'bHC-
HblX Ha .,
HH n04eMY.)
PaccToHHHe 150 KM MOTOLlHKJlHCT npoexaJI co cpeJJ.HeH CKO-
KM
aBT06yc - co CKOpOCTblO 60
, rpY30BaH Ma-
654.
655.
KM
pOCTblO 75
,
WHHa _ co CKOpOCTblO 50 K; H BeJIOCHneJJ.HCT - co CKOpOCTblO
20
. CKOJIbKO 4aCOB nOTpe60BaJIOCb Ka>KJJ.oMY BHJJ.Y TpaHcnop-
Ta,
T06bl npoexaTb BeCb nYTb? CKOpOCTb H BpeMH, 3aTpal/eHHOe
Ha 150 KM nYTH,- 06paTHo nponopLlHOHaJIbHble BeJIHl/HHbI.
(06'bHCHH n04eMY.)
newexoJJ. 3aTpaTHJI Ha nYTb 2,5 l/, JJ.BHraHCb co CKOpOCTblO
3,6
. CKOJIbKO BpeMeHH 3aTpaTHT newexoJJ. Ha TOT >Ke nYTb,
lJ
6 4 5 KM ';)
eCJlH ero CKOpOCTb YJJ.eT ,
.
123
656. Bce l/JIeHbl 6pHraJJ.bl pa60TalOT B O}J.HHaKOBOM TeMne. 4eTBepo
ycneBalOT BbmOJIHHTb onpeJJ.eJIeHHYIO pa60TY 3a 32 l/. CKOJIbKO
l/aCOB nOTpe6yeTcH JJ.JIH BbmOJIHeHHH 3TOH pa60TbI, eCJIH l/HCJIO
pa60l/Hx: I) YMeHbWHTCH B 2 pa3a; 2) YBeJIHl/HTCH B 2 pa3a;
3) YBeJIHl/HTCH B 4 pa3a?
657. 1-lT06bl BblBe3TH TOBap, npH6blBWHH Ha >KeJIe3HoJJ.opO>KHYIO CTaH-
LlHIO, nOTpe6yeTcH 21 aBTOMaWHHa rpY30noJJ.'beMHOCTblO 2,5 T.
CKOJIbKO nOTpe60BaJlOCb 6bl aBTOMaWHH rpY30noJJ.'beMHOCTblO
3,5 T?
658. 15 pa60l/HX 3aKOHl/HJIH OTJJ.eJIKY KBapTHp B HOBOM JJ.OMe 3a
24 JJ.HH. 3a CKOJIbKO JJ.H£:H BbmOJIHHJIH 6bl 3TY pa60TY 18 pa-
60l/HX?
6.
659. KaKHe H3 cJIeJJ.YIOIllHx BeJIHl/HH HBJIHIOTCH 06paTHo nponopLlHo-
HaJlbHblMH (npoBepb):
I) nJIOlllaJJ.b OCHOBaHHH npHMoyrOJIbHOrO napaJIJIeJIenHneJJ.a H
BbiCOTa, eCJIH ero 06'beM 120 CM 3 ;
2) BpeMH naXOTbl H KOJIHl/eCTBO roplOl/erO, H3pacxoJJ.oBaHHoro
TpaKTopoM;
3) KOJIH4eCTBO TKaHH, KynJIeHHOH Ha 50 p. H LleHa I M;
4) BpeMH, 3aTpal/eHHOe Ha BcnawKY 600 ra 3eMJIH, H KOJIH-
l/eCTBO TpaKTOpOB (oJJ.HHaKoBoH MOIllHOCTH)?
660. KOBW HOBoro npoeKTHpyeMoro waralOlllero 3KcKaBaTopa BMe-
llI,aeT 100 M 3 neCKa, CKOJIbKO BarOHOB MO>KeT HanOJlHHTb TaKoH
3KcKaBaTOp, eCJIH OH CJJ.eJIaeT 100 nOJJ.HHTHH? rpY30noJJ.'beM-
HOCTb BarOHa 40 T, a I M 3 neCKa BeCHT 1,5 T.
661 *. Ha peMoHTe JJ.oporH pa60TaJIO 15 l/eJIOBeK, H OHH JJ.OJI>KHbl
6blJIH 3aKOHl/HTb pa60TY 3a 12 JJ.HeH. Ha nHTblH JJ.eHb YTPOM no-
JJ.OWJlH ellle HeCKOJIbKO pa60l/HX, H OCTaBwaHCH pa60Ta 6blJIa
BWnOJIHeHa 3a 6 JJ.HeH. CKOJIbKO pa604Hx npH6wJIo JJ.onOJIHH-
TeJIbHO?
JQnH caMonpOBepKH
662. KaKHe 4HCJIa Ha3blBalOTCH B3aHMHO 06paTHbIMH? npHBeJJ.H npH-
MepbI.
663. HaHJJ.H l/HCJIa, 06paTHbie JJ.aHHbIM: 8, ; , 1+ H 0,3.
664. PaCCKa>KH, KaK Bbll/HCJIHeTCH l/aCTHOe JJ.Byx 06blKHoBeHHblx
JJ.p06eH.
124
665.
666.
667.
Bbll/HCJlH (y C THO) :
I)
.
2 ) O:
8 . 5 9
.
:7
4 ' 4 6
.
6:
7 . 5 3
Bbll/HCJIH:
27. 3
I) 32 'S
5. 1
17. 1 14
2. 1
4""'5. 9 ""6
Bbll/HCJIH:
0,9 (3:2,25--{- (I
++ ))--4 :0,8.
Bbll/HCJIH 3Hal/eHHe Bblpa>KeHHH 1
:x+O,4, eCJlH x=l; x= : ;
3)
:6
. 5
5. s
8 .
. 5
1.
4) I s' 9
1 . 2
1"2'3
1
4:1"3
1 .
2) 4(2".14
3
36:3""'5
. 1
0,13.2""6
2 11
3) 8:2 9 -2\5
(
: 1,5+2+ )
4 3 I
""'5: 0 ,6+ 1""'5.""6
668.
1
x=13'
669. CKOJIbKO CTOHT I Kr KOH4>eT, eCJIH 3a 1+ Kr 3anJIaTHJIH
1
2- p.?
4
KM 3a
I
2 l/.
670.
2 2 1
TaHH npowJIa 2""'5 KM 3a "3 4, aneTH 4"2
CKOpOCTb Koro 60JIbWe H BO CKOJIbKO pa3?'
PaCCKa>KH, KaK:
I) HaHTH l/HCJIO no ero JJ.P06H (no ero npoLleHTaM);
2) cpaBHHTb l/HCJIa C nOMOlll,blO l/aCTHOro (oTHoweHHH);
3) HaHTH H3MeHeHHe BeJIHl/HHW B npoLleHTax.
671.
672.
HaHJJ.H l/HCJIO: 3 1 .
I) 112% KOToporo paBHbl 56; 2) 7 KOToporo paBHbl 2 14 ,
3) 4% KOToporo paBHbl 0,8; 4) 0,7 KOToporo paBHbl 3,43.
CKOJIbKO npoLleHToB COCTaBJIHIOT:
I) 4,5 OT 36; 2) 28 OT 25?
Bbll/HCJIH H3MeHeHHe BeJIHl/HHbl B npoLleHTax:
l) OT 125 JJ.O 105; 2) OT 3,2 JJ.O 5,6.
B napKe H3 400 JJ.epeBbeB 280 JIHCTBeHHbIX. CKOJIbKO npo-
LleHTOB COCTaBJIHIOT JIHCTBeHHble JJ.epeBbH B napKe?
673.
674.
675.
125
676. Ha copeBHoBaHHHx no JIerKOH aTJIeTHKe B TOJIKaHHH HJJ.pa
Yl/aCTBOBaJIH 24 cnopTcMeHa, 3TO COCTaBJIHeT 15% OT Bcex
JIerKOaTJIeTOB, npHHHBwHX Yl/aCTHe B copeBHoBaHHHx. CKOJIbKO
cnopTcMeHoB Yl/aCTBOBaJIO B copeBHoBaHHHx?
677. B npOWJIOM roJJ.Y KOJIX03 c06paJI C 1 ra B cpeJJ.HeM 160 Ll
KapTo4>eJIH, a B 3TOM roJJ.Y 220 Ll. Ha CKOJIbKO npoLleHToB
YBeJIHl/HJICH B cpeJJ.HeM ypO>KaH KapTo4>eJIH C Ira?
678. KorJJ.a npOJIO>KHJIH + HOBOH 3JIeKTpOJIHHHH H eUle 21 KM,
TO OCTaJIOCb npOJIO>KHTb elll,e 20% BceH JIHHHH, Bbll/HCJIH JJ.J1HHY
BceH 3JIeKTpOJIHHHH.
679. B 3KCKYPCHH npHHHJIH Yl/aCTHe 28 Y4eHHKOB, 18 pOJJ.HTeJIeii
H 4 Yl/HTeJIH. COCTaBb KpyroBYIO JJ.HarpaMMY pacnpeJJ.eJIeHHH
Yl/aCTHHKOB 3KCKypCHH.
680. 4TO TaKoe nponopLlHH? HanHwH KaKYIO-JIH60 nponopLlHIO H
YKa>KH ee KpaHHHe H cpeJJ.HHe l/JIeHbI.
681. C4>OPMYJIHpyii OCHOBHoe CBOHCTBO nponopLlHH.
682. PewH ypaBHeHHe:
48 73 7 25
I) 9 X = 45 ; 2) y: 1 9 ="'4; 3) 14:z=15; 4) -;--9'
5) O,25:3=x: 1+.
683. CKa>KH, KaK xapaKTepH3YIOTCH: I) npHMO nponopLlHOHaJIbHble
BeJIHl/HHW; 2) 06paTHo nponopLlHOHaJIbHble BeJIHl/HHW.
684. 3a 1,5 l/ CaMOJIeT npOJIeTeJI 810 KM, KaKoe paCCTOHHHe OH
npOJIeTHT 3a 2,2 l/, eCJIH CKOpOCTb nOJIeTa He H3MeHHTcH?
685. 3 Hacoca oJJ.HHaKoBoii MOIll,HOCTH 3anOJIHHIOT 6acceHH 3a 8 l/.
CKOJIbKO nOTpe6yeTcH BpeMeHH, l/T06bl 3anOJIHHTb 6acceHH,
eCJIH pa60TaTb 6YJJ.YT 4 TaKHX Hacoca?
686. nJIaH 3eMeJIbHOrO Yl/aCTKa KBaJJ.paTHoii 4>OpMbl BbmOJIHeH B
MacwTa6e I: 500, CTopoHa Yl/aCTKa Ha nJIaHe 16 CM. Bbll/HCJlH
nJIOlll,aJJ.b Yl/aCTKa.
687. Hal/epTH OKPY>KHOCTb, paJJ.HYc KOTOpOH 4,3 CM.
688. KaK Bbll/HCJIHTb JJ.JIHHY OKPY>KHOCTH H nJIOUlaJJ.b KPyra?
689. D.HaMeTp nJIOlll,aJJ.KH, HMelOUleH 4>OPMY Kpyra, 15 M. Bbl4HCJIH
ee nJIOUlaJJ.b H JJ.JIHHY OKPY>KHOCTH. (OTBeT oKpyrJIH JJ.O eJJ.HHHLl.)
5. n0J10)l(HTEJ1bHbIE H OTPHU,ATEJ1bHbIE
q H CJ1A.
npSlMOYfOJ1bHASI CHCTEMA
KOOPJJ.HHAT
5.1. OTpHLl.aTeJl bHble ..HCJ1 a
PaCCMOTpHM npHMepbI.
n p H M e p I. 3a Hal/aJIO OTC4eTa npH H3MepeHHH TeMnepaTypbl
npHHHMaeTCH TeMnepaTypa TaHHHH JIbJJ.a (HJIH 3aMep3aHHH BOJJ.bI).
OHa 0603Ha4aeTCH l/HCJIOM O. ECJIH BOJJ.a npeBpaTHJlaCb B JIeJJ., TO no-
Ka3aHHH TepMOMeTpa HH>Ke HYJIH (XOJIOJJ.). nOCJIe Toro KaK JIeJJ. paCTa-
eT, nOKa3aHHH TepMOMeTpa 6YJJ.YT Bblwe HYJIH (TenJIO). nOKa3aHHe
Bblwe HYJIH 3anHCbIBaeTCH l/HCJlOM CO 3HaKOM «+ », a nOKa3aHHe HH>Ke
HYJIH - CO 3HaKOM «-». 3anHcb +4° (4HTaeTcH nAIOC 'leTblpe zpaay-
ca) 03Hal/aeT 4 rpaJJ.yca TenJIa, a-3° (l/HTaeTcH: MUHYC TpU zpaay-
ca) 03Hal/aeT 3 rpaJJ.yca xOJIoJJ.a (pHC. 5.1).
II p M M e p 2. YpoBeHb MOpSI npMHMMaeTcSI 3a HYJIeBOM H 0603HaQa-
eTCSI QMCJIOM O. KaKOe-JIM60 MeCTO Ha 3eMHoM nOBepXHOCTM MO-
>KeT 6bITb MJIIi BbIllIe, MJIM HM>Ke YPOBH5I MOpSI (pMC. 5.2). HanpMMep,
BepllIMHa ropbI Ka36eK Ha KaBKa3e B03BbIlliaeTCSI Ha
5033 M HaJJ. ypOBHeM MOpH, a caMoe rJIy60Koe MeCTO KacnHHCKoro
MOpH HH>Ke ypOBHH MOpH Ha 1025 M. BblCOTbl HH>Ke ypOBHH MOpH
0603Hal/a1OTCH l/HCJIaMH co 3HaKOM «-», a BblCOTbl Bblwe YPOBHH MO-
M
5
5 I
I
I
I
o I
I
I
I
300
o
100
o YpOB8Hb
MOpA
-100
-5 - 5
-200
-300
PHC. 5.1
PMC. 5.2
126
127
OTMeTb Ha JIYl/e TOl/KH, KOTopble
0,5; 4,2; 1,8; 3,4; 2,7. cooTBeTcTBYIOT l/HCJIaM 2;
699. YnpocTH Bblpa>KeHHe H Bbl4HCJIH ero 3Hal/eHHe: 0 A B M C 3
1 2 I , I I I ..
l) Ta+Ta, a=7 2) 2 5 0 1 2 3 2
"3X+""6X, x=4
n+
_ 6 PHC. 5.4
4 6 n, n-- 5 2
19 ""6m-Tm, m=12 0
7 3 2 F N E D 0 A B M C
-a--a a-2 I
x-
x x=
I I I I I I I I 1"'- -1
8 4' -"3 -3 -2 -1 0 1 2 3
700. Bbll/HCJIH: 9 6' 19 -2
I) 0,306.405 2) 12,8.0,11 3) 398.1,05 PHC. 5.5 -3
,
3,024:4,5 210,9:222 123,22 : 404 PHC. 5.6
701. Bbll/HCJIH:
I) (0,75+2,5). 1
' 2) (2
-1+ ): 1,5; 3) 3:2-+ 0 ,5-5;
4)
: (
_
) _
, ( 5. 3 3
8 4 6 14 ,5) 6. 10 +8) .24; 6) 3-}-+
.3-5.
702. JJ.0 nOHH>KemIH LleH nHcbMeHHblH CTOJI CTOHJI 64 P 0
BYIO ' npeJJ.eJIH HO-
LleHY, eCJlH CTOHMOCTb CTOJIa CHH3HJIH Ha 15%.
703. Pa3JJ.eJIH Kpyr Ha TPH ceKTopa TaK, l/T06b1 yrOJI nepBoro ceKTO-
pa COCTaBJIHJI 10% nOJIHOrO
45 % 0 yrJIa, a yroJI BToporo ceKTopa
o nOJIHOrO yrJIa.
704. newexoJJ. weJI C oJJ.HHaKoBoH CKOpOCTblO CHal/aJIa 3 4
I ' a 3aTeM
ewe 3- l/ 3a 3
2' 2 l/ OH npoweJI paccTOHHHe Ha 2 KM 60JIbWe
l/eM 3a 3 l/. KaKoe paCCTOHHHe npoweJI newexoJJ.';) ,
705*. KaKoH BeJIHl/HHbI yrOJI 6 .
o pa3YlOT l/aCOBaH H MHHYTHaH CTpeJIKH
eCJIH l/aCbl nOKa3blBalOT 12 l/ 10 MHH? '
5.2. I.JHCJ10BaSi OCb
nOJIO>KHTeJIbHble H OTpHLlaTeJIbHble l/HCJIa BMeCTe C l/HCJIOM 0 MO>K-
HO H306pa3HTb TOl/KaMH Ha n p HMOH n. JIH 3Toro
6 ' OTMeTHM Ha npHMoH
KaKYIO-JIH 0 TOl/Ky 0 H npHMeM 3T TOl/K 3
COOTBeTcTByeT l/HCJIY 0 TOl/Ka 0 P Y3 Y a HalJaJlO OTClJeTa. OHa
. a JJ.eJIHT npHMYIO Ha JJ.Ba npOTHB -
nOJIO>KHO HanpaBJIeHHblX JIYl/a. Ha OJJ.HOM H3 3THX JIYl/eH 6YJJ.eM H30
-
:>K5a:b ::O>KHTeJIbHble l/
CJla, KaK Mbl Y>Ke paHee JJ.eJIaJIH. Ha PHCYH-
. , pH
ep, TOl/KOH A H306pa>KeHO l/HCJlO I, TOl/KOH B-
4HCJIO 2, T04KOH M - l/HCJIO 2,5 H TOl/KOH C - l/ 3 H
nOJIYl/eHHbI u 6 HCJIO . a JJ.pyroM H3
x JIYl/eH YJJ.eM H306pa>KaTb oTpHLlaTeJIbHble l/H CJIa TI T 6
130 . · 0 bl
H306pa3HTb l/HCJIO -1, B03bMeM TOl/KY D, TaKYlO, LITO OD=OA
(pHC. 5.5), TorJJ.a oTpe30K OD paBeH eJJ.HHH4HOMY oTpe3KY. 4T06bl
H306pa3HTb l/HCJlO - 2, OTMeTHM TOl/KY E, TaKYIO, 4TO OE = OB. T04-
Ka F H306pa>KaeT l/HCJIO - 3, npHl/eM OF= OC; TOl/Ka N H306pa>KaeT
4HCJIO -2,5, npHl/eM ON=OM, H T. JJ.. TaKHM 06pa30M Ha npHMoH
MO>KHO H306pa3HTb H nOJIO>KHTeJIbHble, H oTpHLlaTeJIbHble 4HCJIa, H
l/HCJIO O. nOJIYl/eHHYIO npHMYIO Ha3b1BaIOT IJHCJlOBOH npHMoH HJIH
IJHCJlOBOH OCblO. HanpaBJIeHHe JIY4a OA (pHC. 5.5), Ha KOTOpOM H30-
6pa>KeHbl nOJIO>KHTeJIbHble l/HCJIa, Ha3b1BaeTCH nOJlmKHTeJlbHblM Ha-
npaBJleHHeM, a npoTHBOnOJIO>KHOe HanpaBJIeHHe - oTpHUaTeJlbHblM
HanpaBJleHHeM Ha l/HCJIOBOH OCH. nOJIO>KHTeJIbHOe HanpaBJIeHHe Ha
l/HCJIOBOH OCH (06bll/HO CJIeBa HanpaBo HJIH CHH3Y BBepx, pHC. 5.6)
YKa3blBaeTcH CTpeJIKoH.
4HCJlOBaH OCb - 3TO npHMaH, Ha KOTOpOH: I) Bbl6paHO Ha4aJIO
OTC4eTa; 2) Bbl6paH eJJ.HHHl/HbIH oTpe30K; 3) YKa3aHo (06bll/HO)
nOJIO>KHTeJIbHOe HanpaBJIeHHe.
4HCJIO, KOTOPOMY COOTBeTCTByeT JJ.aHHaH T04Ka Ha 4HCJIOBOH
OCH, Ha3blBaeTCH KOOpJl.HHaTOH 3TOH TOl/KH, n03TOMY l/HCJIOBYIO OCb
Ha3blBalOT elll,e KOOpJl.HHaTHOH OCblO HJIH KoopJJ.HHaTHOH npHMOH. ECJIH
x - KoopJJ.HHaTa TOl/KH P, TO 3anHCblBalOT P (x) (l/HTaeTCH: TO'iIW P
C KOOpaUHaTOU x). Ha pHcYHKe 5.5 OTMe4eHbl TOl/KH 0 (0), D (-I),
B (2), M (2,5), N (- 2,5) H JJ.p. ECJIH H3BeCTHa KoopJJ.HHaTa TOl/KH, TO
MO>KHO OTMeTHTb 3TY TOl/KY Ha KoopJJ.HHaTHOH OCH.
A.
706. 06'bHCHH, nOJIb3YHCb TeKCTOM:
I) KaK H306pa>KaIOTCH nOJIO>KHTeJIbHble H oTpHLlaTeJIbHble
l/HCJIa;
131
C B H E G
. . - T .
2 -1 0
0
A C E B
I . I . I - -
30 20 -10 o 10
0)
F D G
I . - J . I
300 -200 -100 0 100
(2)
PHC. 5.7
A D F
· · I. J
2
...
D
J .
20 30
C E
. I .
200 300
.
..
707.
2) LITO TaKOe
UCAOBaR OCb;
3) lITo TaKoe KOopOUHaTa TO
KU;
4) LITO 03Hal/a1OT 3anHCH A ( -5 ), B (78),
C (-100,3),
D( -
),
r.ll.e Ha l/HCJIOBOH OCH pacnOJIO>KeHbl CJIeJJ.YIOUlHe TOl/KH:
A (8), B ( - 9), C (0), D ( - 567), E (222,2), F ( - 19,8)?
3anHwH KOOpJJ.HHaTbi TOl/eK, 0603Ha4eHHblx 6 Y KBaMH Ha
cYHKe 5.7. PH-
Hal/epTH l/HCJIOBYIO OCb, npHHHB 3a eJJ.HHHl/HbIH OTpe30K I CM
OTMeTb Ha HeH TOl/KH K (- 2), L (3,5), N (0), F (- 4,5), S (0,5):
Hal/epTH l/HCJIOBYIO OCb, npHHHB 3a eJJ.HHHl/HbIH oTpe30K 5 CM.
OTMeTb Ha HeH T04KH A (-I), B (
), C (-0,9), D ( 1
) ,
F (0,2), G (
) , H ( - 0,3).
708.
709.
710.
711.
6.
Hal/epTH l/HCJIOB Y IO OCb , OTM e T b Ha U
HeH TOl/KY A (- 2) H TOl/KH
B, C, D H E, eCJIH:
I) TOl/Ka B YJJ.aJIeHa OT TOl/KH A B OTpHLlaTeJIbHOM HanpaB-
JIeHHH Ha 2 eJJ.HHHLlbl;
2) TOl/Ka C YJJ.aJIeHa OT T04KH A B nOJIO>KHTeJIbHOM HanpaBJIeHHH
Ha 3 eJJ.HHHLlbl;
3) T04Ka D YJJ.aJIeHa OT C
TOl/KH B nOJIO>KHTeJIbHOM HanpaB-
JIeHHH Ha 1,5 eJJ.HHHLlbl;
132
4) TOl/Ka E YJJ.aJIeHa OT TOl/KH D B oTpHLlaTeJIbHOM HanpaB-
JIeHHH Ha 3,5 eJJ.HHHLlbI.
3anHwH KoopJJ.HHaTbl T04eK B, C, D H E.
712. Hal/epTH BepTHKaJIbHYIO WKaJIY TepMoMeTpa. OTMeTb Ha HeH 3°
XOJI OJJ. a , 8 0 XOJIoJJ.a, 1 0 TenJIa. 5° X OJIoJJ.a, 4° TenJIa, 7 0 XOJIoJJ.a.
5.3. npOTHBOn0J10>KHble IJHCJla
npOl/TH TeKCT naparpa<t>a, 3aTeM nOBTopH, onHpaHCb Ha 3aJJ.al/Y
713.
PaccMoTpHM, HanpHMep, napbl l/HCeJI
1 1
I H -1, -2 H 2, 2,9 H -2,9, T H -T'
4HCJla B Ka>KJJ.oH H3 TaKHX nap OTJIHl/aIOTCH JJ.Pyr OT JJ.Pyra
TOJIbKO 3HaKOM. 4HCJIa B 3THX napax H306pa>KaIOTCH Ha l/HCJIOBOH
OCH TOl/KaMH, KOTopble pacnOJIO>KeHbl Ha OJJ.HHaKOBOM paCCTOHHHH, HO
B npOTHBOnOJIO>KHbIX HanpaBJIeHHHX OT Hal/aJIa OTCl/eTa (pHC. 5.8).
TaKHe l/HCJIa Ha3b1BaIOTCH npOTHBOnOJlO)l(HblMH IIHCJlaMH. 4HCJIO I
npOTHBOnOJIO>KHO l/HCJIY - I, H, Ha060poT, l/HCJIO - I npoTHBono-
JIO>KHO l/HCJIY I. npOTHBOnOJIO>KHbIMH HBJIHIOTCH TaK>Ke l/HCJIa - 2 H 2,
1 1
2,9 H -2,9. 3 H -3' -10 H 10 H T, JJ.. 4HCJlO 0 Cl/HTaeTCH npOTH-
BOnOJIO>KHbIM caMOMY ce6e,
HaTypaJIbHble l/HCJla (l. 2, 3, 4, ...), l/HCJIa, HM npoTHBOnOJIO>K-
Hble (- I, - 2, - 3, - 4, ..,), H 4HCJIO 0 Ha3blBalOTCH LleJlblMH l/HC-
JlaMH. UeJIbie l/HCJIa, oTpHLlaTeJIbHble H nOJIO>KHTeJIbHble JJ.p06Hble
l/HCJIa Ha3b1BaIOTCH pau.HOHaJIbHblMH l/HCJlaMH.
ECJIH 0603Hal/HTb 6YKBOH a JII060e paLlHOHaJIbHOe 4HCJIO, TO
l/HCJIO, npOTHBOnOJIO>KHOe eMY, 0603Hal/aeTCH - a. HanpHMep, eCJlH
a=3, TO -a= -3; eCJIH a= -5, TO -a= -( -5)=5 (TaK KaK
l/HCJlO, npoTHBOnOJIO>KHOe l/HCJlY -5, paBHo 5). TaKHM >Ke 06pa30M
-( -1,8)= 1,8; -0=0 H T. JJ.. HTaK, eCJIH a - OTpHLlaTeJIbHOe 4HCJIO,
TO - a eCTb nOJIO>KHTeJIbHOe; eCJIH a - l/HCJIO nOJlO>KHTeJIbHOe, TO
- a - l/HCJIO OTpHLlaTeJIbHoe.
-29
....r .
-3 -2
2,9
.1..1...-.
2 3
. . . .
-1 1 0 1
-3" 3"
PHC. 5.8
133
A.
713. I) npOl/TH BHHMaTeJIbHO a63aLl TeKCTa, KOTOpblH Hal/HHaeTCH co
CJIOB: «PaCCMOT
HM, HanpHMep...Ji>. Ha30BH JJ.PyrHe l/HCJIa, KOTO-
pble Ha l/HCJIOBOH OCH OJJ.HHaKOBO YJJ.aJIeHbl OT Hal/aJIa OTCl/eTa H
paCnOJIO>KeHbl B npOTHBOnOJIO>KHbIX HanpaBJIeHHHX. KaK Ha3bl-
BalOTCH 3TH l/HCJIa?
2) npOl/TH CJIeJJ.YIOUJ,HH a("i3aLl TeKCTa. npHBe.lI.H npHMepbl LleJIblX
l/HCeJI, oTpHLlaTeJIbHblX H nOJIO>KHTeJIbHbIX .lI.P06HbIX l/HCeJI.
3) OPOl/TH TeKCT .lI.0 KOHLla:
a) 06'bHCHH, n04eMY -(-7)=7; -(-3,8)=3,8; -0=0.
6) HaHJJ.H -x, eCJIH x=4,7; x= 125' x= -II. x= -0 2
B) HaH.lI.H l/HCJIO, npOTHBOnOJIO>KHOe
aHHOMY: ' , .
5
-5; 8; T; -0,78; 0; -968; 0,01; -0,001.
714. KaKHe H3 paBeHcTB BepHble:
I) -(+2)= -2 2) -7= -(+7)
-(-5)=-5 +0,2=-(-0,2)
=-(-
) -1+=-(-1 : ) -0=+0
715. npOl/TH, 3aMeHHH MHOrOTOl/He CJIOBaMH «paBHOJi> HJIH «He paB-
HOJi>, l/T06bl nOJIYl/eHHOe npeJJ.JIO>KeHHe 6blJIO BepHblM:
I) -(-15)...15 2) -1,4...-(-1,4) 3) -(+1)...1
6...-(-6) -(+0,3)..,-0,3 -8...-(+8)
716. Ha30BH OTBeTbl:
I) -(+12) 2) -(-32) 3) -(-5,9) 4) -(+97)
-( -3,8) -( -4,7) -( +0,06) -(-100)
717. HaHJJ.H 3Ha4eHHH Bblpa>KeHHH - m, eCJIH m= I; m=0,7;
m=2 3 . m - O ' m - 5 . m - 4
" -, - - I - -ge
718. HanHwH l/HCJIa, npoTHBOnOJIO>KHble Ka>KJJ.OMY H3 CJIeJJ.YIOIlI,Hx
tIHCeJI, H OTMeTb HX Ha l/HCJIOBOH OCH: - 2; 4; - 2,5; 1,5; O.
719. 3anHllIH l/HCJIa, npoTHBOnOJIO>KHble H 06paTHble Ka>KJJ.OMY H3
CJIeJJ.YIOIlI,HX 4HCeJI: 3' J...,.
. 0 I . 7
, 2' 3' " .
3) -( -4)=4
- ( + 10) = - 10
6.
720. 3anHllIH KoopJJ.HHaTbl TOl/eK A, B , C H D ( 5 9)
pHC.. H l/HCJIa,
A C B D
-3 -2 -1 0 I.
2 3 4
PHC. 5.9
134
npOTHBOnOJIO>KHble 3THM KoopJJ.HHaTaM.
721. KaKoe l/HCJIO -k, eCJIH:
I) k - nOJIO>KHTeJIbHOe l/HCJIO;
2) k = 0;
3) k - oTpHLlaTeJIbHOe l/HCJIO?
722. .llaHbl T04KH M (-3) H N (2,5). 3anHwH KoopJJ.HHaTbl TOl/eK
A, B H C:
I) eCJIH KoopJJ.HHaTbl TOl/eK A H M H KoopJJ.HHaTbl TOl/eK B
H N - npoTHBOnOJIO>KHble l/HCJIa;
2) eCJIH TOl/Ka C pacnOJIO>KeHa JIeBee TOl/KH B H YJJ.aJIeHa
OT Hee Ha 4 eJJ.HHHLlbI.
3 -I
'
723. 3anHwH 4HCJIa, npoTHBOnOJIO>KHble 4HCJIaM -; 2 '
-2,5; -0,5; -0,1. 3anHwH l/HCJIa, 06paTHble nOJIY4eHHblM
l/HCJIaM.
724. BblJJ.eJIH H3 CJIeJJ.YIOIlI,HX paLlHOHaJIbHblX l/HCeJI: I) LleJIble nOJIO-
>KHTeJIbHble 4HCJIa; 2) LleJIble oTpHLlaTeJIbHble l/HCJIa; 3) LleJIble
4HCJIa; 4) JJ.p06Hble nOJIO>KHTeJIbHble l/HCJIa; 5) JJ.p06Hble oTpHLla-
TeJIbHble 4HCJIa; 6) JJ.p06Hble l/HCJIa:
5 3 4 9' 12 1
-3; II; -4' 8; 15; -2; 0; \3; 12; 7; -25; -"9; 43;
9
25; -60,
725*. 06'bHCHH, KaKHe H3 CJIeJJ.YIOIlI,HX npeJJ.J10>KeHHH BepHbl:
I. Ka>KJJ.oe 4HCJIO paBHo cBoeMY npoTHBOnOJIO>KHoMY l/HCJIY.
2. HaHJJ.eTcH 4HCJIO, KOTopoe npoTHBOnOJIO>KHO caMoMY ce6e.
3. HH OJJ.HO l/HCJIO He paBHO CBoeMY npOTHBOnOJIO>KHOMY 4HCJIY.
4. ECJIH a= -b, TO b= -a.
5. ECJIH a= -b H b=c, TO a=c.
726*. HaHJJ.H 3Ha4eHHH 6YKB, TaKHe, 4T06bl paBeHcTBo 6blJIO BepHblM:
I) -x=-7 2) -x=9,2 3) -x=-(+6)
-m=-(+4) -y=-10 -s=7,4
-a= -1,5 -( -n)= I -1= -9.1
5.4. MO.Zl.YJ1b IIHCJ1a
.llJIHHa OTpe3Ka HHa4e Ha3blBaeTCH paCCTOHHHeM Me>KJJ.Y ero KOH-
LlaMH. PaCCTOHHHe Bblpa>KaeTCH BcerJJ.a nOJIO>KHTeJIbHbIM l/HCJIOM. Pac-
CTOHHHe OT Ha4aJIa OTCl/eTa JJ.O T04KH, H306pa>KalOllI,eH l/HCJIO, Ha-
3blBaeTCH MO,ll,YJleM HJIH a6COJIIOTHblM 3Hal/eHHeM tIHCJIa. HanpHMep,
135
C B 0 A D 732
I J I I I I I I I .
-4 -3 -2 -1 0 2 3 4
PMC. 5.10
paCCTOHHHe OT Ha4aJIa OTC4eTa JJ.O T04KH A (2) Ha l/HCJIOBOH OCH paBHO
2 eJJ.HHHLlaM (pHC. 5.10), n03TOMY MOJJ.YJIeM l/HCJIa 2 HBJIHeTCH caMO
l/HCJlO 2. TOl/Ka B (- 2) YJJ.aJIeHa OT Hal/aJIa OTCl/eTa Ha 2 eJJ.HHHLlbl;
CJIeJJ.OBaTeJIbHO, MOJJ.YJIeM l/HCJIa - 2 HBJIHeTCH TaK>Ke l/HCJIO 2, TO
eCTb l/HCJIO, npOTHBOnOJIO>KHOe tIHCJIY - 2. AHaJIOrHl/HO nOJIYl/HM:
MOJJ.YJIb 4HCJIa - 3,5 paBeH 3,5; MOJJ.YJIb l/HCJIa 4 paBeH 4 H T. JJ.. Mo-
JJ.YJIb l/HCJIa 0 eCTb caMO l/HCJIO O.
I-haK, MOaYlleM nOIlO3ICUTellbHOZO 'IUClla U 'IUClla 0 1l8/I.ReTCIl ca-
MO 'IUCIlO, a MOaYlleM oTpuqaTellbHOzo 'IUClla - npoTu80noll03ICHoe
eMY 'IUCIlO.
MOJJ.YJIb l/HCJIa a 0603Hal/aeTCH TaK: I a I, HanpHMep, 1 - 21 = 2;
12/=2; 1-3,5/=3,5; 101=0; I : 1=
; I
1=
H T. JJ.,
A.
727. npOl/TH TeKCT naparpa<t>a H CKa>KH:
I) Ha KaKOM paccToHHHH OT Hal/aJIa OTCl/eTa HaXOJJ.HTCH TOl/KH
C (- 7) H D (4,5);
2) tIeMY paBHbl MOaYAU 4HCeJI 9 H - 5;
3) 4eMY paBeH: 1-121; 12,61 H 101.
728. OnpeJJ.eJIH paccToHHHe OT Hal/aJIa OTCtIeTa JJ.O TOl/eK A, B, C H D
(pHC. 5.1I).
729. OnpeJJ.eJIH paccToHHHe OT Hal/aJIa OTC4eTa JJ.O TOl/eK M (-5,5),
N (6,9), P ( - 15) H R (l00) Ha l/HCJIOBOH OCH.
730. HaHJJ.H MOJJ.YJIH CJIeJJ.YIOIlI,Hx l/HCeJI (oTBeTbl 3anHWH B BHJJ.e pa-
BeHcTBa) :
257
-5; 8; -0,7; 0; "3; 6; 3,9; -1000; -1"8; -0,01.
3 K 9 2.
7 1. aKHe 4HCJIa HMelOT MOJJ.YJIb, paBHblH 5; 6; 57; 0; 10; IT'
- I; - IO?
A C B D
I I I I.
-4 -3 -2 -1 0 2 3
PMC. 5.11
136
TYPHCTbI nocTpoeHbI Ha JIHHellKY
)1JUI nOJJ.IDI.TH
IlJJIara no 06e CTO-
pOHbI OT Hero. PaCCTO
HHe OT
IlJJIara JJ.O nepBoro TYPHCTa
(H cnpaBa H CJIeBa) paBHO 2 M,
Me)K,Il,Y Ka)K,Il,bIM CJIeJJ.ylOIIJ;HM (H
cnpaBa H CJIeBa) paBHO 0,5 M.
1) YKa)KH pacnOJIO)KeHHe TYPHC-
TOB Ha KooPJJ.HHaTHoll OCH, eCJIH
<i)JIar HaXOJJ.HTCSI: B T04Ke Haqa-
JIa OTCqeTa.
2) OnpeJJ.eJIH KoopJJ.HHaTbI ToqeK,
B KOTOpbIX HaXOMTC5I TYPHC-
TbI BTOpOll CJIeBa H BTopoll cnpa-
Ba OT IlJJIara; Hall)J.H pac-
CTO
HH
OT 3THX TypHCTOB JJ.O
IlJJIara.
3) HaiI,II,H MOJJ.YJIH KoopJJ.HHaT TOqeK, B KOTOpbIX HaX0)151TC
TYPHCTbI TpeTHll CJIeBa H IDITbIll cnpaBa OT <i)JIara,
4) 4TO elll,e MO>KHO onpeJJ.eJIHTb no YCJIOBHIO 3aJJ.a4H?
..' '.cr-or.'
(
e'
.
dr-,-
A I-:::='
,-'
'. ,-
ti I/. r;'.
<-!ccc
733. Bbll/HCJIH:
I) 1-71+3.6
1-601: 12-4
2) 5.101 +8
3) \-3
I ++
4+- I - I
I
eCJIH a= -4;
1101 + 75:5
Bblpa>KeHHH lal-I,5,
734.
Bbl4HCJIH 3Ha4eHHe
a= -10,1; a=2.
Bbll/HCJIH 3Ha4eHHe Bblpa>KeHHH
x=3,6,
Ixl :4, eCJIH x= -0,84; x=O;
735.
6.
736.
HaHJJ.H 3Ha4eHHH 6YKB TaKHe. l/T06bl paBeHcTBo 6blJIO BepHblM:
I) Ix I =3 2) I-xl =5 3) It I =0 4) 3 Ixl =3
Iml=1 l-zl=1 lal=-7 Izl-I=O
BbI4HCJlH:
I) 1-101+1-101-121.1- 5 1; 2)
3) 1-201+2.1-101-1- 12 1; 4)
Bbl4HCJIH 3Hal/eHHe Bblpa>KeHHH
a= -345,6 H b= -100,37.
H3BeCTHo, 4TO Ixl = Iyl. MO>KHO JIH CKa3aTb, LITO x=y?
nOl/eMY?
737.
1-51-131+3.1-11;
131 + 1-211 :3-1-91.
lal-lbl+2Ial, eCJIH
738.
739.
137
740. H3BecTHo, LITO S = - t. MO>KHO JIH CKa3aTb, 4TO 1 S 1 = It I?
nOl/eMY?
5.5. CpaBHeHHe IfHCeJ1
Mbl YMeeM cpaBHHBaTb nOJIO>KHTeJIbHble l/HCJIa. HanpHMep, 3 < 7,
10 5< 10 6 2 3 4 4 5 6
, , , ""'5<""'5' "'7>9' ""6<"'7 (CM.
2.5). 3HaeM TaK>Ke,
LITO l/HCJIO 0 MeHbllle JII060ro nOJIO>KHTeJIbHOro l/HCJIa. Tenepb BbIHC-
HHM, KaK CpaBHHTb tIHCJIa, eCJIH CpeJJ.H HHX HMelOTCH H OTpHLlaTeJIb-
Hble. CpaBHHM, HanpHMep, nOKa3aHHH TepMoMeTpa:
_50 H 3 0 , -10 0 H 7°, _1 0 H 0 0 , _7 0 H 0 0 .
TaK KaK TeMnepaTypa - 50 H H >K e TeMnepaTypbl 3°, - 10 0 H H >K e
7 0 I 0 0 0 7 0 0
,- H H >K e H - H H >K eO, TO C4HTaeM, tITO tIHCJIO -5
Me H b W e l/HCJIa 3, -10 Me H b W e 7, -I Me H b W e 0 -7
Me H b W e O. 3anHllleM: '
-5<3, -10<7, -1<0, -7<0,
3Hal/HT, 1l1060e oTpuqaTellbHoe 'lUCIlO MeHbwe HYllll U MeHbwe 111060-
20 no1l0
uTellbH020 'lUClla.
CpaBHHM OTpHLlaTeJIbHble nOKa3aHHH TepMOMeTpa, HanpHMep:
_10 0 H _5°, _2 0 H _7 0 , _3° H _1°.
BHJJ.HM, LITO TeMnepaTypa _10 0 H H >K e _50 _7° H u '" e
3 0 ' rl In _2 0 ,
- H H >K e - i 0. n03TOMY Cl/HTaeM TaK>Ke, LITO
-10< -5, -7< -2, -3<-1.
3aMeTHM TaK>Ke, LITO MOJJ.YJIb MeHbwero l/HCJIa 60JIbWe l/eM
60JIbWerO l/HCJIa: ' MOJJ.YJIb
1-101>1-51,1-71>1-21,1-31>1-11.
n03TOMY MO>KHO CKa3aTb, LITO U3 aBYX oTpuqaTellbHblX 'lUcell MeHb-
we TO, MOaYllb ICOTOp020 60llbwe.
ECJIH l/HCJIa OTMeTHM Ha 4HCJIOBOH OCH, TO YBHJJ.HM, LITO H3
JJ.BYX l/HCeJI MeHbwe TO, H306pa>KeHHe KOToporo pacnOJIO>KeHO JIeBee
(npH nOJlO>KHTeJIbHOM HanpaBJIeHHH CJIeBa HanpaBO). HanpHMep Ha
pHcYHKe 5.12 TOl/Ka A ( - 3) pacnOJIO>KeHa JIeBee TOl/KH B ( - i ,5),
n03TOMY - 3 < - 1,5, Tenepb npeJJ.J10>KeHHe «a - &lUCAO nOA03ICU-
reAbHoe» MO>KHO 3anHcaTb KOpOTKO B BHJJ.e a> o. A npeJJ.JIO>KeHHe
«a - &lUCAO orpUl{areAbHoe» - B BHJJ.e a < O. ECJIH >Ke l/HCJIO a He
nOJIO>KHTeJIbHOe, TO OHO MO>KeT 6blTb OTpHLlaTeJIbHblM HJIH paBHblM
HYJIIO. TorJJ.a 3anHweM: a
O,
ECJIH l/HCJIO a He OTpHLlaTeJIbHOe, TO OHO MO>KeT 6blTb nOJIO>KH-
TeJIbHblM HJIH paBHblM HYJIIO, 3anHweM: a
O. 3HaK
03Ha-
4aeT «MeHbwe HJIH paBHO», a 3HaK
- «60JIbWe HJIH paBHO».
HanpHMep, B HepaBeHCTBe a
-5 3Hal/eHHe a MO>KeT 6blTb paBHblM
_ 5 HJIH HMeTb JII060e JJ.Pyroe 60JIbWee - 5.
A.
741. BHHMaTeJIbHO npOl/TH TeKCT naparpa<t>a H CKa>KH, KaKoe H3 JJ.BYX
4HCeJI 60JIbllle, eCJIH:
I) OJJ.HO 4HCJIO nOJIO>KHTeJIbHOe H JJ.Pyroe oTpHLlaTeJIbHOe;
2) OJJ.HO l/HCJIO nOJIO>KHTeJIbHOe H JJ.Pyroe paBHo HyJIIO;
3) OJJ.HO 4HCJIO OTpHLlaTeJIbHOe H JJ.pyroe paBHo HYJIIO;
4) 06a 4HCJIa oTpHLlaTeJIbHble.
742. KaKaH H3 JJ.BYX TOtIeK JIeBee JJ.pyroH (nOJIO>KHTeJIbHOe HanpaBJIe-
HHe l/HCJIOBOH OCH CJIeBa HanpaBo):
I) A(-2) H B(I); 2) C(-3,4) H D(O); 3) F(-7) H 0(-3);
4) M (-5,3) H N (-I); 5) K (1,5) H L (3); 6) R (2,5) H S (-4,1)?
743. KaKoe H3 JJ.BYX 4HCeJI 60JIbwe:
I) -4 H -0,5; 2) -2 H -3,8; 3) -18 H -15; 4) -9
H -9,OI?
744. KaKHe H3 CJIeJJ.YIOIlI,HX HepaBeHcTB BepHbl:
I) -10<-1 2) 69,7 >52,4 3) -(-3»5
17 <0 -1,5<3,2 -0,5 >-1
41 > -(-9) 2 < -(-2) -9 <0
745. OTMeTb 4HCJIa Ha l/HCJIOBOH OCH H 3anHWH HX B nopHJJ.Ke B03-
paCTaHHH: -3,8; I; -3,5; 2,3; 3,8; -1.
746. 3anHwH l/HCJIa B nopHJJ.Ke B03paCTaHHH:
I) -8; -10; 0; 5; I; -I; 3; -7; -19; 8; 15;
2) - 1000; 1000; 500; - 500; 0; 900; - 800; 700;
3) -1,4; 3,5; 2,97; -2,99; -3,01; 0; 0,5.
3anHwH l/HCJIa B nopHJJ.Ke y6bIBaHHH:
I) 25; -19; 32; 16; -8; -5; 2; I; 0; -15;
2) -32,7; 57,6; -12,8; 12,9; -\5,7; 3,01; 2,02;
3) -300,01; -300,02; 0,07; -0,08; 0,059; -0,005.
A B 0 C 747.
D
I I I .
-3 -2 -1 0 2 3
PMC. 5.12
138
139
748.
HanHWH BCe LleJIble l/HCJIa, KOTOpble paCnOJIO>KeHbl Ha l/HCJIOBOH
OCH Me>KJJ.Y tIHCJIaMH:
I) -2,5 H 3,8; 2) -20,4 H 1,5; 3) -100,3 H -95,2,
HanHwH weCTb KaK MO>KHO 60JIbWHX nOCJIeJJ.OBaTeJIbHbIX LleJIblX
l/HCeJI, KOTopble MeHbwe l/HCJIa 2,6.
HanHwH weCTb KaK MO>KHO MeHbWHX nOCJIeJJ.OBaTeJIbHbIX Lle-
JIblX l/HCeJI, KOTopble 60JIbWe l/HCJIa - 4,1.
Me>KJJ.Y KaKHMH coceJJ.HHMH LleJIblMH l/HCJIaMH HaXOJJ.HTCH l/HCJIO
-891-) 0 27 ) _ 2 1-)
3' ,. 8 .
3anHllIH npeJJ.JIO>KeHHH B BHJJ.e HepaBeHCTBa:
I) - 5,6 - OTpHLlaTeJIbHOe 4HCJIO;
2) 19 - nOJIO>KHTeJIbHOe l/HCJIO;
3) a - OTpHLlaTeJIbHOe l/HCJIO;
4) b - nOJIO>KHTeJIbHOe l/HCJIO;
5) x - He OTpHLlaTeJIbHOe l/HCJIO;
6) Y - He nOJIO>KHTeJIbHOe 4HCJIO;
7) x He MeHbwe 10;
8) t MeHbwe HJIH paBHO -I.
756.
Ha l/HCJIOBOH OCH (pHC. 5.(5) l/HCJIa a H c npOTHBOnOJIO>KHble.
MO>KHO JIH CKa3aTb:
I) LITO 4HCJIO b nOJIO>KHTeJIbHOe (oTpHLlaTeJIbHoe);
2) KaKoe H3 JJ.aHHbIX 4HCeJI HMeeT HaH60JIbWHH MOJJ.YJIb, KaKoe
HaHMeHbwHH?
HaHJJ.H LleJIOe 3Hal/eHHe x, TaKoe, tIT06bl HepaBeHcTBo 6blJIO
BepHblM:
I) -5<x
0
-1
x
3
Ixl<5
749.
750.
751.
752.
6.
753. 3anHwH B nopHJJ.Ke y6blBaHHH l/HCJIa, npOTHBOnOJIO>KHble JJ.aH-
HbIM:
3,85; -7,8; -0,1; 0,01; 3,7.
Ha l/HCJIOBOH OCH (pHC. 5,13) OTMetIeHbl T04KH, cooTBeTcTBYIO-
lll.He l/HCJIaM C H d. MO>KHO JIH CKa3aTb (n04eMY?):
I) KaKoe H3 JJ.aHHbIX 4HCeJI 60JIbWe'
2) MOJJ.YJIb KaKoro tIHCJIa 60JIbwe? '
Ha l/HCJIOBOH OCH (pHC. 5.14) l/HCJIa m H n npOTHBOnOJIo>t{-
Hble, MO>KHO JIH CKa3aTb:
I) KaKoe H3 4HCeJI 60JIbWe;
2) rJJ.e pacnOJIO>KeHO Hal/aJIO OTCtIeTa Ha 4HCJIOBOH OCH?
754.
755.
757.
2) -4,7<x
I
-3
x<2,4
Ixl
3
3) 102<x< 105
-47<x< -40
Ixl
1
758. HanHwH caMoe MaJIeHbKoe: I) LleJIOe JJ.BY3Hal/HOe 4HCJIO; 2) Lle-
JIoe l/eTblpex3Ha4Hoe 4HCJIO,
759*. HanHwH HaHMeHbwee LleJIOe Tpex3Ha4Hoe 4HCJIO, B KOTOpOM
LlH<t>pa JJ.eCHTKOB 0, a CYMMa LlH<t>P eJJ.HHHLl H COTeH paBHa 10.
760*. H3BecTHo, LITO a H b - nOJIO>KHTeJIbHble, a m H n - OTpHLla-
TeJIbHWe l/HCJIa. KaKoe HepaBeHCTBO BepHoe, KaKoe HeBepHoe?
B KaKOM CJIyl/ae Ha Bonpoc OTBeTHTb HeB03MO>KHO?
1) a>m 2) b>m 3) m>b 4) -m>O
a>O m>n b>a -b >0
b<O a<n O>n 0 >-a
b<n n<O O<b -n <0
J].JUI nOBTopeHHH
761. Bbll/HCJIH (y C THO):
I) 2 (500-350)+200:2-400;
2) (750+ 150):2-150+(100+50).40;
3) 0,25.100+50+2000.0,01-0,02.1000.
762. BbmOJIHH JJ.eHcTBHH:
I) 3+'-h-+(
0 ++):11; 2) :
.6+1 :
' C75 - :0 );
3) (2+-1
):
+(+++ ).+;
4) (0,75:0,03-23+): 1+'
.
HaHJJ.H p% OT 4HCJIa a, eCJIH:
1) p=30 H a=25; 2) p=45 H a=84; 3) p= 102 H a=3,4.
HaHJJ.H 4HCJIO x:
I) 10% KOToporo paBHbl 15; 2) 15% KOToporo paBHbl 4,05;
3) 3,2% KOTOpOro paBHbl 0,128.
I I . m
c d
763.
PHC. 5.13 PMC. 5.14
764.
I I I . n
a b c
PHC. 5.15
140
141
765. YMeHbwH l/HCJIO 120:
I) Ha 2,5%; 2) Ha 15%; 3) Ha 7,5%.
766. YBeJIH4b l/HCJIO 160:
I) Ha 80%; 2) Ha 6,2%; 3) Ha 42%,
767. PewH ypaBHeHHe:
I 5 3 1
I) x-3""6=212 2) 7"""x=l 14
4,75+y= 1,8 x:5,8=4
x 5
3) 0,3 =1:2
3,4:0,03=x:O,6
768. Pa60l/HH H3rOTOBHJI 3a 2,4 l/ 30 JJ.eTaJIeH. CKOJIbKO TaKHX JJ.eTa-
JIeH H3rOTOBHT pa60tIHH 3a 8 l/, eCJIH TeMn pa60Tbl ero OCTaHeT-
CH npe>KHHM?
769*. (C Tap H H H a H 3 a JJ. a l/ a JI. <1>, MarHHLlKoro.) EJJ.HH 4e-
JIOBeK BbInbeT KaJJ.b nHTHH B 14 JJ.HeH, a co >KeHOIO BbInbeT
TOe>Ke KaJJ.b B 10 JJ.HeH, H BeJJ.aTeJIbHO eCTb, B KOJIHKO JJ.HeH
>KeHa ero cnoc06HO BbInbeT TOe>Ke KaJJ.b?
5.6. nepeCelJeHHe npHMblx
npOl/TH BHHMaTeJIbHO TeKCT naparpa<t>a. OOBTOpH, onHpaHCb Ha
3aJJ.al/Y 770.
D.Be npHMble, HMelOllI,He OJJ.HY 061l1,YlO T04KY, Ha3blBalOTCH nepece-
KaJOUJ,HMHCH npHMbIMH, a HX 061l1,aH TOtIKa Ha3blBaeTCH TOIJKOH
nepeCelJeHHH. Ha pHcYHKe 5.16 npHMble S H t nepeCeKalOllI,HeCH,
A - T04Ka HX nepeCel/eHHH.
n pH Me 4 a H He. PaHbwe npHMYIO (H JIY4) Mbl 0603Ha4aJIH JJ.BY-
MH 3arJIaBHblMH JIaTHHCKHMH 6YKBaMH. npHMaH 0603HatIaeTCH TaK>Ke
OJJ.HOH CTp04HOH JIaTHHCKOH 6YKBOH, HanpHMep S, t, u, v, TaKHM
0603Hal/eHHeM Mbl 6YJJ.eM nOJIb30BaTbCH B JJ.aJIbHeHweM.
npH nepeCel/eHHH JJ.BYX npHMblX 06pa3YIOTCH tIeTblpe yrJIa C 061l1,eH
BeplllHHOH, He Cl/HTaH pa3BepHYTb1X. Ha pHcYHKe 5.17 OHH 0603Ha4eHbl
LlH<t>paMH: L I, L2, L3, L4. ECJIH BeJIH4HHa OJJ.Horo yrJIa H3
HHX H3BeCTHa, TO BeJIHl/HHbl OCTaJIbHblX Tpex Bbl4HCJIHTb JIerKO.
PHC. 5.16
PHC. 5.17
142
n pH Me p. nYCTb L I = 50 0 (pHC. 5.17), HaHJJ.eM BeJIH4HHbl
Tpex OCTaJIbHblX yrJIOB.
06paTHM BHHMaHHe Ha TO, 4TO LI, L2 (LI H L4) 06pa3YlOT
pa3BepHYTblH yrOJI, BeJIH4HHa KOToporo paBHa 180 0 . CJIeJJ.oBaTeJIbHO,
L 2 = 180 0 - L I = 180 0 - 50 0 = 130 0 .
J],aJIee HaHJJ.eM
3= 1800 _ L2= 180 0 -130 0 =50 0 H
L 4 = 180 0 - L 3 = 180 0 - 50 0 = 130 0 ,
3aMeTHM, 4TO
LI=L3 H L2=L4.
A.
770.
I) np04TH a63aLl, KOTOpblH Hal/HHaeTCH CJIOBaMH «D.Be npH-
Mble..,Ji> H cJIeJJ.YlOllI,ee 3a HHM npHMe4aHHe. 4eM xapaKTepH-
3YlOTCH nepeCeKalOllI,HeCH npHMble? Hal/epTH JJ.Be nepeCeKalOllI,He-
CH npHMble, 0603Hal/b 3TH npHMble H TOl/Ky HX nepeCel/eHHH.
2) npOl/TH cJIeJJ.YIOIlI,HH a63aLl H paccMoTpH npHMep. n04eMY
L2=180 0 -LI, L3=180 0 -L2, L4=180 0 -L3?
3) 06'bHCHH, nOl/eMY H3 yrJIOB, KOTopble 06pa3YlOTcH npH ne-
pece4eHHH JJ.BYX npHMblx, HMelOTCH JJ.Be napbl paBHblx yrJIOB.
Hal/epTH npHMble a H b, KOTopble nepeceKalOTcH B T04Ke M.
H3Mepb nOJIY4eHHble yrJIbl H Bbll/HCJIH HX CYMMY.
Ha30BH napbl nepeCeKalOIlI,HXCH npHMblx, H306pa>KeHHb1X Ha PH-
cyHKe 5.18, H HX TOl/KH nepece4eHHH.
OJJ.HH H3 yrJIOB, 06pa30BaBwHxcH npH nepeCel/eHHH JJ.BYX npH-
. I ) 149 °' 2) 62° Bbl4HCJIH BeJIH4HHbl OCTaJIbHblX
MbIX, paBeH. , '
yrJIOB.
771.
772.
773.
b D
c
a A
B
d
8
c
o
PHC. 5.18
143
H p/
F
E CD
M
K
PHC. 5.19
774.
775.
Bbll/HCJIH Te yrJIbl ( p HC 5 19 )
. . ,BeJIH4HHbl KOTOpblX He JJ.aHbl.
OnpeJJ.eJIH Ha rJIa3, CKOJIbKO TOl/eK ne p eCel/eHua
rln HMelOT npH-
Mble, H306pa>KeHHble Ha pHcYHKe 5.20.
HatIepTH TPH npHMble TaK, l/T06bl nOJIYtIHJIOCb: I) OJJ.Ha; 2) TPH
TOl/KH nepece4eHHH.
Hal/epTH JJ.Be nepeceKalOlllHecH npHMble, l/T06bl OJJ.HH H3 06pa30-
BaBWHXCH yrJIOB 6bIJI paBeH: I) 430; 2) 900; 3) 115 0 .
776.
777.
c
PHC. 5.20
144
N
p
M
PMC. 5.21
K
PHC.
c
t
PMC. 5.23
PMC. 5.24
6.
778.
Bbll/HCJIH Te yrJIbl (pHC. 5.21-5.22), BeJIH4HHbl KOTOpblX He
JJ.aHbl.
Ha4epTH l/eTblpe npHMble, l/T06bl nOJIYl/HJIOCb: I) OJJ.Ha; 2) l/e-
Tblpe; 3) weCTb T04eK nepeCel/eHHH.
Ha30BH npHMble, KOTopble nepeceKalOTCH (pHC. 5.23).
779.
780.
5.7. nepneH.ll.HKYJ1HpHble npHMble
ECJIH npH nepeCel/eHHH JJ.BYX npHMblX OJJ.HH H3 nOJIY4eHHblX yrJIOB
6YJJ.eT npHMblM (90°), JIerKO y6eJJ.HTbcH, LITO OCTaJIbHble yrJIbl TO>Ke
6YJJ.YT npHMblMH (pHC. 5.24).
llBe npHMble, 06pa3YlOIlI,He npH nepeCel/eHHH npHMble yrJIbl, Ha-
3bIBalOTCH nepneH)l.HKYJUlpHblMH I . npHMble S H t Ha pHcyHKe 5.24
nepneHJJ.HKYJIHpHbI. 3anHweM: s..L t (l/HTaeTcH: npHMble S H t nep-
neH-JJ.HKYJIHpHbl). TaK >Ke MO>KHO 3anHcaTb: t..L S,
4T06bl nOCTpoHTb nepneHJJ.HKYJIHpHble npHMble, MO>KHO HCnOJIb30-
BaTb tIepTe>KHblH yrOJIbHHK. Ha pHcYHKe 5.25 nOKa3aHO, KaK nocTpo-
HTb npHMYIO, KOTopaH npoxoJJ.HT l/epe3 TOl/KY A H nepneHJJ.HKY-
JIHpHa npHMoH s. OKa3b1BaeTcH, LITO lJepe3 )l.aHHYIO TOIJKY MO)KHO
npOBeCTH TOJlbKO O)l.HY npHMYIO, KOTopaH nepneH)l.HKyJlHpHa )l.aHHOH
npRMOH.
ECJIH OTpe3KH JIe>KaT Ha nepneHJJ.HKYJIHpHbIX npHMblX, TO 3TH OT-
pe3KH TaK>Ke Ha3blBalOTCH nepneHJJ.HKYJIHpHbIMH. HanpHMep, CTOpOHbl
KBaJJ.paTa, HMelOlliHe 06111YIO TOl/KY, nepneHJJ.HKYJIHpHbl.
I OT JlaTHHCKoro CJlOBa perpendicular is - oTBecHblii.
145
C Q
fJ A
11
(f
i S
PMC 5.25 PHC. 5.26
s
4epe3 TOtIKy P, He JIe>KaIUYIO Ha npHMOH (pHC. 5.26), npoBe-
JJ.eHbl TPH npHMble a, b H c, KOTopble nepeceKalOT npHMYIO S, npH4eM
npHMaH a nepneHJJ.HKYJIHpHa npHMoH s. OTpe30K PA Ha3blBaeTCH
nepneH.II.HKYJUlpOM K npHMOH s. OKa3b1BaeTcH, tITO nepneH.II.HKYJlHp
KOpOlJe Bcex .II.PyrHx oTpe3KoB, npOBe.ll.eHHblX H3 .II.aHHOH TOIJKH K
npHMoH. Ha pHcYHKe 5.26 oTpe30K PA Kop04e oTpe3KoB PB H PC.
D.JIHHa nepneHJJ.HKYJIHpa, npoBeJJ.eHHOro H3 JJ.aHHOH T04KH Ha npHMYIO,
Ha3blBaeTCH paCCTOHHHeM OT TOIJKH .11.0 npHMoH. 3Hal/HT, paccTOH-
HHeM OT TOl/KH P JJ.O npHMoH S Ha pHcYHKe 5.26 HBJIHeTCH JJ.JIHHa OT-
pe3Ka PA (HO He JJ.JIHHbl OTpe3KOB PB HJIH PC).
A.
781. OpoBepb no TeKCTY. nOMHHWb JIH Tbl:
I) KaKHe npHMble Ha3blBalOTCH nepneH.OuKYARpH.btJlf.U;
2) CKOJIbKO nepneHJJ.HKYJIHpHbIX npHMblx MO>KHO npoBecTH l/epe3
JJ.aHHYIO TOl/Ky K JJ.aHHoH npHMoH;
3) LITO Ha3blBaeTCH paCCTORH.UeJlf. OT TOl/KH JJ.O npHMoH.
782. OnpeJJ.eJIH Ha rJIa3, KaKHe npHMble Ha pHcYHKe 5.27 nepneHJJ.H-
KYJIHpHbl. OpoBepb ce6H, nOJIb3YHCb yrOJIbHHKOM HJIH TpaHcnop-
THpOM.
783. npHBeJJ.H npHMepbl nepneHJJ.HKYJIHpHbIX npHMblx H3 oKPY>KalOllI,eH
Te6H 06cTaHOBKH.
146
b
a
CD
o
o
o
784.
PHC. 5.27
Hal/epTH npHMYIO m H OTMeTb TOl/KY A BHe 3TOH npHMoH.
C nOMOIUblO yrOJIbHHKa npOBeJJ.H npHMYIO n, npOXOJJ.HIlI,YIO 4epe3
TOl/KY A TaK, l/T06bl BbInOJIHHJIOCb YCJIOBHe n.1. m, CKOJIbKO
TaKHX npHMblX MO>KHO npOBecTH l/epe3 TOtIKY A?
)lJIHHa KaKoro OTpe3Ka HBJIHeTCH paccTOHHHeM OT T04KH P JJ.O
npHMOH S (pHC. 5.28)? H3Mepb ero.
Ha4epTH npHMYIO t H OTMeTb T04KY B BHe 3TOH npHMoH. H3Mepb
paccToHHHe OT TOl/KH B JJ.O npHMoH t.
Hal/epTH JJ.Be nepeCeKalOllI,HeCH npHMble a H b. BHe 3THX npH-
MblX OTMeTb T04KY M, H3Mepb paccToHHHH OT TOl/KH M JJ.O
npHMblx a H b.
Hal/epTH npHMYIO a. OTMeTb npH nOMOIUH yrOJIbHHKa TOl/KH K, L
H M, TaKHe, 4T06bl paccTOHHHe OT TOl/KH K JJ.O npHMoH a paBHH-
JIOCb 2 CM, OT T04KH L JJ.O npHMoH a 4,3 CM H OT TOl/KH M JJ.O
npHMoH a 3, I CM.
785.
786.
787.
788.
6.
789.
HatIepTH npHMYIO t H OTMeTb JII06ble JJ.Be TOl/KH. npOBeJJ.H 4e-
pe3 3TH T04KH npHMble,nepneHJJ.HKYJIHpHble npHMOH t. BcerJJ.a JIH
HaHJJ.YTcH JJ.Be TaKHe npHMble?
KaKoe H3 CJIeJJ.YIOIUHx npeJJ.J10>KeHHH
BepHoe, a KaKoe HeT?
I. D.Be nepeceKalOIUHecH npHMble
BcerJJ.a nepneHJJ.HKYJIHpHbl.
2. D.Be nepneHJJ.HKYJIHpHble npHMble
Ha nJIOCKOCTH BcerJJ.a nepeceKalOTcH.
3. HeKoTopble nepeceKalOIUHecH
npHMble nepneHJJ.HKYJIHpHbI.
4. ECJIH npHMaH U nepneHJJ.HKYJIHp-
Ha npHMoH v, TO npHMaH v
nepneHJJ.HKYJIHpHa npHMOH U. P
5. ECJIH a.1.b H b.l.c, TO ai-c. PHC. 5.28
790:
147
N
t
K
0 c
b
PHC. 5.29 PHC. 5.30
791. Ha pHcYHKe 5,29 s .1./, LAOB= 1000 H LBOC=34 0 .
Bbll/HCJIH LAOD H LDOB,
792. Ha pHcYHKe 5.30 KM .1. LN, L POM + L LOR = 75 0 H
LKOR=58°. Bbll/HCJIH LPOM H LLOP.
5.8. napanJ1eJ1bHble npHMble
llBe npHMble Ha nJIOCKOCTH MorYT He nepeceKaTbCH, TaKHe npH-
Mble Ha3blBalOTCH napaJlJleJlbHblMH I . l-1TaK, aBe HenepeceICalOU4ueCR
npRMble. lle3fCaU4ue B OaHOU U TOU. 3fCe n/lOCICOCTU. Ha3blBa1OTCR
napUllllellbHblMU.
/ ,,;
'f
... r
--::,
npeJJ.CTaBJIeHHe 0 napaJIJIeJIbHblX npHMblX JJ.aeT HaM, HanpHMep,
JJ.JIHHHbIH npHMOH Yl/aCTOK >KeJIe3HOJJ.opO>KHbIX peJIbCOB. ECJIH npHMble
S H t napaJIJIeJIbHbl (pHC. 5.31), TO 3anHlllYT: S 1/ t (4HTaeTCH: npHMble
S H / napaJIJIeJIbHbl) HJIH /1/ S.
npHMble b H C Ha pHcYHKe 5.32 nepneHJJ.HKYJIHpHbl K OJJ.HOH H
TOH >Ke npHMoH a. npHMble b H C He MorYT nepeceKaTbCH B Ka-
KOH-JIH60 T04Ke. lleHcTBHTeJIbHO, B 3TOM CJIYl/ae nOJIYl/HJIOCb 6bl, 4TO
l/epe3 T04KY HX nepece4eHHH npOBeJJ.eHbl JJ.Ba nepneHJJ.HKYJIHpa K
npHMOH a. A 3TO HeB03MO>KHO (
5.7). 3Hal/HT, eCJlH JJ.Be npRMbie B
nJlOCKOCTH nepneHJJ.HKYJlRpHbI TpeTbeii npRMoii, TO OHH napaJlJleJlbHbI.
ECJIH OTpe3KH JIe>KaT Ha napaJIJIeJIbHblX npHMblX, TO 3TH OTpe3KH
I OT rpelJeCKOrO CJlOBa parallelos - pHJlOM Hl!YIUHii.
148
c
PHC. 5.31
PMC. 5.32
Han p HMe p n p OTHBOJIe>KaIUHe
TaK>Ke Ha3blBalOTCH napaJIJIeJIbHblMH. ,
CTOpOHW npHMoyrOJIbHHKa napaJIJIeJIbHW.
llJIH nOCTpOeHHH napaJIJIeJIbHblX npHMblX (OTpe3KOB) HCnOJIb3yeM
JIHHeHKY H yrOJIbHHK (pHC. 5.33). nOCTpOHM, HanpHMep, JJ.Be na-
paJIJIeJIbHble npHMble, OJJ.Ha H3 KOTOpblX npoxOJJ.HT 4epe3 JJ.aHHYlO u TOl/-
KY A. llJIH 3Toro YCTaHaBJIHBaeM yrOJIbHHK OJJ.HHM KpaeM K JIHHeHKe H
npOBOJJ.HM npHMYIO BJJ.OJIb JJ.Pyroro KpaH yrOJIbHHKa. 3aTeM nepeJJ.BHra-
eM BJJ.OJIb JIHHeHKH (JIHHeHKY He JJ.BHraeM) yrOJIbHHK JJ.O T04KH A.
lJ.epe3 3TY TOl/KY npOBOJJ.HM npHMYIO. nOJIYl/eHHble JJ.Be npHMble na-
paJIJIeJIbHW. u
'Iepe3 TO"lKY, Jle)Kalll.YIO BHe npRuMOH, MO
HO npOBeCTH TOJlbKO
OJJ.HY npRMYIO, napaJlJleJlbHYIO JJ.aHHOH npRMOH.
A
o -....--7
r - ----;l' ,,/
I /,,/
v........,,/
,,/
t
.....
I
PMC. 5.33
149
JX:
CD CD
u
CD 0
PMC. 5.34
3TO npeJJ.JIO>KeHHe Bblpa>KaeT OCHOBHOe CBoifcTBO napaJIJIeJIbHblX
npHMblX.
793.
794.
795.
796.
K
I
I
I
I
I
DJ--
B
PMC. 5.35
150
A.
6.
797.
nOCTpOH JJ.Be npHMble b H C, napaJIJIeJIb-
Hble TpeTbeH npHMOH a, MorYT JIH npH-
Mble b H C nepeceKaTbCH? (OTBeT 060c-
HYH, onHpaHCb Ha OCHOBHoe CBOHCTBO
napaJIJIeJIbHblX npHMbIX.)
Ha4epTH 4eTblpe npHMble TaK, l/T06bl:
798.
I) He 6blJIO HH OJJ.HOH TOl/KH nepece4eHHH; 2) nOJIY4HJIHCb TPH
TOl/KH nepeCel/eHHH; 3) nOJIY4HJIOCb nHTb TOl/eK nepeCel/eHHH.
799*. KaKHe pe6pa npHMoyrOJIbHorO napaJIJIeJIenHneJJ.a (pHC. 5.35):
I) napaJIJIeJIbHbl; 2) nepneHJJ.HKYJIHpHbl?
.lI.JlH nOBTopeHHH
800. Ha30BH 4HCJIa, npoTHBOnOJIO>KHble CJIeJJ.YIOIlI,HM:
1
_ 4 8 . 32' -5-; -0,8; 7,3.
'" 7
801. Ha30BH MOJJ.YJIH CJIeJJ.YIOIlI,HX 4HCeJI:
11
51; -0,7; -5,8; \"2; -15.
802.
Ha30BH pe3YJIbTaT:
I) -(-9); 2) -(+1,6); 3) -(-0,04); 4) -(+6,8);
5) -(-23),
HatJepTH l/HCJIOBYIO OCb H OTMeTb Ha HeH TOl/KH A ( - 3,5),
B (1,5), C ( - 0,8), D ( - 2,6), E (2,2).
Bbl4HCJIH (y C THO) :
I ) 50-38 . 2) 4(30+70) ; 3) 1,7+2,5+0,3 . 4) 4,2-0,6
8+4' 130-50 6,5-1,5' 2 (5-4,8) .
BbI4HCJIH:
( 2 1 ) 3. ( I 27 ) I
.
I) 2,5: 0,75'"3-\"2 ; 2) 3].3,5. I IT- 55 '15- 3 '
3) (1i-(15:3
-1O,5:1+'
4 )+ITs)' :3 .
Bb;4HCJIH 3Ha4eHHe Bblpa>KeHHH Ixl-\yl. eCJIH:
2 5
I) x=7,03 H y= 1.93; 2) x="3 H Y= -\"2.
PaCnOJIO>KH 4HCJIa B nopHJJ.Ke B03pacTaHHH:
_ 5 4' 53' -5 , 41; 5,31; -5,39; -6.
" "
5
ABTOM06HJIb, npoexaB 50 KM 3a "6 l/, YBeJIHl/HJI CKOpOCTb
Ha 8 KM . CKOJIbKO KHJIOMeTpOB npoeJJ.eT aBTOM06HJIb 3a 2 4 nOCJIe
lJ
Toro KaK YBeJIHl/HJI CKOpOCTb?
B JJ.
M TOprOBJIH nOCTynHJIO B npoJJ.a>KY 800 naJIbTO. B nepBblH
JJ.eHb npoJJ.aJIH 15% Bcex naJIbTO, a BO BTOpOH 25% TOro, LITO
OCTaJIOCb. CKOJIbKO npoLleHTOB Bcex naJIbTO, nOCTynHBwHx B
npoJJ.a>KY, OCTaJIOCb HenpOJJ.aHHbIMH?
B WKOJIe 1000 Y4alllHXCH. BeCHOH 3aKOH4HJIH WKOJIY 10% Y4a-
lllHXCH, OceHblO 3a Cl/eT nepBOKJIaCCHHKOB l/HCJIO Yl/alllHXCH B
151
npOBepb no TeKCTY, nOMHHWb JIH Tbl:
I) KaKHe npHMble Ha3blBalOTCH napaJlJleJlbHblMU;
2) KaK pacnOJIaralOTCH B nJIOCKOCTH JJ.Be npHMble, KOTopble
nepneHJJ.HKYJIHpHbl K OJJ.HOH H TOH >Ke npHMoH;
3) KaK nOCTpoHTb napaJIJIeJIbHble npHMble;
4) OCHOBHoe CBOHCTBO napaJIJIeJIbHblX npHMblx.
npHBeJJ.H npHMepbl napaJIJIeJIbHblX npHMblX H3 oKPY>KalOllI,eH Te-
6H 06CTaHoBKH.
OnpeJJ.eJIH Ha rJIa3, KaKHe npHMble Ha pHcyHKe 5.34:
I) napaJIJIeJIbHble; 2) nepeCeKalOllI,HecH; 3) nepneHJJ.HKYJIHpHble.
(npH He06xoJJ.HMOCTH HCnOJIb3YH yrOJIbHHK H JIHHeHKY.)
Hal/epTH npHMYIO S H OTMeTb BHe 3TOH npHMOH TOl/Ky A.
N 4epe3 TOl/KY A npoBeJJ.H npHMYIO t nep-
neHJJ.HKYJIHpHO npHMoH S H npHMYIO U
nepneHJJ.HKYJIHpHO npHMoH t. KaK pacno-
JIaraeTCH npHMaH U OTHOCHTeJIbHO npH-
MOH s?
803.
804.
805.
806.
807.
808.
809.
810.
WKOJIe YBeJIHl/HJIOCb Ha 10%. CKOJIbKO Yl/aIll,HXCH Tenepb Y4HT-
CH B WKOJIe?
81 1*. nOCJIe copeBHoBaHHH no npbl>KKaM B BblCOTY MHwa, BOBa, JIe-
HH, BaJIepa H llHMa CpaBHHJIH CBOH pe3YJIbTaTbI. OKa3aJIOCb,
LITO: I) MHwa npblrHYJI Bblwe BOBbl, HO HH>Ke JIeHH; 2) JJ.Ba
MaJIbtIHKa HMeJIH OJJ.HH H TOT >Ke pe3YJIbTaT; 3) BaJIepa, KOTOpblH
B3HJI BblCOTY B 1,3 M, npblrHYJI HH>Ke BOBbl; 4) MHwa npblrHYJI
Ha 20 CM Bblwe BaJIepbl; 5) llHMa npblrHYJI HH>Ke JIeHH Ha
5 CM, HO Bblwe BOBbl Ha 10 CM. KaKYIO BblCOTY B3HJI Ka>KJJ.bIH
H3 MaJIbl/HKOB?
5.9. KOOp.ll.HHaTHaR nJ10CKOCTb
Mbl 3HaeM, LITO nOJIO>KeHHe TOl/KH Ha KoopJJ.HHaTHOH OCH onpeJJ.e-
JIeHO OJJ.HHM l/HCJIOM, KOTopoe Ha3blBaeTCH KoopJJ.HHaTOH 3TOH TOl/KH.
KaK onpeJJ.eJIHTb nOJIO>KeHHe TOl/KH B nJIOCKOCTH? 4T06bl OTBeTHTb
Ha 3TOT Bonpoc, nOJJ.YMaH, KaK Tbl HaHJJ.ewb CBoe MeCTO B KHH03aJIe
HJIH KaK onpeJJ.eJIHWb nOJIO>KeHHe waxMaTHOH <t>HrYPbl Ha waxMaTHOH
JJ.OCKe. KaKHe JJ.aHHble He06xoJJ.HMbl Te6e Jl.JIH 3Toro?
npOBeJJ.eM JJ.Be nepneHJJ.HKYJIHpHble KoopJJ.HHaTHble OCH TaK, KaK
nOKa3aHO Ha pHcYHKe 5.36. fOpH30HTaJIbHYIO OCb Ha3blBalOT OCblO
a6ClI,HCC H 0603Hal/a1OT 6YKBOH x, BepTHKaJIbHYIO OCb Ha3blBalOT
OCblO OPAHHaT H 0603Hal/a1OT 6YKBOH y. TOl/KY nepeCel/eHHH oceH
Ha3bIBalOT HalJaJlOM KOOPAHHaT. OCb a6CLlHCC H opJJ.HHaT BMeCTe 06pa-
3YIOT npHMoyrOJlbHYIO cHCTeMY KOOPAHHaT. nJIOCKOCTb, Ha KOTOpOH
HMeeTCH CHCTeMa KoopJJ.HHaT, Ha3blBaeTCH KoopJJ.HHaTHOH nJIOCKOcTblO.
Ha KoopJJ.HHaTHOH nJIOCKOCTH OTMeTHM TOl/KY M (pHC. 5.37).
npoBeJJ.eM H3 3TOH TOl/KH nepneHJJ.HKYJIHpbl Ha OCH KoopJJ.HHaT. Ha
OCH a6cLllfcc nOJIYl/HM T04Ky A C KoopJJ.HHaToH - 2, Ha OCH
0pJJ.HHaT TOl/KY B C KoopJJ.HHaToH 3, 4HCJIO - 2 Ha3blBaeTCH a6cLlHc-
coif, H 4HCJIO 3.- oPAHHaToif TOl/KH M.
A6cLlHcca H opJJ.HHaTa BMeCTe Ha3blBalOTCH KOOPAHHaTaMH TOIJKH.
152
y
y C
4
M
3 B
2
1
A 0
-3 -2 1 0 1 2 3 4 x
E 1 -1
-2 D
-3
6
5
4
3
2
1
-7-6 -5 -4-3 -2 -1 0
-1
-2
-3
-4
-5
-6
1234567
x
PHC. 5.36
PHC. 5.37
KoopJJ.HHaTbl T04KH 3anHCblBalOTCH B cK06Kax, HanpHMep M ( - 2; 3)
a 2 H 3 ) n p H 3anHCH KOOPJJ.H-
(4HTaeTcH: TO'iKa M C KOOp UHaTaMU- . u
HaT T04KH He06xoJJ.HMO CTporo C06JIIOJJ.aTb CJIeJJ.YIOIlI,HH nopHJJ.OK: a6c-
lI,HCCa BcerAa nHweTCH Ha nepBOM MeCTe, a OPAHHaTa - Ha BTOpOM.
n03TOMY rOBopHT, LITO KoopJJ.HHaTbl T04KH 06pa3YIOT ynoPH
O ( ;
: ) -
HYIO napy l/HCeJI. Ha pHcYHKe 5.37 OTMel/eHbl erne T04KH "
D (2' - 2) E ( - 3; - I) H 0 (0; 0).
ECJIH T
4Ka JIe>KHT Ha OCH a6CLlHCC, TO ee opJJ.HHaTa paBHa HY-
JIIO' eCJIH TOl/Ka JIe>KHT Ha OCH opJJ.HHaT, TO ee a6CLlHcca paBHa H 2
JI O IO ) ,
, np u M e p A H B 3anHweM: A (-, ,
TorJJ.a KoopJJ.HHaTbl T04eK, Ha rl ,
B (0; 3). u na p ar p a
Tenepb Tbl MO>KeWb OTBeTHTb Ha nOCTaBJIeHHblH B Ha4aJIe -
<t>a Bonpoc: l/T06bl onpeJJ.eJIHTb nOJIO>KeHHe JII060H T04KH Ha nJIOC-
KOCTH, Hy>KHO 3HaTb ee KoopJJ.HHaTbI.
OCH KoopJJ.HHaT pa36HBalOT nJIOC-
KOCTb Ha 4eTblpe 4aCTH, KOTopbIe Ha3bl-
BalOTCH KooPAHHaTHblMH lJeTBepTHMH.
HYMepaLlHH l/eTBepTeH H 3HaKH KOOp-
JJ.HHaT no 4eTBepTHM nOKa3aHbI Ha PH-
CYHKe 5.38. 3TO HY>KHO 3anOMHHTb.
y
II
'I8TB8pTb
(-; +)
I
'I8TBBpTb
(+; +)
o
x
812.
A.
npoBepb no TeKcry, nOMHHWb JIH
Tbl:
I) . KaK Ha3bIBalOTCH nepneHJJ.HKY-
JIHpHble OCH KoopJJ.HHaT;
PHC. 5.38
III
'I8TB8pTb
(-;-)
IV
'I8TB8pTb
(+; -I
153
v I
A- e--
- f-C
x
B
CeBep
A -I- B
3anaA E 1 -1 BOCTOK -
0
-
D C
lOr
PMC. 5.39
PHC. 5.40
2) KaK onpe
eJIHeTCH nOJIO)KeHHe T04KH Ha nJIOCKOCTH;
3) KaK 3anHCblBalOTCH Koop
HHaTbl T04KH;
4) KaK pa3JJ.eJIeHa Koop
HHaTHaH nJIOCKOCTb Ha 4eTBepTH.
813. Ha pHCYHKe 5.39 X H Y - nepneHJJ.HKYJIHpHble wocceHHble JJ.O-
porH, WTPHXH - KHJIOMeTpOBble CTOJI6bl. Ha KaKOM paccTOHHHH
OT wocce X H OT wocce Y CTOHT JJ.OM, 0603Ha4eHHblH T04KOH A?
TOl/KOH B? T04KOH C? T04KOH D?
814. TYPHCTbI pa3pa60nUIH MaprnpYT rroxo,J:(a (pHC. 5.40). CTapT H
4>HHHW B T04Ke 0, OCTaHOBKH B T04Kax A, B, C, D H E, Oxa-
paKTepH3YH Y4aCTKH MapwpYTa no HX JJ.JIHHe H no HanpaBJIe-
HHIO, eCJIH Ka>KJJ.OMY JJ.eJIeHHIO cooTBeTcTByeT I KM.
815. Ha pHcYHKe 5.41 X H Y - onywKH JIeca (3anOBeJJ.HHKa), rJJ.e na-
ceTCH JIOCb, B T04Kax, 0603Ha4eHHblX 6YKBaMH, 6blJIH 3aMe4e-
Hbl CJIeJJ.bl OT ero KonblT,
I) Ha KaKOM paCCTOHHHH OT JIeCHblX onyweK x H Y n06bl-
BaJl JIOCb?
2) 3anHwH KoopJJ.HHaTbl T04eK, B KOTOpblX n06bIBaJI JIOCb.
816. Ha30BH a6cLlHccbl H opJJ.HHaTbl T04eK (pHC. 5.42), 0603Ha4eH-
HblX 6YKBaMH.
817. 3anHwH KoopJJ.HHaTbl T04eK (pHC. 5.43), 0603Ha4eHHblx 6YK-
BaMH. B KaKOH 4eTBepTH HaXOJJ.HTCH Ka>KJJ.aH H3 3THX T04eK?
818. LITO nOKa3blBaeT MOJJ.YJIb a6CLlHCCbl (opJJ.HHaTbl) TOl/KH? LITO no-
Ka3blBaeT 3HaK a6CLlHCCbl (opJJ.HHaTbl) T04KH?
819. KaKoH 3HaK HMeeT a6cLlHcca (opJJ.HHaTa) T04KH, eCJIH T04Ka
HaXOJJ.HTCH: I) B I 4eTBepTH; 2) BO II 4eTBepTH; 3) B III 4eT-
BepTH; 4) B IV l/eTBepTH?
154
.VIKMI
5
5- 4 K
3
2
f- -1 L
R 0
-=E.:4:33?..= 1...:1 _1_2_3_4_5 X(KM
-
p-
M
-"
N
'5
I
y
TS - -3
I 2 __,R
I I
I I
I K
-3 -2 -1 0 2 3' x
I -1 V I
I
!.p - - - -2 I
I
-a I
-4 - ..lM
PHC. 5.41
PMC. 5.42
V G
5
I
4
H
D 3
2 A
K E
1
4 3 2 1 0 1 2 4
6
-6 -
M' x
F
1
S
--
-C :t
3 L B
..
R I I I I
5 rl I I
I I
PHC. 5.43
820. 4TO MO>KHO CKa3aTb 0 KoopJJ.HHaTax T04KH, eCJIH T04Ka JIe>KHT
Ha: I) OCH a6cLlHcc; 2) OCH opJJ.HHaT?
6.
821. HanHWH KoopJJ.HHaTbl BepWHH npHMoyrOJIbHHKa (pHC. 5.44). Bbl-
4HCJIH ero nJIOl.llaJJ.b H nepHMeTp.
155
v
v
D
c
L
"
"
A B
K
PHC. 5.44 PHC. 5.45
V
H A E
F
PHC. 5.46
822. HanHwH KoopJJ.HHaTbl BepwHH KBaJJ.paTa (pHC. 5.45), Bbl4HCJIH
ero nJIOl1.laJJ.b H nepHMeTp.
823. Bbl4HCJIH nJIOlll,aJJ.b H nepHMeTp 4JHrypbl, H306pa>KeHHOH Ha PH-
cYHKe 5.46.
CaMOCTOHTenbHaH pa60Ta 4
T e M a. nOCTpoeHHe TOIfKH no ee KoopJJ.HHaTaM.
113 npeJJ.blJJ.YllI,erO naparpa4>a Tbl Y3HaJI, KaK onpeJJ.eJIHTb
KoopJJ.HHaTbl T04KH Ha KoopJJ.HHaTHoH nJIOCKOCTH. Tenepb pewH 06paT-
HYIO 3aJJ.a4Y: HaY4Hcb CTpOHTb T04KY Ha nJIOCKOCTH, eCJIH H3BeCTHbl
ee KoopJJ.HHaTbl.
1. Ha4epTH cHCTeMY KoopJJ.HHaT. 3a eJJ.HHH4HbIH OTpe30K Ha
06eHx OCHX npHMH OTpe30K JJ.JIHHOH I CM. Ha Ka>KJJ.OH OCH B 060HX
HanpaBJ1eHHHX OTJIO>KH no 5 TaKHX eJJ.HHHLl.
156
2. nOCTpOH Ha KoopJJ.HHaTHoH nJIOCKOCTH T04KY A (2; - 3).
)lJIH 3Toro:
I) Ha OCH a6cLlHcc HaHJJ.H T04KY C a6cLlHccOH 2, 4epe3 Hee
npoBeJJ.H npHMYIO, nepneHJJ.HKyJIHPHYIO OCH a6cLlHcc (oJJ.HoBpeMeHHo
napaJIJIeJIbHYIO OCH opJJ.HHaT, CM.
5,8).
2) Ha OCH opJJ.HHaT HaHJJ.H T04KY C opJJ.HHaToH - 3, 4epe3 Hee
npoBeJJ.H npHMYIO, nepneHJJ.HKYJIHPHYIO OCH opJJ.HHaT (oJJ.HoBpeMeHHo
napaJIJIeJIbHYIO OCH a6cLlHcc).
3) T04KY nepece4eHHH npoBeJJ.eHHblx npHMblx 0603Ha4b 6YKBOH A.
3TO H eCTb HCKOMaH T04Ka, TaK KaK ee a6cLlHcca paBHa 2 H opJJ.HHaTa
paBHa - 3,
3. Ha ::noH >Ke KoopJJ.HHaTHoH nJIOCKOCTH nOCTpoH T04KH
8(1; 4), C(-5; 2), D(-2; -5), £(4; -3), F(O; -4), G(5: 0).
A.
824. B KaKoH KoopJJ.HHaTHoH 4eTBepTH HaXOJJ.HTCH T04Ka A ( - 2; 5);
8(10; 35); C(I; -I); D(-8; -39); £(0,03; -0,001);
F(-1207; 3541)?
825. Ha4epTH cHCTeMY KoopJJ.HHaT. 3a eJJ.HHH4HbIH oTpe30K Ha 06e-
HX OCHX B03bMH oTpe30K JJ.JIHHOH I CM, nOCTpOH T04KH
A(2; 3), 8(-2; 4), C(5; -I), D(-4; -3), £(2; -5),
F (4; 4), G (0: -2), H (3; 0), P (-1,5; -4), M (-4,5; 2,5),
K (3,7; 0,4), N (1,2; -1,2).
826. BbIHCHH:
I) rJJ.e HaXOJJ.HTCH Bce T04KH C a6cLlHccOH, paBHoH 0;
2) rJJ.e HaXOJJ.HTCH Bce T04KH C opJJ.HHaToH, paBHoH 0;
3) KaKHM 0611.lHM CBOHCTBOM 06JIaJJ.a1OT a6cLlHccbl Bcex T04eK,
pacnOJIO>KeHHblX CJIeBa OT OCH opJJ.HHaT;
4) B KaKHX 4eTBepTHx HaXOJJ.HTCH TOl/KH C nOJIO>KHTeJIbHOH a6c-
LlHCCOH;
5) B KaKHX 4eTBepTHx HaXOJJ.HTCH T04KH C oTpHLlaTeJIbHOH op-
JJ.HHaTOH;
6) KaKHM 0611.lHM CBOHCTBOM 06JIaJJ.a1OT opJJ.HHaTbl Bcex T04eK,
JIe>KaI1.lHX Bblwe OCH a6cLlHcc.
827. Ha KoopJJ.HHaTHoH nJIOCKOCTH Ha4epTH npHMYlO, npoxoJJ.HI1.lYIO
4epe3 T04KH C (-4; -2) H D (4; 3), OTMeTb Ha ::nOH npHMoH T04-
KH, a6cLlHccbl KOTOpbIX: -3, - I, 0, 2, 3anHwH KoopJJ.HHaTbl
nOJIY4eHHblx T04eK.
828. Ha KoopJJ.HHaTHoH nJIOCKOCTH Ha4epTH npHMYIO, npoxoJJ.HIlI,YIO 4e-
pe3 T04KH A (- 2; -3) H 8 (2; 4), OTMeTb Ha 3TOH npH-
157
MOH T04KH, 0pJJ.HHaTbl KOTOpbIX: -2, 0, I, 3. 3anHwH KOOPJJ.H-
HaTbl nOJIY4eHHblx T04eK.
829. Ha KoopJJ.HHaTHoH nJIOCKOCTH Ha4epTH npHMYIO, KOTopaH npo-
XOJJ.HT 4epe3 T04KH K(-2; -I) H L(-2; 4), H npHMYIO,
KOTopaH npoxoJJ.HT 4epe3 T04KH A (0; 0) H B (I; - I). 3anHwH
KoopJJ.HHaTbl T04KH nepece4eHHH 3THX npHMblx.
830. Ha KoopJJ.HHaTHoH nJIOCKOCTH Ha4epTH:
I) TpeyrOJIbHHK KLM, eCJIH K (-4; -I), L (2; 0) H M (0; 3);
2) KBaJJ.paT ABCD, eCJIH A(-2; I), B(3; I), C(3; -4) H
D(-2;4);
3) npHMoyrOJIbHHK KLMN, eCJIH K (2; -3), L (2; 4), M (-2; 4)
H N (-2; -3),.
6.
831. LI.aHbl KoopJJ.HHaTbl Tpex BepwHH npHMoyrOJIbHHKa KLMN:
K ( - 1,5; - 2), L ( - 1,5; I) H M (3; I),
I) Ha4epTH 3TOT npHMoyrOJIbHHK.
2) OnpeJJ.eJIH KoopJJ.HHaTbl T04KH N H KoopJJ.HHaTbl cepeJJ.HH
CTOpOH npHMoyrOJIbHHKa.
3) Bbl4HCJIH nJIOl1.laJJ.b H nepHMeTp npHMoyrOJIbHHKa,
832. LI.aHbl KoopJJ.HHaTbl JJ.BYX BepwHH KBaJJ.paT3 ABCD:
A ( - I; - 1,5) H B ( - I; 2).
I) Ha4epTH KBaJJ.paT ABCD (JJ.Ba CJIY4aH).
2) OnpeJJ.eJIH KoopJJ.HHaTbl BepwHH C H D.
3) Bbl4HCJIH nJIOl1.laJJ.b H nepHMeTp KBaJJ.paTa.
833. KaK paCnOJIO>KeHbl Ha KoopJJ.HHaTHoH nJIOCKOCTH T04KH, y KOTO-
pblX:
I) a6cLlHcca paBHa 3; 2) 0pJJ.HHaTa paBHa --2;
3) op
HHaTa paBHa 5; 4) a6cLlHcca paBHa --4?
834. rJJ.e HaXOJJ.HTCH Ha KoopJJ.HHaTHoH nJIOCKOCTH T04Ka P (x; y),
eCJIH:
I) x>O, y>O; 2) x<O, y<O; 3) x>O, y=O; 4) x=O,
y>O; 5) x<O, y>O; 6) x=O, y<O?
835*. rJJ.e pacnOJIO>KeHbl T04KH Ha KoopJJ.HHaTHoH nJIOCKOCTH, KO-
opJJ.HHaTbl KOTOpblX YJJ.oBJIeTBOpHIOT YCJIOBHIO:
I) x=O H Iyl> 10; 2) y::::;;;;O H Ixl> 10;
3) Ixl < I H y>O?
836*. Ha KoopJJ.HHaTHoH nJIOCKOCTH JJ.aHa JIHHHH (pHC. 5.47).
I) HaHJJ.H Ha JIHHHH T04KY, a6cLlHcca KOTOpOH paBHa - 3.
158
y
4
-4 -3 -2 -1 0 1 2
-1
x
-2
PHC. 5.47
2) HaHJJ.H Ha J1HHHH T04KY, opJJ.HHaTa KOTOpOH paBHa 0,
3) npH KaKOM 3Ha4eHHH a6cLlHccbl x opJJ.HHaTa Y HMeeT HaH-
60JIbWee (HaHMeHbwee) 3Ha4eHHe?
)lJUI nOBTopeHHH
837. Bbl4HCJIH (y C THO):
I) 3 4 2) 2 . 5
4.""'5 "3'""6
. 4 3.
2.""'5 9
0,509.100 40.0,8
7 . 3
3) 8'"'4
6 .
7. 3
6,03: 0,3
5 . 5
4) !f'g
8 3
9 4
0,8:0,01
838. BbI4HCJIH:
I ) 2....!....+....!....+....!....
482
-
+....!....
684
1 1
0,2+4+2"
2 ) I....!....-
+
639
333
10+""'5-4
2 1
"3- 0 ,4-""6
313
3) 4+10-8
4 9 1
1""'5-10+ 1 4
3 5
I 25 + 2---
, 4 12
839. Bbl4HCJIH 3Ha4eHHe Bblpa>KeHHH:
I) (646: 19+77):(52.47-2407);
2) 22,5:3,75+208,45+2,5:0,004;
3) (0,39+ 19,52:32+(12,51 +0,99).
): 15;
4) ((7,3-5+):1++1+:0,5), (2+-0,75)
159
840*. TeKcTbl CJleJJ.YIOI1.lHX 3aJJ.a4 He 3aKOH4eHbI. 3anHWH ce6e B TeT-
pa.n.b nOJIHblH TeKCT 3a.n.a4 H peWH HX:
1
I. 24 Y4eHHKa 6 KJIacca 3aHHMalOTCH cnopTOM. "6 H3 HHX XOJJ.HT
Ha JIbl>KaX ...
2. HeKoTopble Y4eHHKH 6 KJIacca HMelOT JJ.OMa JIH60 nTHLl,
JIH60 4epenax. Y Bcex nTHLl H 4epenax BMeCTe 24 JIanKH ...
3. B JJ.OMe >KHBeT 60 4eJIOBeK, KOTopble 06pa3YIOT ceMbH H3
2 4eJIOBeK, H3 3 4eJIOBeK, H3 4 4eJlOBeK H H3 5 l/eJIOBeK.
CeMeH, COCTOHI1.lHX H3 3 4eJIOBeK, B JJ.Ba pa3a 60JIbWe, 4eM
ceMeH, COCTOHIlI,HX H3 5 4eJIOBeK ...
841*. (CTapHHHaH 3a.n.a4a H3 ApMeHHH.) OJJ.HH KY-
neLl npoweJI 4epe3 TPH ropo.n.a, H B3blCKaJIH C Hero B nepBOM
ropoJJ.e nOWJIHHbl nOJIOBHHY H TpeTb HMYl1.leCTBa, H BO BTOpOM
ropo.n.e nOJIOBHHY H TpeTb (c TOro, 4TO Y Hero OCTaJIOCb), H B
TpeTbeM ropoJJ.e CHOBa B3WCKaJIH nOJIOBHHY H TpeTb (c Toro,
4TO Y Hero 6bIJIO), H, KorJJ.a OH npH6blJI JJ.OMOH, Y Hero OCTaJIOCb
II .n.axeKaHOB (JJ.eHe>KHbIX eJJ.HHHLl). 11TaK, Y3HaH, CKOJIbKO
Bcero .n.axeKaHOB 6blJIO BHa4aJIe Y KynLla.
5.10. rpacl>HKH
Ha KoopJJ.HHaTHoH nJIOCKOCTH MO>KHO H306pa>KaTb 3aBHCHMOCTH
Me>K.n.y pa3JIH4HblMH BeJIH4HHaMH. PaccMoTpHM npHMepbI.
n p H M e p I. Te.Mneparypy U3.MepJlJlU 8 rel.f.eHUe CYTOK. [Jo pe-
3YJlbTara.M U3.MepeHUJl nOJlYI.f.UJlU CJleaylOU{YIO ra6Jluu,y (npOTOKOJl
Ha6JllOaeHUU) :
BpeMH CYTOK, 4 0 2 4 6 8 10 12 14 16 18 20 22 24
TeMnepaTypa, °C 5 2 0 -3 -4 -2 2 6 8 5 4 3 3
::ha Ta6JIHLla Bblpa>KaeT 3aBHCHMOCTb Me>KJJ.Y BpeMeHeM H TeMne-
paTYPoH. LI.JIH rpa4>H4ecKoro H306pa>KeHHH 3TOH 3aBHCHMOCTH no-
CTpOHM npHMoyrOJIbHYIO cHcTeMY Koop.n.HHaT. Ha OCH a6cLlHcc 6YJJ.eM
OTKJla.n.bIBaTb 3Ha4eHHe BpeMeHH, a Ha OCH opJJ.HHaT - TeMnepaTY-
pbI. LI.aJIee, nOCTpOHM Ha KoopJJ.HHaTHOH nJIOCKOCTH Bce T04KH, KO-
opJJ.HHaTaMH KOTOpblX HBJIHIOTCH COOTBeTCTBYlOllI,He 4HCJIa H3 JJ.aHHOH
Ta6JIHLlbl (pHC. 5.48). nOJIY4HJIH 13 T04eK. ECJIH 6bl Mbl H3MepHJIH
TeMnepaTYPY 4al1.le, CKa>KeM 4epe3 Ka>KJJ.ble 15 MHH HJIH 5 MHH, TO
160
t:C
8
6
4
4
2
0 2 4 6 8 10 12 14 16 18 20 22 24 t,
-2
4
-4
PHC. 5.48
nOJlY4HJIH 6bl 3Ha4HTeJIbHO 60JIbWe T04eK. TeMnepaTypa H3MeHHeTCH
HenpepblBHO, HO HenpepblBHO H3MepHTb ee HeB03MO>KHO. ECJIH JJ.O-
nYCTHTb, 4TO pe3KHX CKa4KOB TeMnepaTypbl He 6blJIO, Mbl MO>KeM
nOJIY4eHHble T04KH coeJJ.HHHTb HenpepblBHOH JIHHHeH. TaK no-
JIY4HJICH rpa4»HK TeMnepaTYPbl (pHC. 5.49).
60JIee T04HO TaKYIO JIHHHIO MO>KeT Bbl4epTHTb TepMorpa4>-
npH60p, KOTOpblH aBTOMaTH4eCKH 3anHCblBaeT TeMnepaTYPY
(pHC. 5.50).
t
C
8
t 'i
6
4
2
o
16 18 20 22 24
-2
-4
PMC. 5.49
161
c--
PHC. 5.50
nOJIY4eHHblH rpa!}>HK HarJIH.lI.HO onHCblBaeT H3MeHeHHe TeMnepa-
Typbl B Te4eHHe CYTOK. C nOMOl1.lblO 3Toro rpa!}>HKa MO>KHO OTBeTHTb
Ha MHorHe BOnpOCbI. HanpHMep:
I. KaKaH TeMnepaTypa B03JJ.yxa 6blJIa B I 4 JJ.HH, TO eCTb B
13.00?
Ha OCH a6CLlHCC HaHJJ.eM 4HCJIO 13, 3aTeM Ha rpa!}>HKe OTMe-
THM T04KY A, a6cLlHcca KOTOpOH paBHa 13. HaHJJ.eM 0P.lI. HH3T )'
T04KH A. OHa paBHa 4.
l1TaK, B I 4 JJ.HH TeMnepaTypa B03JJ.yxa 6blJIa 4 0 .
2. B KaKoe BpeMH TeMnepaTypa B03JJ.yxa 6blJIa 7°?
Ha OCH opJJ.HHaT HaHJJ.eM l/HCJIO 7. Mbl BHJJ.HM, 4TO TaKYIO op-
JJ.HHaTY Ha rpa!}>HKe HMelOT JJ.Be T04KH: B H C. HaH.lI.eM a6cLlHc-
Cbl 3THX T04eK. OHH npH6JIH3HTeJIbHO paBHbl 15 H 17,
l1TaK, TeMnepaTypa B03JJ.yxa 7 0 6blJIa OKOJIO 15 4 H OKOJIO 17 4.
n p H Me p 2. BeAocuneaUCT eXaA C nOCTOJlHHOU CIWpOCTblO
20 K: . H306pa3U.M. 2patjJuK 3T020 aBUJlCeHUJl,
nYTb, npoHJJ.eHHblH BeJIOCH, neJJ.HCTOM MO>KHO Bbl4HCJIHTb no q>op-
MYJIe 5=201. Tenepb HaHJJ.eM nYTb, KOTOpblH npoexaJI BeJIOCH-
neD,HCT, HanpHMep, 3a I 4, 2 4, 3 4, ... :
BpeMH, 4 I 2 3 4 5
nYTb, KM 20 40 60 80 100
Ha OCH a6cLlHcc 6YJJ.eM OTKJla.lI.bl-
BaTb 3Ha4eHHH BpeMeHH, Ha OCH op-
JJ.HHaT - paccToHHHe. Ha KoopJJ.HHaT-
HOH nJIOCKOCTH OTMeTHM TOl/KH, KOOp-
162
S,KM
100
t,
80
60
40
20
o
2 3
4
5
PHC. 5.51
JJ.HHaTaMH KOTOpblX CJIY>KaT COOTBeTCTBYlOllI,He 4HCJIa H3 Ta6JIHLlbI.
OKa3bIBaeTCH, nOJIY4eHHble T04KH pacnOJIO>KeHbl Ha OJJ.HOH npHMOH
(pHC. 5.51). CoeJJ.HHHM T04KH OTpe3KaMH H nOJIY4HM rpact»HK .lI.BH-
)KeHHH BeJIOCHneJJ.HCTa. KaK H B npeJJ.blJJ.YLUeM npHMepe, no 3TOMY
rpa!}>HKY Mbl MO>KeM pewaTb 3aJJ.a4H JJ.BYX THnOB: HaHTH npoHJJ.eHHblH
nYTb, 3HaH BpeMH, H, Ha060pOT, HaHTH BpeMH, 3HaH npoHJJ.eHHblH nYTb.
A.
842. np04TH npHMep I H 06'hHCHH:
I) KaK nocTpoHTb 2patjJuK Te.M.nepaTypbl;
2) JJ.JIH 4ero MO>KHO HCnOJIb30BaTb rpa!}>HK TeMnepaTypbl.
843. nOCTpOH rpa!}>HK TeMnepaTypbl no Ta6JIHLle:
BpeMH, 4 0 I 2 3 4 5 6 7
TeMnepaTypa, °C 4 2 -I -2 -3 -3 -1 -I
844. Ha pHcYHKe 5.52 H306pa>KeH rpa!}>HK TeMnepaTypbl B03JJ.yxa.
I) KaKaH TeMnepaTypa B03JJ.yxa 6blJIa B 17 4? B 6 4? B 12 4?
2) B KaKoe BpeMH TeMnepaTypa B03JJ.yxa 6blJIa _4°? Oo? 2°?
3) B KaKoe BpeMH TeMnepaTypa B03JJ.yxa 6blJIa caMoH HH3KOH?
KaKoH?
163
t
C
,..
u
i-- 4 L,.oo ......
" ......
-2 ,
.....
1/ \ .
0 2 4 6 /8 10 12 14 16 18 20 22 24
- 2 "-
"
......
1-6 1/
PHC. 5.52
845.
4) B KaKoe BpeMH TeMnepaTypa B03JJ.yxa 6blJIa caMOH BbICOKOH';I
KaKoH? '
5) KorJJ.a TeMnepaTypa B03JJ.yxa 6blJIa HH>Ke HYJIH? Bblwe HYJIH?
6) Ha CKOJIbKO rpaJJ.ycOB H3MeHHJIaCb TeMnepaTypa B npOMe-
>KYTKe BpeMeHH OT 16 4 JJ.O 22 4? OT 2 4 JJ.O 6 4?
7) B K
KOH npOMe>KYToK BpeMeHH TeMnepaTypa nOHH>KaJIacb?
B KaKOH nOBblWaJIaCb?
8) KaK H3MeHHJIaCb TeMnepaTypa B npOMe>KYTKe BpeMeHH OT
6 4 JJ.O 14 4? OT 12 4 JJ.O 20 4?
np04TH npHMep 2 H 06'hHCHH:
I) KaK nOCTpoHTb rpa4>HK JJ.BH>KeHHH;
2) JJ.JIH 4ero MO>KHO HCnOJIb30BaTb rpa4>HK JJ.BH>KeHHH.
nOCTpOH rpa4>HK JJ.BH>KeHHH no Ta6JIHLle:
846.
BpeMH, 4 0 1 2 3 4 5
PacCTOHHHe OT nYHKTa
Bbl6blTHH, KM 0 2 4 6 8 10
847. Ha pHcYHKe 5.53 rpa4>HK JJ.BH>KeHHH aBTOMaWHHbl.
I) Ha KaKOM paccToHHHH OT MeCTa OTnpaBJIeHHH 6blJIa aBTO-
MaWHHa l/epe3 I 4, 4epe3 2,5 l/, 4epe3 4 4 nOCJIe Ha4aJIa
JJ.BH>KeHHH?
164
2) 3a KaKoe BpeMH aBTOMaWHHa npoexaJIa 75 KM? 100 KM?
225 KM?
3) KaKoe paccToHHHe npoexaJIa aBTOMaWHHa Bcero H CKOJIbKO
nOTpe60BaJIOCb JJ.JIH 3Toro BpeMeHH?
848. Ha pHcYHKe 5.54 rpa4>HK JJ.BH>KeHHH BeJIOCHneJJ.HCTa.
I) Ha KaKOM paccToHHHH OT MeCTa OTnpaBJIeHHH 6blJI BeJIOCH-
neJJ.HCT 4epe3 2 4, 4epe3 3 4 H 4epe3 3,5 4 nOCJIe Ha4aJIa
JJ.BH>KeHHH?
2) LJepe3 CKOJIbKO 4aCOB BeJIOCHneJJ.HCT 6blJI Ha paccToHHHH
5 KM, 15 KM, 40 KM OT MeCTa OmpaBJIeHHH?
3) KaKoe paccToHHHe npoexaJI BeJIOCHneJJ.HCT Bcero H CKOJIbKO
nOTpe60BaJIOCb JJ.JIH 3Toro BpeMeHH?
4) C KaKoH CKOpOCTblO exaJI BeJIOCHneJJ.HcT?
6.
849, Ha pnCYHKe 5.55 JJ.Ba rpa<fJHKa TeMilepaTypbI. JhlHHSI IIOTOJIllI,e -
rpa<fJHK TeMilepaTypbl B MocKBe, JIHHH5I nOTOHbllle - B neTep-
6ypre B TOT )Ke JJ.elIb, OIIpeJJ.eJIH, IIOJIb3YSlCb rpaqmKOM:
I) B KaKoe BpeMH B 3THX ropoJJ.ax TepMOMeTp nOKa3aJI - 40;
3°; 60;
S,KM
-2F II'
_I II"
2?O /
f-1
O 1/
7
r-.,po 7
/
I-
/
1/
V
,'I
0 1 2 3 4 5 .1
I I I
PHC. 5.53
SI(M
I
35
130 If
J
II
25 J
I
20
15 If
J
10 If
J
I
5
J t;<f
IT
0 1 2 3 4
I
PHC. 5.54
165
I
f--t
C
-8
-6 .....-I. .At 0c
'
-4 & C. '/1J::
1'lJp
-2 "" 'l3 yp /,
) '/ I
t.4
° '\. 2 4 6 8 )°7 12 14 16 18 20
--=2
'\ / VI
-=-4 '\.
\. "- ....../ /
-0
"- ,
8 )
"- ,/
-=10
PHC. 5.55
2) KaKYIO TeMnepaTYPY nOKa3aJl TepMOMeTp B Ka>KJJ.OM ropoJJ.e
B 2 4; B 6 4; B 10 4; B 16 4;
3) B KaKOH nepHOJI. BpeMeHH TeMnepaTypa B03JI.yxa B ropo-
JJ.ax 6blJIa HH>Ke HYJIH; Bblwe HYJIH;
4) B KaKoe BpeMH TeMnepaTypa B03JI.yxa B 3THX ropoJJ.ax no-
HH3HJIaCb; nOBbICHJIaCb;
5) B KaKoe BpeMH TeMnepaTypa B03JJ.yxa B 3THX ropoJJ.ax
6blJIa OD,HHaKOBoH, KaKOH HMeHHO;
6) B KaKOU npOMe)!{YTOK BpeMeHH TeMnepaTypa B03JJ.Yxa B Moc-
KBe 6bIJIa HU>Ke, 4eM B IIeTep6ypre, B IIeTep6ypre HH>Ke, qeM
B MocKBe;
7) KaKaH caMaH BblCOKaH TeMnepaTypa 6blJIa B 3THX ropoJJ.ax;
caMaH HH3KaH;
8) KaK H3MeHHJIaCb TeMnepaTypa B03JJ.yxa B Ka>KJJ.OM H3 ropo-
JJ.OB B npOMe>KYTKe BpeMeHH OT 6 4 JI.0 14 4;
9) KaKYro TeMnepaTYPY B03,IJ;yxa nOKa3aJI TepMOMeTp B IIeTep-
6ypre, eCJIH B MOCKBe OH nOKa3aJI 4°.
166
850. Ha pHcYHKe 5.56 rpa4>HK JJ.BH>KeHHH rpynnbl TYPHCTOB.
I) Ha KaKOM paccToHHHH OT Ha4aJIa nYTH 6blJIH TYPHCTbl 4e-
pe3 t 4aCOB, eCJIH t= I; t= 1,5; t=3; t=3,5; t=7; t=9,5?
2) CKOJIbKO BpeMeHH nOTpe60BaJIOCb TypHcTaM, 4T06bl npH-
6blTb B nyHKT Ha3Ha4eHHH, OTCTOHI1.lHH Ha paccToHHHH S KM,
eCJIH 5=2; s=9; s= 10; s= 16; s= 18?
3) 4eMY paBHHJIaCb nepBOHa4aJIbHaH CKOpOCTb TypHCTOB H KaK
JJ.OJIro WJIH OHH C 3TOH CKOpOCTblO?
4) 1.IeMY paBHHJIaCb CKOpOCTb TYPHCTOB B Te4eHHe TpeTberO 4a-
ca JJ.BH>KeHHH?
5) 4epe3 CKOJIbKO 4aCOB nOCJIe Ha4aJIa JJ.BH>KeHHH TypHCTbl cJJ.e-
JIaJIH nepBblH npHBaJl H KaK JJ.OJIro OHH oTJJ.bIXaJIH?
6) C KaKOH CKOpOCTblO JJ.BHraJIHCb TYPHCTbl nOCJIe nepBoro npH-
BaJIa H CKOJIbKO KHJIOMeTpOB WJIH OHH C 3TOH CKOpOCTblO?
7) C KaKoH CKOpOCTblO JJ.BHraJIHCb TypHCTbl nOCJIe BToporo npH-
BaJIa H KorJJ.a npH6wJIH Ha MeCTO Ha3Ha4eHHH?
8) C KaKoH cpeJJ.HeH CKOpOCTblO JJ.BHraJIHCb TypHCTbl, TO eCTb
CKOJIbKO KHJIOMeTpOB B l/ac npoxOJJ.HJIH 6bl OHH, eCJIH 6bl He 6bl-
JIO npHBaJIOB H CKOpOCTb Ha npOTH>KeHHH Bcero nYTH 6blJIa 6bl
nocTOHHHOH?
S,KM
2
4
2 G)
/
0
8 /
E /
6 /
F
4 V
2 J
C /
0 / D
8 - -
6 /6
/
4 I
2 It., t,'I
0
2
2
2 3
5 6 7 8 9 10
PHC. 5.56
167
S,KM
10
0
1\
0 \ \
/ I \
,
1\
0 I \
o J
1/ 1\
t,'I
0 "
8
60
4
2
234:> 6 78
PHC. 5.57
If S,KM
-tbo )
f. 7 1/
t 1/
-80 J
"
I 1/
II
1--60 ¥'
I
y /
J / V
40 I /1/
1/ j h. V
1/
ICJi :::J /
I I -
'l
-20 I II V .
J / " .,
J I '/ RI. ---
Ii
,
II t,
'1 'f
PHC, 5.58
S,KM
800
400
2 3
PHC. 5.59
5
789
851. Ha pHcYHKe 5.57 JJ.aH rpa4>HK JJ.BH>KeHHH aBTOM06HJIH 3a OJJ.HH
pa604HH JJ.eHb. OnHwH D,BH>KeHHe aBTOM06HJIH no 06pa3LlY npe-
.lI.blJJ.YllI,eH 3aJJ.a4H.
852. COCTaBb 3aJJ.a4H no rpa4>HKaM .lI.BH>KeHHH (pHC. 5.58). PewH HX.
853. Ha pHcYHKe 5.59 rpa4>HKH nOJIeTa JJ.BYX CaMOJIeTOB, KOTopble
O.lI.HOBpeMeHHo BblJIeTeJIH C a3poJJ.poMa. no PHCYHKY COCTaBb
3aJJ.a4H H pewH HX.
854*. COCTaBb rpa4>HK JJ.BH>KeHHH aBTOM06HJIH rAM, npeJJ.nOJIaraH,
4TO CKOpOCTb JJ.BH>KeHHH aBTOM06HJIH Me>KJJ.Y OCTaHOBKaMH no-
CTOHHHaH. npH nOCTpoeHHH rpa4>HKa BbmOJIHH CJIeJJ.YIOIlI,He YCJIO-
BHH:
1) B MeCTO OTnpaBJIeHHH MaWHHa B03BpaTHJIaCb 4epe3 8 4;
2) 4aCOBaH OCTaHOBKa 6blJIa Ha paccToHHHH 50 KM OT MeCTa
OTnpaBJIeHHH 4epe3 2 4 nOCJIe Ha4aJIa .lI.BH>KeHHH;
3) B caMYIO JJ.aJIbHIOIO T04KY (100 KM) OT MeCTa OmpaBJIeHHH
aBTOM06HJIb npH6blJI B Ha4aJIe weCToro 4aca, OCTaHOBKH TaM He
6bIJIO;
4) nOCJIe nOJIY4acoBoH e3JJ.bl Ha 06paTHoM nYTH 6blJIa CJJ.eJIaHa
nOJIY4acoBaH OCTaHOBKa B 80 KM OT MeCTa OTnpaBJIeHHH;
5) nOCJIe 3Toro MaWHHa B03-
BpaTHJIaCb B nYHKT OTnpaB-
JIeHHH. OCTaHoBKH 60JIbWe He
6bIJIO.
169
A C D B E
, . I I . I I . I I . I I , ..
-5 -4 -3 -2 -1 0 1 2 3 4 5
PHC. 5.60
y
4
3
C D
2
A
1
E
4_
3_ -=2_ _-1 0 1- _2_ 1- 3 4 x
F l' G
2
f-3
B
4
PHC. 5.61
.lI.JUI caMonpoBepKH
855. KaKHe 4HCJIa Ha3blBalOTCH npoTHBOnOJIO>KHbIMH? fJpHBeJJ.H npH-
Mepbl npoTHBOnOJIO>KHbIX 4HCeJI.
856. 4TO TaKoe MOJJ.YJIb 4HCJIa?
857. Ha30BH JJ.Ba pa3JIH4HblX oTpHLlaTeJIbHblX 4HCJIa. OnpeJJ.eJIH, Ka-
Koe H3 HHX MeHbwe.
858. Ha4epTH 4HCJIOBYIO OCb H OTMeTb Ha HeH T04KH K (-3,4),
L (1,2), M (-I), N (3), P (-1,5), R (-2,6).
859. 3anHwH KoopJJ.HHaTbl T04eK, 0603Ha4eHHblx 6YKBaMH (pHC.
5.60), H YKa>KH paCCTOHHHH OT Ha4aJIa OTC4eTa JJ.O 3THX T04eK.
860. LI.aHbl 4HCJIa: 2,9; -3,7; -5; 2,29; -0,5; -3,4 H O.
170
I) PaCnOJIO>KH HX B nopHJJ.Ke B03paCTaHHH.
2) HaHJJ.H 4HCJIa, npoTHBOnOJIO>KHbIe JJ.aHHbIM.
3) HaHJJ.H HX MOJJ.YJIH.
861. HaHJJ.H Bce LleJIble 4HCJIa, npH KOTOpblX paBeHcTBo -5:::;;;x<2,4
BepHo.
862. 4eMY paBHHeTcH -b, eCJIH b= -5; b= 1,8; b=O;
863. Bbl4HCJIH 3Ha4eHHe Bblpa>KeHHH lei +3,2, eCJIH
b=4,9?
2
c=""'5;
c= -1,8; c=O.
864. BbI4HCJIH:
I) -(-4,9) 2)
-( + 15)
3) 1-4,81+5,4
31,6-1-17,91
4) 1 1 7 2 1.: ::
I
.IOI
2
-( -+ )
-(+2
)
865. 3anHwH B BHJJ.e HepaBeHcTBa:
I) - 3,8 - OTpHLlaTeJIbHOe 4HCJIO;
2) 7 - nOJIO>KHTeJIbHOe 4HCJIO;
3) x He nOJIO>KHTeJIbHOe;
4) Y He OTpHLlaTeJIbHoe.
866. KaKHe npHMble Ha3blBalOTCH nepeCeKalOl.llHMHcH? nepneHJJ.HKY-
JIHpHbIMH? napaJIJIeJIbHblMH?
867. Ha4epTH: I) nepeCeKalOllI,HeCH npHMble d H e; 2) napaJI-
JIeJIbHble npHMble r H s; 3) nepneHJJ.HKYJIHpHble npHMble m H n.
868. 4TO Ha3bIBaeTCH paccToHHHeM OT T04KH JJ.O npHMoH?
869. Ha4epTH npHMYIO a. no 06e CTOpOHbl 3TOH npHMoH OTMeTb
T04KH A H B. 113Mepb paccToHHHe OT 3THX T04eK JJ.O
npHMoH a,
870. KaK Ha3bIBalOTCH KoopJJ.HHaTbl T04KH Ha nJIOCKOCTH H 4TO OHH
nOKa3bIBa1OT?
871. 3anHwH KoopJJ.HHaTbl T04eK A, B, C, D, E, F, G H H (pHC. 5.61).
872. B KaKoH KoopJJ.HHaTHoH 4eTBepTH pacnOJIO>KeHbl T04KH
R(-30; I), S(2; -43), T(IO; 10), U(-9,5; -20)?
873. Ha4epTH cHcTeMY KoopJJ.HHaT H oTpe30K, KOHLlbl KOToporo
T04KH K (-4; 2) H L (3; -1,5). OTMeTb Ha oTpe3Ke T04KH, a6-
CLlHCCbl KOTOpblX paBHbl - 2; - 1,4; 0; I; 2.
874. Ha4epTH cHcTeMY KoopJJ.HHaT H npHMYIO no JJ.BYM ee T04KaM
A(3; 5) H B (-I; -3). OTMeTb Ha npHMoH T04KH, OpJJ.fi-
HaTbl KOTOpblX paBHbl - 1,5; 0; 0,5; I; 2.
875. Ha KoopJJ.HHaTHoH nJIOCKOCTH Ha4epTH npHMoyroJIbHHK, BepwH-
Hbl KOToporo A ( - 2; 2), B (5; 2), C (5; - 3) H D ( - 2; - 3).
171
t
C
S,KM
/' .......
7 "- .......
/ " t,
2 4 6..1 8 10 12 14 16 18 20 22,24
"7 ,
L/
0 /
/
V
1/
J
/ t,"
6
4
2
120
10
80
60
40
20
o
-2
-4
-6
23456
PHC. 5.63.
PHC. 5.62
876. nOCTpOH rpa<J>HK TeMnepaTypbl no Ta6JIHLle:
BpeMH, 4 0 I 2 3 4 5 6 7
TeMnepaTypa, °C 0 -2 -4 -5 -5 -3 0 2
877. Ha pHcYHKe 5.62 H306pa>KeH rpa<J>HK TeMnepaTypbl.
I) KaKaH TeMnepaTypa B03JJ.yxa 6blJIa B 1 4? B 6 4? B 22 4?
2) B KaKoe BpeMH TeMnepaTypa B03JJ.yxa 6blJIa _4°? OO? 2°?
3) B KaKoe BpeMH TeMnepaTypa 6blJIa caMOH HH3KOH? CKOJIb-
KO rpaJJ.ycoB?
4) B KaKoe BpeMH TeMnepaTypa 6blJIa caMoH BbICOKOH? CKOJIb-
KO rpaJJ.ycoB?
5) KaK H3MeHHeTCH TeMnepaTypa B npOMe>KyTKe BpeMeHH OT
6 4 .lI.0 16 4?
878. nOCTpOH rpa<J>HK JJ.BH>KeHHH no Ta6JIHLle:
BpeMH JJ.BH>KeHHH, l/ 0 I 2 3 4
PaccToHHHe OT MeCTa
OTnpaBJIeHHH, KM 0 15 30 45 60
879. Ha pHcYHKe 5.63 rpa<J>HK JJ.BH>KeHHH MOTOpOJIJlepa.
I) KaKoe paccToHHHe npoexaJI MOTOpOJIJIep?
2) 4epe3 CKOJIbKO 4aCOB MOTOpOJIJIep HaXOJJ.HJICH Ha paccToHHHH
20 KM, 40 KM, 70 KM, 100 KM OT nYHKTa OmpaBJIeHHH;I
3) Ha KaKOM paccToHHHH OT nYHKTa OTnpaBJIeHHH HaXOJJ.HJICH
MOTOpOJIJIep l/epe3 1,5 4, 3 4 nOCJIe Hal/aJIa JJ.BH>KeHHH?
4) Ha KaKOM KHJIOMeTpe 6blJIa OCTaHOBKa H KaK JJ.OJIro OHa
npoJJ.oJI>KaJIacb?
172
6. .D.EACTBHSI C PALlHOHAJ1bHbIMH
lJHCJ1AMH
6.1. CnO)f(eHHe .ll.BYX oTpHu.aTeJ1bHblX IfHCeJ1
Tbl, HaBepHoe, 3HaeWb TaKHe Hrpbl (HanpHMep, JJ.OMHHO), B KOTO-
pblX BblHrpaBwHH MO>KeT nOJIY4HTb onpeJJ.eJIeHHOe 4HCJIO (BblHrpaH-
HbIX) 04KOB, a npoHrpaBwHH - wTpa<J>Hblx 04KOB. ECJIH, HanpHMep,
HrpOK 3a OJJ.HY Hrpy BblHrpaJI 3 04Ka, a 3a BTOPYIO 5 04KOB, TO 0611.lee
4HCJIO BblHrpaHHblx 04KOB HaXO.lI.HM CJIO>KeHHeM: 3+5=8, HJIH MO>KHO
3anHcaTb: (+3)+( +5)= +8. ECJIH >Ke HrpOK 3a OJJ.HY Hrpy nOJIY4HJI
3 wTpa<J>Hblx 04Ka, a 3a BTOPYIO 5 WTpa<J>HbIX 04KOB, TO 0611.lee
4HCJIO wTpa<J>Hblx 04KOB HaXOJJ.HM TaK>Ke CJIO>KeHHeM. ECJIH Bblpa3HTb
4HCJIa wTpa<J>Hblx 04KOB OTpHLlaTeJIbHblMH 4HCJIaMH, TO nOJIY4HM
(-3)+( -5)= -8, 4HCJIO -8, a TaK>Ke Bblpa>KeHHe (-3)+( -5)-
3TO cYMMa l/HCeJI - 3 H - 5. 4HCJIa - 3 H - 5 - CJIaraeMble. nep-
Boe CJIaraeMOe 06bl4HO 3anHCbIBalOT 6e3 CK06KH: -3+("7"5).
nOJIb3YHCb 3THM npHMepOM, MO>KHO CJIO>KHTb JJ.Ba JII06blX OTpHLla-
TeJIbHblX 4HCJIa, HanpHMep:
-6+(-5)=-11, -1+(-15)=-16,
-47+(-3,2)= -50,2,
- : +( -
)= -I
'
Mbl BHJJ.HM, 'ITO CYMMa iJ8YX oTpuqaTellbHblX 'IUcell eCTb 'IUCIlO
oTpuqaTellbHoe, Jlf.OiJYllb ICOTOpOZO pa8eH CYMMe JIf.OiJYlleu CIlCZZaeJlf.blX.
3Ha4HT, 4T06bl CJIO>KHTb JJ.Ba OTpHLlaTeJIbHblX 4HCJIa, HY>KHO:
I) CIlO3ICUTb Jlf.OiJYIlU CllaZaeJlf.blX;
2) nepeiJ nOIlY'IeHHblJIf. 'IUCIlOJlf. nOCTa8UTb 3HaIC «-».
CJIO>KeHHe oTpHLlaTeJIbHblX 4HCeJI MO>KHO H306pa3HTb C nOMOl1.lblO
4HCJIOBOH OCH (pHC. 6.1).
J- -5 I- -2
I.
E] -6 -5 -4 -3 -2 -1 0
-2 + (-5) =-7
PHC. 6.1
173
np04TH TeKCT naparpaQJa H 06"bHCHH, KaK Hal'hH CYMMy
- 8 + ( - (9). CQJOPMYJIHPYI'i npaBHJIO CJIO>KeHHH JJ.BYX OTpHLla-
TeJIbHblX l/HCeJI.
CJ!O>KeHHe KaKHX 4HCeJI H306pa>KeHO Ha pHcYHKe 6.2?
Bbl4HCJIH (y C THO) OKOHl/aTeJIbHbIH pe3YJIbTaT Hrpbl, eCJIH 3a
JJ.Be napTHH nOJIY4eHbl:
I) - 5 04KOB H - 2 04Ka, 2) - 3 04Ka H - 7 04KOB,
- 3 04Ka H - I 04KO, - 6 04KOB H - 9 04KOB,
- I 04KO H - 7 04KOB; - 10 04KOB H - I 04KO.
KaKHM JJ.el'icTBHeM Tbl HaWeJI oTBeTbl?
B npHBeJJ.eHHblx HH>Ke npHMepax YMeHbweHHe TeMnepaTypbl
Bblpa>KeHO oTpHLlaTeJIbHblM 4HCJlOM. OnpeJJ.eJIH (y C THO), KaK
H3MeHHJIaCb TeMnepaTypa B pe3YJIbTaTe JJ.BYX H3MeHeHHH:
1) _6 0 H _5°; 2) _50 H _60; 3) _1° H _1°; 4) -14 0
H _20; 5) _3° H _90; 6) -13 0 H _40; 7) _7 0 H -120;
8) _50 H -13 0 .
KaKHM JJ.eHcTBHeM Tbl HaWeJI OTBeTbl?
Bbl4HCJlH (y C THO):
I) -10+(-20) 2)
-30+(-50)
-10+( -90)
BbI4HCJIH:
. I- I- ,
-6 -5 -4 -3 -2 -1 0
(0
I- ,-
-35 -30 -25 -20 -15 -10 -5
0
174
880.
881.
882.
883.
884.
885.
I) -32,7+( -ll,4)
-100,01 +( -238,1)
-3005+( -2937)
-675+(-67,5)
3) -0.78+(-1,96)
-5,9+(-67)
- 0,08 + ( - 0,28)
-7,42+( -0,88)
A.
-17+(-13)
-21 +( -12)
-39+( -II)
3) -200+(-300)
-350+( -140)
-900+( -100)
2) -98,7+(-56,09)
-12,32+(-29,8)
-400+( -57,9)
-82+( -99)
4) -3,07 +( -1,03)
-11,1+(-11,1)
-3,49+( -0,51)
-12,9+( -1,99)
..
886. BbI4HCJIH:
I) -
+(-
) 2)
-2++( -+)
-
+( --i-)
- : +( -
) 3) -
+( -
)
-3++( -1+) -5++(-2
)
-
+(-
) -
+(-
)
-I,6+(-n), eCJlH n=4;
887. Bbl4HCJIH 3Ha4eHHe Bblpa>KeHHH
1 1
n= 15,7; n= I 2"""; n="3'
6.
888.
BbI4HCJIH:
I) -3 ++( -0,85) 2)-0,75+( -I + ) 3)-0,6+( -4 + )
-4++(-2,5) -2++(-3,8) -6,15+( -2+)
-0,1 +( -3 1 3 0 ) -I -i-+( -2,2) -3
+( -0,4)
889.
K CYMMe 4HCeJI -8 +- H -2
npH6aBb 4HCJIO, npOTHBonO-
2
JIO>KHOe 4HCJIY 1"""3'
Bbl4HCJIH 3Hal/eHHe Bblpa>KeHHH - m + ( - n), eCJIH: I)
1
H n=86; 2) m=O,47 H n= 1,03; 3) m=""6 H
4) m= -l,l Ii n= -4,7.
890.
m=275
1
n=-'
3 '
6.2. CnO>KeHHe .ll.BYX "IHCeJ1 C pa3HblMH 3HaKaMH
JIonYCTliM, 4TO HrpOK nOJIY4HJI 3a OJJ.HY napTHIO Hrpbl 5 wTpaQJ-
HblX 04KOB, a 3a BTOpylO napTHIO 5 BblHrpblwHblX 04KOB. B 3TOM
CJIY4ae HH TO HH JJ.pyroe 4HCJIO He liMeeT nepeBeca H B HTore pe3YJIbTaT
Hrpbl paBeH HyJIIO. 3anHweM: -5+5=0 (pHC. 6.3.). TaKHM >Ke
06pa30M nOJIY4HM, HanpHMep: -10+ 10=0, 7+(-7)=0 H T. JJ..
Bo061l1,e cyMM.a iJ8YX npoTU80nO/l.03ICHbl.X tluce/l. paBHa HY/l.IO.
+5
. -5
I.
0 5 -5 -4 -3 -2 -1
PHC. 6.2 --5 + 5 = 0
. I
[9]
PHC. 6.3
I f
2
175
J. +7 - I
-5
.
-5 -4 -3 -2 -1 0 0
-5+7=2
PHC. 6.4
I
-7 .L
+5
I
-3 51 -1 0 2 3 4 5
5 + (-7) =-2
PHC. 6.5
PaCCMOTpHM Tenepb CJIO>KeHHe JJ.BYX 4HCeJI C pa3HblMH 3HaKaMH,
MOJJ.YJIH KOTOpblX He paBHbI. JIonYCTHM, 4TO HrpOK nOJIY4HJI 3a OJJ.HY
napTHIO 5 wTpaQJHblx, a 3a BTOpylO 7 BblHrpblwHblX 04KOB. TaK KaK
4HCJIO BblHrpblwHblX 04KOB HMeeT HaJJ. 4HCJIOM WTpaQJHblX 04KOB
nepeBec B 2 04Ka, TO B pe3YJIbTaTe JJ.BYX napTHH HrpOK HMeeT
2 BblHrpblwHblX 04Ka. 3anHweM: - 5 + 7 = 2 (pHC. 6.4), ECJlH
>Ke _HrpOK nOJIY4HJI 3a OJJ.Hy napTHIO 5 BblHrpblwHblX, a 3a BTOPYIO
7 wTpaQJHblx 04KOB, TO B HTore OH HMeeT 2 wTpaQJHblx 04Ka. 3anH-
weM: 5+(-7)= -2 (pHC, 6.5).
nOJIb3YHCb 3THM npHMepoM, MO>KHO CJIO>KHTb JII06ble JJ.Ba 4HCJIa
C pa3HblMH 3HaKaMH, MOJJ.YJIH KOTOpblX He paBHbl, HanpHMep:
-50+40= -10,
25+(-5)=20,
-0,2+0,3=0,1,
8+( -9)=-1.
Mbl BHJJ.HM, 4TO CYMMa TaKHX 4HCeJI MO>KeT 6blTb KaK nOJIO>KHTeJIb-
HOH, TaK H oTpHLlaTeJIbHoH. OT 4ero 3aBHCHT 3HaK CYMMbl? 04eBHJJ.Ho,
3HOIC CYMMbI. c08naiJaeT co 3HaICOM TOZO C/l.aZaeMOZO. MOiJY/l.b ICOTOpO-
zo 60/l.bwe. Ho 4T06bl HaHTH MOJJ.YJIb CYMMbl, HY>KHO U3 60/l.bWeZO
MOiJY/l.JI 8b1.tleCTb MeHbwuii.. HTaK, 4T06bl CJIO>KHTb JJ.Ba 4HCJIa C pa3-
HblMH 3HaKaMH, MOJJ.YJIH KOTOpblX He paBHbl, HY>KHO:
I) U3 60/l.bWeZO MOiJY/l.JI 8b1.tleCTb MeHbwuii. MoiJY/l.b;
2) nepeiJ nO/l.YtleHHbI.M tlUC/l.OM nOCTa8UTb 3HaIC TOZO C/l.azaeMOZO.
MOiJY/l.b ICOTOpOZO 60/l.bwe.
176
ECJIH OJJ.HO H3 CJIaraeMblX paBHO HyJIIO, TO CYMMa paBHa JJ.Pyro-
MY CJIaraeMOMY, HanpHMep, 0+( -5)= -5; -7,8+0= -7,8.
A.
891.
06"bHCHH, KaK Bbl4HCJIHTb CYMMY: I) -9+6; 2) 4+(-4);
3) -8+0; 4) 7+(-1),
npoBepb no TeKCTY, nOMHHWb JIH Tbl npaBHJIO CJIO>KeHHH
JJ.BYX 4HCeJI C pa3HblMH 3HaKaMH, MOJJ.YJIH KOTOpblX He paBHbI.
OT 4ero 3aBHCHT 3HaK CYMMbl?
CJ!O>KeHHe KaKHX 4HCeJI H306pa>KeHO Ha pHCYHKe 6.6?
Bbl4HCJIH OKOH4aTeJIbHblH pe3YJIbTaT Hrpbl, eCJIH 3a JJ.Be napTHH
nOJIY4eHbl:
I) -3 04Ka H +2 04Ka, 2) -2 04Ka H +2 04Ka,
+ 5 04KOB H - 3 04Ka, 0 04KOB H - 4 04Ka,
- 7 04KOB H + 5 04KOB; + 8 04KOB H - 2 04Ka.
3anHwHTe peweHHe B BHJJ.e CYMMbl 4HCeJI.
Bbl4HCJIH (y C THO):
I) -20+60 2)
20+( -8)
30+(-50)
3) 27+(-19)
- 32 + 18
-18+ 18
892.
893.
894.
-100+500
620+( -700)
-400+800
4) 10+(-10)
-30+( -18)
- 100 + 0
895. BbI4HCJIH:
I) 37,5+28,1 2) -19,2+(-3,3)
427,5+( - 11,9) - 11 ,3 + ( - III ,49)
-32.9+( -100,1) 15067 +( -(1069)
3) -O,U7 +(-1,23) 4) 3,92+(-1,92)
5,64+( -12,8) -1,47 + 1.48
-2,19+ 7,06 0,75+( -1,87)
J- I -L
-4 -3 -2 _1 0 2 3 4
(0 ,- -L
-30 -20 -10 0 10 20 30 40 50
CD
PHC. 6.6
177
896. BbI4HCJIH:
-..!..+
2) I
+( _ I
) 902.
I)
7 7
+( -
) -.2...+.2... 903.
8 8
-++( -+ ) 1+( -+ )
3) _..!..+..!.. 4) 2 ++( -3
) 904.
4 6
- 1 ++( -
) -4+( -I + ) 905.
_..!..+-.!. 2 .2...+( -
) 906.
8 4 12 9
897. Bbll/HCJIH 3Hal/eHHe Bblpa}l{eHHH a+( -3,5),
eCJIH a=O,84;
a=7,3; a= -4 7' a=2..!..
, , 6 .
KaK H3MeHHTCH 4HCJIO, eCJIH K HeMy npH6aBHTb: I) nOJIO}l{HTeJIb-
Hoe 4HCJlO; 2) oTpHLlaTeJIbHOe 4HCJIO?
KaKHM 3HaKOM «+» HJIH «-» H Y }I{HO 3 aMeH U Tb
rl 3Be3.lI.04KH,
4T06b1 paBeHcTBa 6b1JIH BepHblMH:
I) (*10)+(*5)=-5
(*30)+(*10)=40
(*8)+(*9)= I
898.
899.
900.
6.
2) (*5)+(*5)=0
(*10)+(*10)= -20
(*10)+(*10)= 20
KaKHe H3 He p aBe HcTB (p )
aBeHCTB BepHble, KaKHe HeT (06'bHC-
HH n04eMY):
I) 200+(-50»0
6,78+( -19,2)<0
12,6+( - 12,6)=0
3) -20+ 10< -20
10+(-20)< 10
-19+( -19)< -19
901. Bbll/HCJIH:
3
I) -4+2,6
I
-1 2 +( -0,26)
2,05+( -3
)
178
2) -57+87<0
15,7 +0>0
-7+0=0
4) -370+500<500
425+( -275)<0
-3,95+ 1,25<0
2) -0,25+( -
)
2,81+( -I 1
)
-04 +
, 15
1
3) 6+(-0,5)
27
100 +( - 3,4)
7
-12"+ 1,12
Bbl4HCJIH 3Ha4eHHe Bblpa}l{eHHH x+y, eCJIH: I) x= -125
H y=37; 2) x= -23,6 H y= -0,94; 3) x=4
H y= -I T'
1
K CYMMe 4HCeJI 1,25 H -I 4 npH6aBb 4HCJIO, 06paTHoe 4HC-
1
JIY 1""'5'
npH KaKOM YCJlOBHH 6YJJ.YT BepHW paBeHCTBa:
I) -b+a= -b; 2) -a+(-b)= -a; 3) a+b=a?
MO}l{eT JIH cYMMa a+ b 6blTb MeHbllle a? (npHBeJJ.H npHMepbI.)
KaKHe H3 CJIeJJ.YIOIlI,HX HepaBeHcTB, r JJ.e a < 0, BepHble, KaKHe
HeT (06'bHCHH n04eMY):
I) 2+a>2 2) a+5,3<5,3
-3+a< -3 - 7 +a> 0
3) a+a>O
a+ a < a.
907. KaKHe H3 CJIeJJ.yIOIlI,HX HepaBeHcTB, rJJ.e a> 0, BepHble, KaKHe
HeT (06'bHCHH n04eMY):
1) -7+a>-7
a+( -1.8)< -1,8
2) a+4<0 3) a+(-a»O
a+( -+» -+ -a+(-a)<O
908. HaHJJ.H CyMMY, CJIaraeMblMH KOTOpOH HBJIHIOTCH 4HCJlO 06paT-
Hoe H 4HCJIO, npOTHBOnOJIO}l{HOe 4HCJIY 3,5.
909. OnpeJJ.eJIH CYMMY, CJIaraeMblMH KOTOpOH HBJIHIOTCH HaH60JIbWee
LleJIOe OTpHLlaTeJIbHOe 4eTblpex3Ha4Hoe 4HCJIO H HaHMeHbwee
LleJIOe nOJIO}l{HTeJIbHOe Tpex3Ha4Hoe 4HCJIO.
JJ,JIH nOBTopeHHH
910. Bbl4HCJIH (y C THO):
I) 47+86+53 2) 4,8+2,6+3,4
3) ..!..+ 3 ..!..+-.!.
494
2..!..+2-+
3 3 7
3
+-.!.+2
6 8 8
184+ 16+ 125
0,75+ 1,99+0,25
3,2+0,8+5,7
373+ 189+ III
911.
1
BeJIOCHneJJ.HCT npoexaJI 47,5 KM H elll,e 1"3 l/ co CKOpOCTblO
V KM . CKOJIbKO KHJIOMeTpOB npoexaJI BeJIOCHneJJ.HcT? CocTaBb
lJ
Bblpa}l{eHHe H Bbl4HCJIH, eCJIH I) v= 12; 2) v=21,3.
179
912. PewH ypaBHeHHe:
I) y+ 793=8100
4361 +x=6083
6756+ t=9478
2) 0,795+a= 1,1
c+6,9= 14,05
1,001 +x= 1,111
3) 3++x=4+
19 17
Z+27"=18
s+
=
16 24
913.
Bbll/HCJIH (y C THO):
I) 10% OT 80, 2)
18% OT 500,
1
"3 OT 24;
3) 40% OT 220,
6% OT 30,
2
T OT 35,
20% OT 150,
15% OT 70,
3
"'4 OT 40;
914. HaHJJ.H 4HCJIO (y C THO):
I) 10% KOToporo paBHbl 20;
3) 5% KOToporo paBHw 10;
5) 31 % KOToporo paBeH 93;
1
7) 8"" KOToporo paBHa 3;
5
9) ""9 KOToporo paBHbl 45;
2) 40% KOToporo paBHw 20;
4) 37% KOToporo paBHw 37;
6) 12% KOToporo paBHbl 36;
2
8) T KOToporo paBHbl 14;
10) 0,7 KOToporo paBHbl 2,8.
915. Bbl4HCJIH (y C THO), CKOJIbKO npOLleHTOB COCTaBJIHeT OJJ.HO 4HCJIO
OT JJ.pyroro:
I) 6 OT 12, 2) 40 OT 20, 3) 70 OT 700, 4) 0,3 OT 10,
5 OT 30, 9 OT 45, 50 OT 10, 0,8 OT 4,
5 OT 25; 6 OT 30; lOoT 8; 0,5 OT 0,8.
916. 5IHLla B cpeJJ.HeM cOJJ.ep>KaT 12,5% 6eJIKOB H 12% >KHpOB.
CKOJIbKO rpaMMoB 6eJIKOB H >KlipOB cOJJ.ep>KHTCH B JJ.eCHTH
HHLlax, eCJIH OJJ.HO HHLlO BeCHT 60 r?
917. 1..:IT06bl npoexaTb paccToHHHe H3 OJJ.Horo ropoJJ.a B .lI.pyroH,
aBTOM06HJIb H3pacxoJJ.oBaJI 25 JI 6eH3HHa, a MOTOLlHKJI 8,4 JI.
Ha CKOJIbKO npoLleHToB pacxoJJ. 6eH3HHa y MOTOLlHKJIa MeHbwe,
4eM y aBTOM06HJIH?
918. 5pHraJJ.a BbmOJIHHJIa nJIaH Ha 110%, H3rOTOBHB npH 3TOM
66 JJ.eTaJIeH, CKOJIbKO JJ.eTaJIeH HY>KHO li3rOTOBHTb no nJIaHY?
919. UeHa TOBapa JJ.O JJ.ByX CHH>KeHHH 6blJIa 18 p, nepBblH pa3 LleHY
TOBapa CHH3HJIH Ha 15%, a BTOpOH - Ha 12%. nOCTaBb pa3yM-
Hble Bonpocbl H pewH 3aJJ.a4Y.
920. l1JIHHa nepBoH OKPY>KHOCTH 43,96 JJ.M, a JJ.HaMeTp BTOpOH OKPY>K-
HOCTH 12 JJ.M. Ha CKOJIbKO nJIOlll,aJJ.b nepBoro Kpyra MeHbwe
(60JIbwe) BToporo?
180
6.3. 3aKoHbI CJ10>KeHHH.
CnO>KeHHe HeCKOJ1bKHX "IHCeJ1
CJIO>KeHHe paLlHOHaJIbHblX (CM.
5.3), KaK H CJIO>KeHHe HaTy-
paJIbHblX, 4HCeJI nOJJ.4HHHeTCH nepeMeCTHTeJIbHoMY Ii C04eTaTeJIbHOMY
3aKOHaM, 3Hal/HT, eCJIH a, b H C - JII06ble paLlHOHaJIb-
Hble 4HCJIa, TO:
a + b = b + a CQJOPMYJIHPYH 3TH 3aKOHbJ!
(a+ b)+c=a+(b +c)
npHMep I. -5+(-7)=-12,aTaK>Ke -7+(-5)=-12.
n p H M e p 2. (7 +( -8»+( - 10)= - 1+( -10)= -II, a TaK-
>Ke 7 +« -8)+( -10»=7 +( -18)= - II.
nepeMeCTHTeJIbHblH H c04eTaTeJIbHblH 3aKOHbl n03BOJIHIOT rpyn-
nHpoBaTb CJIaraeMble B cyMMe HeCKOJIbKHX 4HCeJI TaK, 4T06bl Bbl-
4HCJIeHHH CTaJIH KaK MO>KHO npollle.
n p Ii M e p 3. Bbll/HCJIHM CyMMY
-9+( -5)+ 10+( -3)+( -7)+9+( -8)+ 1+2.
Bo-nepBbIX, 3aMeTHM, LITO B CYMMe HMelOTCH JJ.Ba npOTliBOnOJIO>K-
HblX 4HCJIa: 9 H - 9, cyMMa KOTOpbIX paBHa o. Hx MO>KHO 3a4epK-
HYTb. l1aJIee crpynnHpyeM l/HCJIa C OJJ.HHaKOBbIMH 3HaKaMIi.
nOJIY4HM:
-?+( -5)+ 10+( -3)+( -7}+?+( -8)+ 1+2=
=( -5)+( -3)+( -7)+( -8)+(10+ 1+2)= -23+ 13= -10
A.
921. Bbl4HCJIH (y C THO), HCnOJIb3YH 3aKOHbl CJIO>KeHHH:
I) (27+19)+11 2) -14+(-6+19)
-17 +( -(2)+( -8) 15+25+( -8)
-1,2+(4,9+5,2) -6,8+( -3,7)+( -1.3)
( -++ 4 ) + ( -
)
+ ( - 2 + ) + ( -
)
922. PaccMoTpH npHMep 3 H 06"bHCHH, KaK npolll,e Bbll/HCJIHTb CYMMY
- 5 + ( - 2) + 5 + ( - 4) + 3
923. Bbl4HCJIH (y C THO) :
I) -3+(-8)+3 2) -2+6+(-3)+2
17 +( -6)+( -17) 19+( - 15)+( -15)+( -(9)
-0,8+(-4,9)+0,8 -2,4+(-0,6)+ 1,3+ 1,7
+ ( -
) + ( - ; ) ++ ( - + ) + ( -+ ) + ( -
)
181
924.
Bbll/HCJIH 3Ha4eHHe Bblpa>KeHHH:
I) -9+(-10)+20+(-1)+ 1;
2) 2+5+(-7)+(-8)+9+(-1);
3) -3+8+(-1)+(-4)+5+3;
4) -2+6+(-4)+ 7 +( -2)+5;
5) 10+(-8)+2+(-9)+(-3)+8.
Bbll/HCJlH 3Ha4eHHe Bblpa>KeHHH:
I) 20+(-30)+(-7)+4+(-3);
2) -48+52+(-36)+(-16)+ 108;
3) -60+(-20)+41 +(-10)+69;
4) 64+(-45)+ 16+(-47)+ 12.
Bbl4HCJIH 3Ha4eHHe Bblpa>KeHHH:
I) -34,03+(-52,07)+86,1;
2) 327,05+(-456,3)+ 198,97;
3) 1569,75+(-569,76)+(-11);
4) 16,9+(-42)+(-11,02);
5 )
+
+( 4 ) .
15 15 -15'
6) 3+( -I + )+( -I : ).
Bbll/HCJlH 3Ha4eHHH Bblpa>KeHHH a+ b + c, eCJIH: I)
b=7 H c= -16; 2) a=O,9, b= -1,4 H c=O,6.
a= -24,
925.
926.
927.
6.
928.
Bbl4HCJIH 3Ha4eHHe Bblpa>KeHHH:
I) 5,95+(-7,64)+4,08+(-1,36)+(-2,18);
2) -8,24+(-3,16)+ 7 +( -0,6)+( -15,94)+8,26;
3) -2
+
+ ( -
) + I-.!....,
2 6 6 3 '
4) -4,1+( -2 I
)+7++(-0,6).
YnpocTH Bblpa>KeHHe H Bbl4HCJIH ero 3Ha4eHHe, eCJIH m= -3'
m=5: '
I) m+(-1,9)+3,2+(-I.4+5,4);
2) -I,5+(m+( -4,2»+3,8+( -3);
3) 19,8+( -5,6)+(m+( -12,8»;
4) (m+20)+(-I0,5)+12,9.
lI.aHbl J./HCJIa 10; -3; 2,3; -5,6; -4 H 7,9. OnpeJJ.eJIH:
I) 4HCJIO, npoTHBOnOJIO>KHOe cYMMe 3TH X 4HCeJI; 2) CYMMY 4H-
ceJI, npoTHBOnOJIO>KHbIX JJ.aHHbIM 4HCJIaM; 3) CYMMY MOJJ.YJIeH
3THX 4HCeJI; 4) MOJJ.YJIb CYMMbl 3THX l/HCeJI.
929.
930.
182
931*. npH KaKOM YCJIOBHH paBeHcTBo x+y+z+I=O BepHo, eCJIH
x H y - npoTHBOnOJIO>KHble 4HCJIa?
6.4. BbI"IHTaHHe
BWl/HTaHHe - 3TO JJ.eHcTBHe, C nOMOUlblO KOToporo no 3aJJ.aHHOH
cyMMe JJ.BYX CJIaraeMblX H OJJ.HOMY H3 HHX HaXOJJ.HM JJ.Pyroe CJlaraeMoe.
BbI4eCTb H3 l/HCJIa a 4HCJIO b - 3Ha4HT HaHTH T.aKoe 4HCJIO x, 4T06bl
x+ b =a. nOKa>KeM Tenepb, 4TO Bbll/HTaHHe paLlHOHaJIbHWX 4HCeJI
BbmOJIHHMO BcerJJ.a, JJ.a>Ke B TOM CJIY4ae, KorJJ.a YMeHbwaeMoe MeHbwe
Bbl4HTaeMoro.
PaCCMOTpHM, HanpHMep, ypaBHeHHe x+5=2, OTKYJJ.a
x=2-5.
( I )
nOKa Mbl He MO>KeM BbmOJIHHTb 3TO BbI4HTaHHe. Ho ceH4ac Ha-
Y4HMCH. B JJ.aJIbHeHWeM Mbl Y3HaeM, 4TO K 06eHM 4aCTHM ypaBHeHlH1
MO>KHO npH6aBHTb OJJ.HO H TO >Ke l/HCJIO, npH6aBHM K 06eHM l/aCTfI'\l
JJ.aHHoro ypaBHeHHH 4HCJlO, npOTHBOnOJIO>KHOe BbI4HTaeMOMY, TO
eCTb 4HCJIO - 5, nOJIY4HM:
(x+ 5)+( -5)=2+( -5).
npHMeHHH B JIeBOH l/aCTH nOJIYl/eHHOrO paBeHcTBa C04eTaTeJIb-
HblH 3aKOH CJIO>KeHHH H Y4HTbIBaH, 4TO cYMMa JJ.BYX npoTHBOnOJIO>K-
HblX 4HCeJI paBHa HYJIIO, nOJIY4HM:
x+(5+( - 5»=2+( -5),
x+0=2+(-5),
x=2+(-5), (2)
x=-3.
npOBepKa nOKa3blBaeT, 4TO l/HCJIO - 3 HBJIHeTCH KopHeM
ypaBHeHHH x + 5 = 2. lI.eHcTBHTeJIbHO, - 3 + 5 = 2,
HTaK, H3 paBeHcTB (I) H (2) 3anHweM:
2-5=2+( -5).
PaCCY>KJJ.aH aHaJIOrH4HO, nOJIY4HM, HanpHMep:
7-10=7+(-10)=-3, npoBepKa: -3+10=7;
0-2=0+( -2)= -2, npoBepKa: -2+2=0;
-9-5= -9+( -5)= -14, npoBepKa: -14+5= -9;
7-(-1)=7+1=8, npoBepKa: 8+(-1)=7;
-3-(-5)= -3+5=2, npoBepKa: 2+(-5)= -3.
183
Mbl BHJJ.HM, 4TO Bbll/HTaHHe paLlHOHaJIbHblX 4HCeJI 3aMeHHeTCH
CJIO>KeHHeM. A HMeHHO: 'lTo6bl U3 OiJHOZO 'lUCJl.a 8bl'leCTb iJpyzoe
'lUCJl.O, HY3ICHO 1C YMeHbwaeMOMY npu6a8UTb 'lUCJl.O, npOTU80nOJl.03IC-
Hoe 8b1.'lUTaeMOMY.
B 6YKBeHHOM BHJJ.e (a H b - JII06ble paLlHOHaJIbHble 4HCJla):
a-b=a+(-b)
TaK KaK JII060e paLlHOHaJIbHOe 4HCJIO HMeeT npOTHBOnOJIO>KHOe
eMY l/HCJIO, TO BbI'IHTaHHe BblnOJlHHMO BcerJJ.a.
ECJIH YMeHbwaeMoe 60JIbWe Bbl4HTaeMoro, TO pa3HOCTb nOJIO-
>KHTeJIbHa. ECJIH >Ke YMeHbwaeMoe MeHbwe Bbl4HTaeMoro, TO pa3-
HOCTb OTpHLlaTeJIbHa. Pa3HOCTb 6YJJ.eT paBHOH HYJIIO, eCJIH YMeHb-
waeMoe H Bbl4HTaeMoe paBHbl
A.
932. np04TH TeKCT naparpaQJa, npoJJ.YMaH npHMepbl H CKa>KH:
1) KaKHM JJ.eHcTBHeM 3aMeHHeTCH BW4HTaHHe (l/TO npH 3TOM
HY>KHO nOMHHTb);
2) KorJJ.a pa3HocTb JJ.BYX l/HCeJI nOJIO>KHTeJIbHaH; oTpHLlaTeJIb-
HaH; paBHa HYJIIO.
933. Bbl4HCJIH (y C THO), 3aMeHHH Bbl4HTaHHe CJlO>KeHHeM:
I) 3-2 2) 20-8 3) -8-(-5) 4) 5-8
0-8 7-12 3-( -7) -5-( -8)
2-3 11-11 -10-2 -2-9
19-10 -5-(-1) -3-0 10-(-5)
934. BbI4HCJIH:
I) -4589-5001 2) -56,7-12,9 3) 3 I
----
8 4
625-1300 32,5-(-19,9) : -( - :
)
423-( -196) 63,7 -92,7 7 2
12"-"3
98-( - 719) -356.8-11,49 1 1
---
7 2
56-( -189) -11,2-4,81 3 5
---
4 6
935. YMeHbwH Ka>KJJ.oe H3 JJ.aHHbIX 4HCeJI Ha 10:
25; 15; 10; 7; 0; -10; -12; 12; -I; l.
936. YBeJIH4b Ka>KJJ.oe H3 JJ.aHHbIX l/HCeJI Ha 10:
0; 5; -2; -1; 10; -10; -15; 3; -21: l.
184
937. Bbl4HCJIH pa3HocTH a-b H b-a, eCJlH:
I) a=-37 H b=42; 2) a=-O,89 H b=II,5;
3) a= -330 H b=280; 4) a=560 H b= -310.
938. Ha CKOJIbKO rpaJJ.ycoB nOHH3HJIaCb (noBWCHJIacb) TeMnepaTY-
pa, eCJIH nOKa3aHHH TepMoMeTpa H3MeHHJIHCb:
I) OT 0° JJ.O 5°; 2) OT 0° JJ.O _5°; 3) OT 3° JJ.O 8°;
4) OT 14° JJ.O llo; 5) OT _2° JJ.O 3°; 6) OT _5° .lI.0 10°;
7) OT _10° JJ.O _2°; 8) OT 4° JJ.O _5°; 9) OT 6° JJ.O _4°.
939. PewH ypaBHeHHe H npoBepb ero KopeHb:
I) x+74= 131; 2) 2,5+u=0,7; 3) -8,7+y=5,19.
6.
940. Bbll/HCJIH (y C THO):
2 1
I) -"3-"3
+-( -
)
3 1
---1-
4 4
1 3
3) ""'5-""'5
5 7
-10+10
- : -( - : )
3 1
2) w-w
-{-( -+ )
I 2
-3+ I "3
4 2
4) -""'5-""'5
+-( -
)
-
-I...!..-
6 6
941. BbI4HCJIH:
7 5
3) -3 16 + 48
942.
I ) 2.!!.-31.. 2) -4
-6
12 4 9 3
12 1
3,08-525" -1,1+37
-1,8+2
5
-( -4,4)
KaKoe H3 CJIeJJ.YlOlliHx HepaBeHcTB
JIH a>O:
I) 3-a<3 2) -a<O 3) a+a=O
5,6-a< 5,6 m-a> m -a+a=O
-4,7-a> -4,7 b-a<b a+a<O
06"bHCHH Ha npHMepax, 4eM OTJIH4alOTCH pa3HocTH a - b H b - a.
Bbl4TH 113 HaHMeHbwero 0JJ.H03Ha4Horo LleJIOrO l/HCJIa HaH60JIb-
wee JJ.BY3Ha4Hoe LleJIOe 4HCJlO.
CaMaH HH3KaH TeMnepaTypa, H3MepeHHaH Ha nOBepXHOCTH 3eM-
185
-4,05-( -2
)
13
7,25- 3 20
(paBeHCTB) BepHoe, ec-
943.
944.
945.
JIH, -89,2 0c. 3TO Ha 70,8 °C Bblwe caMoH HH3KOH TeMnepaTY-
pbl Ha nOBepXHOCTH JIYHbl. B JIa60paTOpHblX YCJIOBHHX nOJIYl/H-
JIH TeMnepaTYPY, KOTOpaH HH}I{e TeMnepaTYPbl JIYHbl Ha
113,14°C. nOCTaBb Bonpocbl K 3aJJ.a4e H pewH ee.
6.5. Bblpa>KeHHH, cO.ll.ep>KaIllHe CJ10>KeHHe
H BbllJHTaHHe
Bblpa}l{eHHe
2+( -5)-( -7)+( -3)-9
COJJ.ep}l{HT JJ.eHCTBHH CJIO}l{eHHH H BbI4HTaHHH. Tbl Y}I{e 3HaeWb, 4TO
Bbl4HTaHHe paLlHOHaJIbHblX 4HCeJI MO}l{HO 3aMeHHTb CJIO}l{eHHeM.
3Hal/HT, nOJIY4HM CYMMY:
2+( -5)-( -7)+( -3)-9=2+( -5)+ 7 +( -3)+( -9).
B TaKOH cYMMe MO}l{HO oT6poCHTb CK06KH H 3HaKH «+» nepeJJ.
OTpHLlaTeJIbHblMH CJIaraeMblMH:
2+( -5)+ 7 +( -3)+( -9)=2-5+ 7 -3-9.
Bblpa>KeHHe 2-5+7-3-9 6YJJ.eM Tenepb nOHHMaTb KaK CYMMY
4HCeJI 2, -5, 7, -3 H -9. 4T06bl Bbll/HCJIHTb :ny CYMMy, LleJIeCO-
06pa3Ho CJIaraeMble crpynnHpoBaTb TaK, l/T06bl B OJJ.HOH rpynne
6blJIH Bce OTpHLlaTeJIbHble, a B JJ.PyroH Bce nOJIO}l{HTeJIbHble CJIarae-
Mble. nOJIY4HM:
2 - 5 + 7 - 3 -9 = 2 + 7 - 5 - 3 -9 =? - 5 - 3 -? = -8.
TaKHM }l{e 06pa30M Bbl4HCJIHM 3Ha4eHHe Bblpa}l{eHHH:
-25+( -30)-10-( -15)+( -50)+ 100=
= -25-30-10+ 15-50+ 100= -115+ 115=0.
A.
946. PaccMoTpH npHMepbl H3 TeKCTa H 06"bHCHH, KaK npollle Bbl-
4HCJlHTb 3Ha4eHHe Bblpa:-KeHHH
-8+( -3)+8-( -4)-1 +6.
947. YnpocTH Bblpa}l{eHHe (l/T06bl He 6blJIO B HeM CK060K) H BbI4HC-
JIH ero 3Ha4eHHe:
I) -(-4)+(-3) 2) -7+(-5)
-6-( -4) 15-( -15)
5-( -8) -10-( -10)
3) -11-(-14)
-20+( -30)
-( - 35)+( - 5)
186
948.
949.
Bbl4HCJIH (y C THO) :
I) -3+2-(-1) 2) 2-(-3)+3
-4-( -5)-5 7 +( -6)-7
1-( -1)+2 0-( -3)+( -5)
0-( -7)+6 4+( -8)+4
Bbl4HCJIH (y C THO) :
I) -3+7-5-6+8
-4+5-10-11 + 12
6-9+2-3+ 10
-5-6-4+5+ 10
BbI4HCJlH:
I) -786-94-(-164);
2) -0,17-(-1,2)-5,43;
3) 16,98+(-9,45)-(-1,03);
4) -5,7 -( -2,8)+( -9,4);
5) 2569-10 456-275324;
6) 3456079-5645020+342600;
7) 56,2 - 72,4 + 4,3;
8) 1567,302-2375,43+500,2+307,928;
9) 19,02 - 23,93 + 5,62 - 0,81;
10) 0,307-0,009+0,998-0,427,
Bbl4HCJlH 3Ha4eHHe Bblpa}l{eHHH 5 - r + t, eCJIH: I)
r= -10 H t= -20; 2) 5= -0,3, r=0,55 H t= I,l.
PewH ypaBHeHHe:
I) x+36= -15 2)
O,84+y=2,6
-7,2+a=3,4
5=7,
3) -5-6-3
4-2+3
6+7-15
-3+ 10-6
2) -2-8-10-4+3+20
30-35+5-6+ 7+ 10
-25+ 19-4-6-2+3
-50-45+85-10+ I +4
950.
951.
952.
693-x=749 3) z-421 = 168
-8,49-a=3,19 x-5,7= -4,3
43,6-y= -5 y-O,63= -1,8
6.
953.
Bbl4HCJIH 3Ha4eHHe Bblpa}l{eHHH:
I) 6-(-(-7)+(-3»-(-(-5)-1-(-8»;
2) -4-(-2)-(-5-(-7)+(-3)-(-8»;
3) -3,28+(-5,62)-(-9)-3,8+0,49;
4) -17,6-2,12+4,32-5,64+0,7.
BbI4HCJIH:
1 )
_...!...+ I ...!...-I ...!...+2'
4 4 4 4 '
1 1 1 2 1
2) 4 2 -3 2 -4"3+ 5 "3-"3;
3 5 2
3) 13-1 20 - 2 ,85+6""6- 2 3'
954.
187
955. Bbl4HCJIH 3Hal/eHHe Bblpa}l{eHHH m + n - k - p, eCJIH:
1) m= -2, n=3, k=7, p=5;
2) m=O,5, n= -0,6, k= -0,2, p=0,6;
1 7 2
3) m=T' n= -10' k=I5' p= -0,7.
956. 113 l/HCJIa - 5 Bbll/TH CYMMY 4HCeJI 7, - 5, - 6 H 2, 3aTeM
C l/HCJIOM - 5 CJIO}l{H CYMMY 4HCeJI, npOTHBOnOJIO}l{HblX JJ.aHHbIM.
957. 3anOJIHH Ta6JIHLlY:
a b la-bl Ib-al lal-Ibl Ibl-Ial
-10 -8
20 70
-50 10
10 -30
958. KaKoe H3 CJIeJJ.YIOIlI,HX HepaBeHcTB (paBeHcTB) BepHoe, ec-
JIH a<O:
I) 5-a<5
-4-a< -4
8-a< 8
5+a<5
2) -a<O
c-a>c
b-(-a)<b
x+a>x
3) a+a=O
-a+a=O
-a+( -a)<O
a-( -a)=O
959.
YnpOCTH Bblpa}l{eHHe:
I) a+5-7-a+9
-x-IO+x-20-20
c-8-3+2-c
PewH ypaBHeHHe:
I) x+7-10=12
8-x+4= 10
5-6-x=20
2) 30-m+7-9=-5
n-5+6-1O=0
-6+8-IO-x= -3
2) a+b-5-b+7-a
8-IO+m-n-m+n
x-y+5-4-x+y
960.
961*, T04Ka C KoopJJ.HHaToH -3 JJ.BH}I{eTCH no 4HCJIOBOH OCH CJIe-
JJ.YIOIllHM 06pa30M: BHa4aJIe 5 eJJ.HHHLl B nOJIO}l{HTeJIbHOM HanpaB-
JIeHHH, 3aTeM 7 eJJ.HHHLl B oTpHLlaTeJIbHOM, CHOBa 10 eJJ.HHHLl
B nOJIO}l{HTeJIbHOM H 8 eJJ.HHHLl B OTpHLlaTeJIbHOM, 3aTeM 3
eJJ.HHHLlbl B OTpHLlaTeJIbHOM H, HaKOHeLl, 13 eJJ.HHHLl B nOJIO}l{H-
TeJIbHOM HanpaBJIeHHH. OnpeJJ.eJIH KOHe4Hoe nOJIO}l{eHHe T04KH
Ha 4HCJIOBOH OCH.
188
6.6. PaccTOHHHe Me>K.ll.Y .ll.BYMH TO"lKaMH
KOOp.ll.HHaTHOH OCH
PaCCMOTpHM, HanpHMep, MOJJ.YJIb pa3HocTH 2 -( - 4) H
MOJJ.YJIb pa3HocTH - 4 - 2 (B 3THX Bblpa}l{eHHHX YMeHbwaeMoe
H Bbl4HTaeMoe nOMeHHJIHCb MeCTaMH). BHJJ.HM, LITO MOJJ.YJIH 3THX
Bblpa}l{eHHH paBHbI. lleHcTBHTeJIbHO,
12-(-4)1 = 161 =6 H 1-4-21 = 1-61 =6.
ECJIH H306pa3HTb YMeHbwaeMoe H BWl/HTaeMOe Ha KoopJJ.HHaT-
HOH OCH (pHC. 6.7), TO YBHJJ.HM, LITO MOJJ.YJIb JJ.aHHbIX pa3HOCTeH
6YJJ.eT paBeH paCCTOHHHIO Me}l{JJ.Y COOTBeTCTBYlOlllHMH T04KaMH, TO
eCTb AB = 6 eJJ.HHHLl. 3Hal/HT, eCJIH nOMeHHTb MeCTaMH YMeHbwa-
eMoe H Bbl4HTaeMoe, TO MOJJ.YJIb pa3HOCTH He H3MeHHTCH. OH paBeH
paCCTOHHHIO Me}l{JJ.Y JJ.ByMH TOl/KaMH KoopJJ.HHaTHOH OCH (KoopJJ.HHaTa
OJJ.HOH T04KH - YMeHbwaeMoe, a JJ.PyroH - BbI4HTaeMoe). I1TaK,
eCJIH KoopJJ.HHaTa OJJ.HOH T04KH a, JJ.PyroH b (HX nOCJIeJJ.OBaTeJIbHOCTb
HeCYllI,eCTBeHHa) H paCCTOHHHe d, TO
d= la-bl.
A.
962. np04TH TeKCT naparpa4>a H 06"bHCHH, KaK Bbl4HCJIHTb pac-
CTOHHHe Me}l{JJ.Y T04KaMH C (5) H D (- 3).
963. Bbll/HCJIH (y C THO) paCCTOHHHe Me}l{JJ.Y JJ.ByMH T04KaMH A H B:
I) A (0), B(I); 2) A (2), B(5);
3) A (-10). B(I); 4) A (0), B(-3);
5) A (-5), B (-10); 6) A (3), (B-3).
964. Bbl4HCJIH paccToHHHe Me}l{JJ.Y JJ.BYMH T04KaMH 4HCJIOBOH OCH:
I) A (-3,5), B(I,4); 2) K(4,3), B(I,8);
3) C(-2,9), D(-5); 4) F(O), G(-5,4);
5) A(-9,1), B(-4,9); 6) P(-O,7), R(2,3),
6.
965. Ha4epTH 4HCJIOBYIO OCb, OTMeTb Ha HeH TOl/KY A ( - 2) H oTpe-
A(-41 6(2)
I I , .
-5 -4 -3 -2 -1 0 2 3
PHC. 6.7
189
y,M
972. BbI4HCJIH:
I) 6.2+
10 7
IT' 115
3 5
3-.4-
4 8
2) 6 +:
2
'2.
7' 14
5. 1
6. 1 24
3) 0,824.3,25
4,75.0,032
172.3,15
4) 13,72:2,45
52,8: 0,22
5,928: 0,039
F
-30
973. Bbl4HCJIH 3Ha4eHHe Bblpa>KeHHH:
X,M
I) 6,6.23,5+ 1,1.237;
3) (
+ D :(
-
) ;
O,75(++2' 1
)
5)
2) 22,22: 55 - 0,06.0,335;
1 I
4) 5(42"-3,6):2 4 ;
A
B
2 7
3-12
2
3'7+ 0 ,25
6) 19 .
28 . 2 - 0 ,25
PHC. 6.8
974. OJJ.Ha cTopoHa TpeyrOJIbHHKa 6,3 CM, BTopaH n CM, a TpeTbH
Ha 3,5 CM JJ.JIHHee BTOpOH, COCTaBb Bblpa}l{eHHe JJ.J]H BbI4HCJIeHHH
nepHMeTpa TpeyrOJIbHHKa. Bbl4HCJIH nepHMeTp, eCJIH:
I) n=4,3, 2) n=6,3,
975. 3anOJIHH T a6JIHLlY:
IY
:-l tftt!j
30K AB, JJ.JIHHa KOToporo 4 eJJ.HHHLlbl. CKOJIbKO TaKHX B03MO>K-
HocTeH?
966. Ha 4HCJIOBOH OCH OTMe4eHbl TOl/Ka L (I) H oTpe30K KL, JJ.J1HHa
KOToporo 6 eJJ.HHHLl. OnpeJJ.eJIH pacnOJIO}l{eHHe TOl/KH K Ha
l/HCJIOBOH OCH.
967. Ha KoopJJ.HHaTHoH nJIOCKOCTH Ha4epTH npHMoyrOJIbHHK ABCD
no ero BepWHHaM: A (-4; 3), B (I; 3), C (I; -4) H D (-4; -4).
Bbll/HCJIH nJIOlll,aJJ.b H nepHMeTp npHMoyrOJIbHHKa.
968. Bbl4HCJlH nJIOlllaJJ.b H nepHMeTp KBaJJ.paTa, eCJIH KOHLlbl OJJ.HOH
CTopoHblero: I) A(0;0)HB(3;0);2) A(-2; -1)HB(-2;4),
969. Bbl4HCJlH nepHMeTp H nJIOlll,aJJ.b npHMoyrOJlbHHKa, eCJIH
H3BeCTHbl TPH ero BepwHHbl: A (-3; 2), B (4; 2) H C (4; -I).
970. KOJIH Ha4epTHJI nJIaH CBoero JJ.OMa (KLMN) H caJJ.a (ABCDEF)
Ha KoopJJ.HHaTHoH nJIOCKOCTH (pHC. 6.8). no 3TOMY nJIaHY KOJIH
cJJ.eJIaJI MHoro Bbll/HCJleHHH, CJJ.eJIaH H TbI.
OTMeTb Ha KoopJJ.HHaTHoH nJIOCKOCTH T04KH, KoopJJ.HHaTaMH KO-
TOpblX 6YJJ.YT COOTBeTCTBYlOlliHe 3Hal/eHHH x H y.
976*. JIJIHHa napKa npHMoyroJIbHOH <J>OPMbl Ha 400 M 60JlbWe ero
WHpHHbl, npH4eM OTHoweHHe JJ.JIHHbl H WHpHHbl paBHO OTHowe-
HHIO 5:3. CKOJIbKO BpeMeHH nOTpe6yeTcH newexoJJ.Y, 4TO-
6bl 060HTH napK, eCJIH ero CKOpOCTb 2 KM ?
'I
6.7. YMHO)f(eHHe .ll.BYX "IHCeJ1
JJ,nH nOBTopeHMH
971. Bbll/HCJIH (y C THO):
I)
.15 2)
:3
2.
4'
6 . 9
7 4 2 . 2
8 5 5'"3
3) 0,7.0,06
100.0,271
95.0,1
4) 43: 1000
93,6:3
0,7:0,01
npaBHJIa JJ.eHcTBHH C paLlHOHaJIbHblMH l/HCJlaMH <J>OPMYJIHPYIOTCH
TaKHM 06pa30M, 4T06bl H3BeCTHble 3aKOHbl JJ.eHcTBHH JJ.JIH nOJIO}l{H-
TeJIbHblX 4HCeJI OCTaJIHCb B CHJIe (npH CJIO}l{eHHH Mbl y}l{e y6eJJ.H-
JIHCb B 3TOM). npH YMHo}l{eHHH paLlHOHaJIbHWX l/HCeJI H3Y4eHHble
3aKOHbl 6YJJ.YT BepHbl, eCJIH Mbl 6YJJ.eM PYKOBOJJ.cTBoBaTbcH CJIe-
lI.yIOIllHM:
190
191
I. llpou38eiJeHue iJ8YX oTpuqaTe/l.bHblX tlUCe/l. eCTb tlUC/l.O no-
/l.03ICUTe/l.bHOe. MOiJY/l.b ICOTOpOZO pa8eH npou38eiJeHuIO MoiJY/l.eU
MH03ICUTe/l.eU.
n p H M e p l. I) (-2).(-3)=2.3=6;
2) (-4).(-0,3)=4.0,3= 1,2;
3) ( _l.... ) . ( _.!.. ) =l......!..= 3.2 =....!....
4 3 4 3 4.3 2'
2. llpou38eiJeHue iJ8YX tluce/l. C pa3HbtMU 3HaICaMU eCTb tlUC/l.O
oTpuqaTe/l.bHoe. MoiJY/l.b ICOTOpOZO pa8eH npou38eiJeHuIO MoiJY/l.eU
MH03ICUTe/l.eu.
npHMep 2. I) (-5).2=-(5.2)=-10;
2) 0,2.(-1,3)= -(0,2.1,3)= -0,26;
3) ( _
) .2= _ (
.2 ) = -
=-
6 10 6 10 6.10 4 .
HTaK, HY}I{HO 3anoMHHTb, LITO npOH3BeJJ.eHHe JJ.BYX OTpHLlaTeJIb-
HblX 4HCeJI BCerJJ.a nOJIO}l{HTeJIbHO, a npOH3BeJJ.eHHe JJ.BYX l/HCeJI C
pa3HblMH 3HaKaMH OTpHLlaTeJIbHO.
3 Ec/l.u XOTlI 6bl OiJUH U3 MH03ICUTe/l.eU pa8eH HY/l.IO. TO U npo-
u38eiJeHue pa8HO HY/l.IO. 06paTHO: ec/l.u npou38eiJeHue pa8HO HY/l.IO.
TO XOTlI 6bl OiJUH U3 MH03ICUTe/l.eU pa8eH HY/l.IO.
n p H Me p 3. I) (-3,4).0=0; 2) 0.(-7)=0;
3) eCJIH (-5).(x+2)=0, TO 06H3aTeJIbHO x+2=O, H, CJIeJJ.OBa-
TeJI bHO, x = - 2.
OTpHLlaTeJIbHblH MHO}l{HTeJIb, CTOHIlI,HH Ha nepBoM MeCTe, 3anH-
CblBaTb B cK06Kax He06H3aTeJIbHo. HanpHMep, npoH3BeJJ.eHHe ( - 3).4
TO }l{e caMoe, 4TO H -3.4, TO eCTb (-3).4= -3.4= -12.
TaKHM >Ke 06pa30M, HanpHMep, MO}l{HO 3anHcaTb: (-5).(-6)=
= -5.( -6)=30.
A.
977. npoBepb no TeKcTY, nOMHHWb JIH Tbl:
I) KaK HaHTH npoH3BeJJ.eHHe JJ.BYX oTpuqaTe/l.bHblx l/HCeJI;
2) KaK HaHTH npoH3BeJJ.eHHe JJ.BYX ttuce/l. c pa3HblMu 3HaKaMU;
3) npH KaKOM yCJlOBHH npOH3BeJJ.eHHe paBHO HYJIIO.
npHBeJJ.H npHMepbl.
978. KaKoH 3HaK =, < HJIH > HY}I{HO nOCTaBHTb BMeCTO 3Be3-
JJ.Ol/KH (oTBeT nOHcHH):
I) -9,8.5 * 0; 2) -9,6.(-4) * 0; 3) -27.0 * 0;
4) -5.(-7)*0; 5) 5,8.4,7*0; 6) 0.(-6)*0?
192
979. Bbl4HCJIH (y C THO):
1) 7.8 2) -5.6
7.( -8) 8.( -5)
-7.8 0.( -3)
980. Bbl4HCJIH (y C THO):
I) _
.2
5 4
.( -
)
-
.( -+ )
4) -
.( -
)
1 1
--.-
2 12
13 0 ,(_+)
981. BbI4HCJIH:
l) 27. ( - 6)
-67.(-15)
-15.0,87
4)
.I..!..
2 3
-I
.( -
)
.( -+ )
3) 0,3.(-0,2)
-4.0,5
1,2.( -100)
4) -0,01.8,5
0,7.( -3)
-6.( -0,4)
3 3
2) --.-
7 5
1 3
2'8
3) -3.
5
4
-9'( - 2)
0.( - : )
6) I : -(-
)
-3}.( - ;)
- I ....!.....l....
2 2
.( -+ )
1
5) 3.(-3)
I 3
--.1-
4 5
3
--=,--( -14)
2) -5.(-62)
169.243
-16,9.3,02
5) -3+.( -f )
-3
.( - : )
5 7
14 15
3) 42.(-10)
32,5.( -4,3)
-1,01.( -0,02)
6) 2+.(-2+)
-5+-( -2+)
7 4
-16'21
982. Bbll/HCJIH 3Ha4eHHe Bblpa}l{eHHH -18y, eCJlH y= I; y= -1,2;
y=0,75; y= -3,92.
983. 3anOJIHH Ta6JIHLlY:
I
I 3 -2 5 -0,5
a -1-
4 4 8
4
-"5 a
984. BbI4HCJIH:
I) -4.(-3,2+5,7)
( -0,8-0,12)-5
- I; (-+- I
)
2) -2,8.(-15)+4,2.(-3,5)
-0,7 .180-8,4.3,6
+.( -
)-
.( - : )
193
985. BbI4HCJIH 3Ha4eHHe BbIpa>KeHHH -4,5 (5-x), eCJIH x= -2;
x= I; x= -0,6; x=7,4,
6_
986. BbI4HCJIH:
I ) I.!..!..' ( - 27 ) 2 ) _I
. 27 3) -..!.2.. ( -2
)
30 82 36 106 20 57
-2
5 .1,5 -6
.( -2,8) -
.24,6
4 1 1
-3""7.( -1,08) -5,6.2 2 -3"3'( -0,01)
987. npeJJ.cTaBb Ka>KJJ.oe H3 l/HCeJI -6; -8; 9; 10; -0,5; -1,2 H
- 1 B BHJJ.e npoH3BeJJ.eHHH JJ.BYX MHO>KHTeJIeH, OJJ.HH H3 KOTO-
pblX paBeH - I.
988. npeJJ.cTaBb Ka>KJJ.oe H3 4HCeJI 1, 4, 9, 25, 36 H 100 B BHJJ.e
npoH3BeJJ.eHHH JJ.BYX paBHbIx MHO>KHTeJIeH. CKOJIbKO HMeeTCH
B03MO>KHOCTeH?
989. npeJJ.cTaBb Ka>KJJ.oe H3 4HCeJI -1, -9, -16, -36 H -100
B BHJJ.e npoH3BeJJ.eHHH JJ.BYX npoTHBonOJIO>KHbIX 4HCeJI.
990. BbI4HCJIH:
I) 5+2,(-0,3)+0,3-1,2,( -+);
2) 8-4.( -2+ )+0,6+ 1,8.( -+ ) ;
3) 1+(4,24-1,2,( -I
)).(-10);
4 ) -4. ( -I2--
.(-6)-4
) '
5 4 20 '
5) (( -0,5-
)-( -+ )).( +++.(-0,5»),
911. BbI4HCJIH 3Ha4eHHe BbIpa>KeHHH 2x-3y, eCJIH:
1 4
I) x= -5 H y= -2; 2) x=""6 H Y=g; 3) x= -1.8 H
5
Y=12'
992. npH KaKHX 3Ha4eHHHX a H b 6YJJ.YT BepHbI HepaBeHcTBa:
1) ab>O; 2) ab<O; 3) ab=O; 4) ab;;;:::O; 5) ab
O?
993*. PewH ypaBHeHHe:
I) 3(x-7)=0
-3 (x+ 7)=0
(x-2) (x-I)=O
2) 0,9 (3,2+z)=0
(t-5).5=0
(t+2) (t-I)=O
194
6.8. nepeMeCTHTeJ1bHblH H cOIleTaTeJ1bHblH
3aKOHbI YMHO)KeHHH
ECJIH cpeJJ.H MHO>KHTeJIeH HMelOTCH OTpHLlaTeJIbHbIe 4HCJIa, TO YMHO-
>KeHHe 3THX 4HCeJI, KaK MbI BHJJ.eJIH, CBOJJ.HTCH K YMHO>KeHHIO HX
MOJJ.YJIeH. Ho MOJJ.YJIH - 4HCJIa nOJIO>KHTeJIbHbIe. 3Ha4HT, JJ.JIH paLlHo-
HaJIbHbIX 4HCeJI OCTalOTCH B CHJIe nepeMeCTHTeJIbHbIH H C04eTaTeJIb-
HbIH 3aKOHbI YMHO>KeHHH. 11TaK, eCJIH a, b H C - JII06ble paLlHo-
HaJIbHbIe 4HCJIa, TO
I ab=ba I
H
(ab)-c=a (be)
BcnOMHH H C<POPMYJIHPYH 3TH 3aKoHbI!
nepeMeCTHTeJIbHbIH H C04eTaTeJIbHbIH 3aKOHbI n03BOJIHIOT B npo-
H3BeJJ.eHHH HeCKOJIbKHX 4HCeJI crpynnHpoBaTb MHO>KHTeJIH B JII060M
nopHJJ.Ke. HanpHMep:
.( -3).( - ; )-+=( -f.
).( -3.+ )= -1.( -1)= I.
npOH3BeJJ.eHHe HeCKOJIbKHX 4HCeJI, OTJIH4HblX OT HYJIH - If.UCAO
OTpUl{aTeAbHoe, eCJIH 4HCJIO oTpHLlaTeJIbHbIX MHO>KHTeJIeH Helf.eTHoe.
ECJIH 4HCJIO oTpHLlaTeJIbHbIX MHO>KHTeJIeH If.eTHOe, TO npOH3BeJJ.e-
HHe - If.UCAO nOAWICUTeAbHoe. npOH3BeJJ.eHHe paBHO HYJIIO, eCJIH
XOTH 6bI OJJ.HH H3 MHO>KHTeJIeH paBeH HYJIIO.
A.
994. np04TH TeKCT H nOKa>KH Ha npHMepax:
I) HCnOJIb30BaHHe nepeMecTuTeAbHozo 3aKOHa YMHO>KeHHH;
2) HCnOJIb30BaHHe COlf.eTaTeAbHOZO 3aKOHa YMHO>KeHHH.
3) KaKHM 06pa30M 3aBHCHT 3HaK npoH3BeJJ.eHHH HeCKOJIbKHX
MHO>KHTeJIeH OT 3HaKOB MHO>KHTeJIeH?
995. KaKoH 3HaK (<< +» HJIH «-») HMeeT npoH3BeJJ.eHHe, eCJIH OHO
COCTOHT H3:
I) OJJ.Horo nOJIO>KHTeJIbHOrO H OJJ.Horo oTpHLlaTeJIbHOrO MHO>KH-
TeJIeH;
2) JJ.BYX oTpHLlaTeJIbHbIX H OJJ.Horo nOJIO>KHTeJIbHOrO MHO>KHTe-
JIeH;
3) JJ.BYX nOJIO>KHTeJIbHbIX H OJJ.Horo oTpHLlaTeJIbHOrO MHO>KHTe-
JIeH;
4) Tpex oTpHLlaTeJIbHbIX H JJ.BYX nOJIO>KHTeJIbHbIX MHO>KHTeJIeH?
195
996. OnpeJJ.eJIH 3HaK npoH3BeJJ.eHHH H Bbl4HCJIH (y C THO) :
1 4
2) -I +5 2 . U
-3
. ( -
) -5
4 13
(3-5.0,9).( -6) -6.( - : ).++3
Bbl4HCJIH 3Ha4eHHe Bblpa>KeHHH - 0,5xy, eCJIH I)
= - 17 H
y= -8; 2) x=0,4 H y= -7,5,
6.
1000. Bbl4HCJIH 3Ha4eHHe Bblpa>KeHHH:
1)
.4,8.( -1+ ),10
I 3 2
""'5'4.(-5).2"3
I) -7.(-5).2
-4.8.( -25)
-2.(-16).(-5)
I
3) 2.(-2).(-3)
-5.(-7).( -+)
4.( -+ ).12
997. BbmOJIHH JJ.eHcTBHH:
I) -
.8.( -2+ )
.( -5).( - ; )
.1
. ( -3
)
4 3 4
3) 1+.( -+ ).( -
)
- 1 5 6 ,( - 115 ).( -18)
- 130 .(_
).2
998. BbmOJIHH JJ.eHcTBHH:
I) -4.27.(-0,05)+1,9
-162+8.3.(-35)
999.
2) -0.5.(-1,9).4
-2.( -4,5)-( -0,8)
1,5.( -12).2
4) +.( -4).( -2)
- : .( - :) .5
_..!-. I J..-. 4
5 4
5 3
2) -""6.7'10
1+.( - 1
).( -
)
-12.( -1
).( - ;2 )
4) - ; .+.( -I
1
; .( -
) .28
3J..-. ( -
) .( -5)
5 16
2 4 1
) -6,1'""'5.(-10).14
1
5,6.( -10).0,5'7
-
.( -+). 1 7 2 '
-
.( - : ).( -+ ).( -
)
196
1001. BbmOJIHH JJ.eHcTBHH:
I) 1,6-3,6,(-0,5),(-+);
2) 54.( -+-+ ).( -+);
3) -8.7.0.(-2)-132.( - 11 2 ),( -
6 );
1 1
4) 12.(-0,25)-(-15)-8'4.(-9).
1002. OnpeJJ.eJIH 3HaK npoH3BeJJ.eHHH abed, eCJIH:
I) a>O, b>O, c<O, d>O; 2) a>O, b>O, c>O, d<O;
3) a<O, b<O, c>O, d<O; 4) a<O, b<O, c<O, d<O;
5) a<O, b>O, c>O, d<O, 6) a=O, b>O, c<O, d>O.
1003. KaKHe 3HaKH MorYT HMeTb 4HCJIa a, b H c, eCJIH:
I) abc>O; 2) abc
O; 3) abc<O; 4) abc
O?
1004*. Bbl4HCJIH 3Ha4eHHe Bblpa>KeHHH label H lal.lbl.lcl,
eCJIH:
I) a= -I, b=2 H c=3; 2) a=2, b= -3 H c= -4;
3) a= -3, b= -2 H c= -4.
KaKoH MO>KHO CJJ.eJIaTb BbIBOJJ.?
6.9. Koa4>4>H.....HeHT
PaCCMOTpHM npOH3BeJJ.eHHe, HanpHMep 5. a.( -3). b. c, KOTopoe
COCTOHT H3 6YKBeHHblx H 4HCJIOBblX MHO>KHTeJIeH. nepeMeCTHTeJIbHblH
H c04eTaTeJIbHblH 3aKOHbl YMHO>KeHHH n03BOJIHIOT B 3TOM Bblpa>KeHHH
OTJJ.eJIbHO crpynnHpOBaTb 4HCJIOBble H OTJJ.eJIbHO 6YKBeHHble MHO>KHTe-
JIH. nOJIY4HM:
5.a.( -3).b .c=( -3.5).(a. b .c)= -15abc.
nOJIY4eHHblH TaKHM o6pa30M MHO>KHTeJIb - 15 Ha3blBaeTCH KoacIJ-
c1JHI\HeHToM. K03<p<pHLlHeHT 06bl4 HO 3anHCblBalOT nepeJJ. 6YKBeHHblMH
MHO>KHTeJIHMH. K03<p<PHLlHeHT I He nHWYT. 3Ha4HT, I.a=a, I.xy=
=xy H T. JJ.. BMeCTO K03<p<PHLlHeHTa - I nHWYT TOJIbKO 3HaK «-».
HanpHMep, BMeCTO -I. ab 3anHWYT: - ab, TO eCTb -I .ab = -ab,
n p H Me p l. YnpOCTHM Bblpa>KeHHe 7m.5a.(-3).n.
3anHCblBaH 4HCJIOBble MHO>KHTeJIH nepeJJ. 6YKBeHHbIMH, nOJIY4HM:
7m.5a.(-3).n= -3.7.5.amn= -105amn,
K 03 <p<p H LlHeHTOM nOJIY4eHHoro Bblpa>KeHHH HBJIHeTCH 4HCJIO - 105
197
1010. CocTaBb BbIpa>KeHHe, K03<pq>HLlHeHTOM KOToporo HBJIHeTCH:
2
I) npOH3BeJJ.eHHe 4HCJIa "3 H 4HCJIa, eMY npOTHBOnOJIO>KHOrO;
2) npoH3BeJJ.eHHe 4HCJIa 0,8 H 4HCJIa, eMY o6paTHoro;
3) npoH3BeJJ.eHHe LleJIbIX 4HCeJI, npH KOTOpbIX HepaBeHCTBO
-3:::;;;x<O,5 BepHo,
n p H Me p 2. YnpocTHM BbIpa>KeHHe -2x.( -y).0,1.5,
TaK KaK -y= -I.y, TO nOJIY4HM:
-2x.( -y).O,I.5= -1.( -2).5.0,I.xy= I.xy=xy.
K03<P<PHLlHeHToM nOJIY4eHHoro BbIpa>KeHHH HBJIHeTCH 4HCJIO I.
1
n p H M e p 3. YnpocTHM Bblpa>KeHHe -"3.mn.3a,
nOJIY4HM:
1 1
-3"mn.3a= -3".3amn= -I.amn= -amn.
1011.
K03<p<pHLlHeHT nOJIY4eHHoro BbIpa>KeHHH paBeH - I.
A.
1005. np04TH TeKCT naparpa<pa, paccMoTpH npHMepbI H 06'bHCHH,
KaK HaHTH K03f/Jf/JUl{UeHT Ka>KJJ.oro H3 CJIeJJ.YlOlUHx BbIpa>KeHHH:
I) -4.x.5.y.2; 2) 5e.0,2; 3) -2s.+t.
1006. Ha30BH K03<p<pHLlHeHTbl CJIeJJ.YIOIlI,HX BWpa>KeHHH:
I) 2xy 2) -5lbe 3) -0,74mn 4) -
ab
ab -x -xy m
-2n 3m xyz 2,lab
1007. OnpeJJ.eJIH K03<p<pHLlHeHTbI (y C THO) :
I) 5a.7 2) 6mn.5 3) 4x.(-2)
-3x.(-3) 5ab.(-2).e -a.(-3).2
-2.0,5x 3xy.( -5).t 3,5.( -2xH -I)
1008. YnpocTH BbIpa>KeHHe:
I) 5,2Is.(-t).2,5 2) -0,4.(-6,3b).2
a. ( - 1 + ) be -
m. ( -
). 2n
ab. ( - 6e) - 5a. ( -
b ).3
6.
1012.
1013.
1014.
1015.
1016.
1009. YnpocTH BbIpa>KeHHe:
I 5
1) -3"3a.( -0,9b)'12
3 6ed. ( -I
) '
f
, 14 15
-0 00IX' ( -66
) '
Z
, 3 15
2) -6,4.( -
x ).( -ITst)
5,25y.( -1+).( - :6 )
2
16-ab.0 le.O 6 d
3 ' ,
198
AnSI nOBTopeHHSI
BbI4HCJIH (y C THO):
I) 8-7-2+3
-15-6+8-10+ 7
6-(7 -5-(1)
2) -5+6+5-7-8-6
16-6+ 10-8-9-10
-1,9+3,2-2,1 +<1,8
BbI4HCJIH:
I) 56,44: 0,068
379,5: 5,5
5,33: 8,2
6,84:0,9
2) -0,84.33,5-9,1.3,4
200,4: 33,4 + 25,5: 68
-15,3+4,87-6,14-3,03
3,2.1,5-4,9+ 15,6
HaHJJ.H 4HCJIa, o6paTHbIe JJ.aHHbIM:
3 9 1 4
7; IT; 0,4; 1,3; 10; 2"3; 1""'5'
BbI4HCJIH:
I) 5.3 2 ) 2 7 13 3 ) 14:2 4 5 4) 04 1 1
24 'S \6:24 ' : 15
7 1 1 . 2. 1 1
Iff: 1 2 53'".4 63'".19 5z:0,ll
OJJ.Ha CTopOHa Y4aCTKa 3eMJIH, HMelOIlI,erO <POPMY TpeyroJIb-
HHKa, 60 M, BTopaH COCTaBJIHeT
O% 3TOH CTOpOHbI, a
TpeTbH - Ha 6 M JJ.JIHHHee BTOpOH. BbI4HCJIH nepHMeTp Y4aCTKa.
Ha BbIc-raBKe JIOwaJJ.eH JJ.eTeH
nOpa3HJIH 60JIbWaH JIOMOBaH
JI ow aJJ.b H MaJIeHbKHH nOHH.
Maccbl 3THX >KHBOTHbIX 6bIJIH
COOTBeTCTBeHHO 1,2 T H 78 Kr.
CKOJIbKO npoLleHToB COCTaB-
JIHeT Macca nOHH OT MaCCbl
JIOMOBOH JIOwaJJ.H?
".
..
11"....-... I
'",
! "
,u I,
\ .",
;
),
.
D
'!!i\
7Bier
f
.f. ..
199
1017. YnpocTH Bblpa>KeHHe:
I) 2x+7x 2) 8a+3a+4 3) 18x-5x-4x
6a-a l,7x-I,2x-3 6y-2,5y+0,8y
9y + 23y 4b + 28b + 7 2,3z + 0,8z - O,5z
1018. B nepBoM Llexe 3aBOJJ.a m pa604Hx, BO BTOpOM - B 2 pa3a
60JIbWe H B TpeTbeM 325 pa604Hx. CKOJIbKO pa604Hx Ha 3a-
BOJJ.e? CocTaBb BbIpa>KeHHe H BbI4HCJIH ero 3Ha4eHHe, eCJIH:
I) m=89; 2) m=246.
1019. Ha KoopJJ.HHaTHoH nJIOCKOCTH nocTpoH npHMoyrOJIbHHK, Bep-
WHHbI KOToporo A (-3; I), B (5; 1), C (5; -4) H D (-3; -4),
BbI4HCJIH nepHMeTp H nJIOlll,aJJ.b 3Toro npHMoyrOJIbHHKa.
OTCIOJJ.a CJIeJJ.yeT: eCJ1H nepe,u, cKo6KOH CTOHT 3HaK «-», TO npH
paCKpblTHH CKo60K HY>KHO 3HaKH, CTOHUI,He nepe,u, CJ1araeMblMH B
cKo6Kax, 3aMeHHTb Ha npOTHBOn0J10>KHble.
n p H M e p 3. -(2x-3y-4z+9)= -2x+3y+4z-9.
PaBeHCTBO OCTaHeTCH BepHblM, eCJIH nOMeHHTb MeCTaMH ero
npaBYIO H JIeBYIO 4aCTH. 3Ha4HT, H3 paBeHcTBa
CJIeJJ.yeT
(a+b) c=ac+bc
ac+bc=(a+b) c.
OHO 03Ha4aeT: eCJIH npoH3BeJJ.eHHH HMelOT 061l1,HH MHO>KHTeJIb
(B npHMepe 4HCJIO c), TO npH CJIO>KeHHH 3THX npOH3BeJJ.eHHH
06WHH MHO>KHTeJIb MO>KHO 3anHcaTb 3a cK06KaMH. B cK06Kax OCTa-
eTCH CYMMa JJ.pyrHx MHO>KHTeJIeH. B TaKOM CJIY4ae rOBopHT: 06U1,HH
MHO)l(HTeJ1b BblHeceH 3a CKo6KH.
n p H M e p 4.
I) 7.8-9.8=(7-9).8= -2.8= -16;
2) 2a-2b=2 (a-b);
3) 3x+9y-3=3 (x+3y-I).
A.
6.10. Pacnpe.ll.eJ1HTeJ1bHblH 3aKOH YMHO>KeHHH.
PacKpblTHe cKo6oK
BcnOMHH, B 4eM COCTOHT pacnpeJJ.eJIHTeJIbHbIH 3aKOH YMHO>KeHHH
OTHOCHTeJIbHO CJIO>KeHHH JJ.JIH nOJIO>KHTeJIbHbIX 4HCeJI.
Y6eJJ.HCb Ha npHMepax, 4TO YMHO>KeHHe H CJIO>KeHHe B3aHMOCBH-
3aHbl pacnpeJJ.eJIHTeJIbHbIM 3aKOHOM YMHO>KeHHH H B TOM CJIY4ae, Kor-
JJ.a cpeJJ.H CJIaraeMblX H MHO>KHTeJIeH HMelOTCH OTpHLlaTeJIbHbIe 4HCJIa.
npOBepb, HanpHMep, 4TO:
I) -3 (-5+ 7)= -3.( -5)+( -3).7;
2) 5 (-2-3)=5.( -2)+5.( -3).
ECJIH a, b H C - JII06bIe paLlHOHaJIbHbIe 4HCJIa, TO
(a+b) c=ac+bc.
1020. np04TH TeKCT naparpa<pa, paccMoTpH npHMepbI H paCCKa>KH,
KaK paCK.pbLTb CK.o6K.u B BbIpa>KeHHHx.
1021. PacKpoH cK06KH:
PacnpeJJ.eJIHTeJIbHbIH 3aKOH YMHO>KeHHH OCTaeTCH B CHJIe He-
3aBHCHMO OT l/HCJIa CJIaraeMbIX B cK06Kax. npH npHMeHeHHH pac-
npeJJ.eJIHTeJIbHoro 3aKOHa rOBopHT: pacKpoeM cKo6KH.
n p H M e p I. PacKpoeM cK06KH B BbIpa>KeHHH
3 ( - 5a + 4b - 3).
3 (-5a+4b-3)= -15a+ 12b-9,
TaK KaK 3.( -5)= -15, 3.4= 12 H 3.( -3)= -9.
n pH M e p 2. -I (3x-5y+2)= -3x+5y-2, TaK KaK
-1.3=-3. -1.(-5)=5 H -1.2=-2.
BMecTo K03<p<pHLlHeHTa - I, CTOHllI,erO nepeJJ. cK06KOH, nHwYT
TOJIbKO 3HaK «-». n03TOMY BbIpa>KeHHe -I (3x -5y+2) MO>KHO
3anHCaTb: -(3x-5y+2).
nOJIY4HM
1) 2(x+l)
3 (t - 2)
-2 (2x+ I)
3) 5 (-1,2a+0,4)
- (8,3z - 5,2)
-2 (-3,5x-4,8)
1022. PacKpoH cK06KH:
I) -2(a-2b+l)
(2a-3b -4).3
-(5x-2y-lO)
3) (2a-4b +3).( - 3)
-(4x-2y+9)
-5 (-a+2b+3)
200
2) -(4a+3}
4(-2x+7)
-( -5n-6)
4) 3 ( -+n-
)
1
-""4 (8x+3)
-2(
y- 152 )
2) 4 (3-2x+5k)
6 (1-3c+d)
(a+b-2)-( -7)
4) -2 (a+2b-5)
-(7 +x-y)
-( -5+2a+b)
201
1023. BbIHecH 061l1,HH MHO>KHTeJIb 3a cK06KH:
I) 9b+9c 2) 3x+3y 3) 5x-5y
4c-4d 8b-8c 6x+6y
-6x-6y 7a+7b -2a+2b
4) 7x-14
6a+9
15y - 10
2. B KaKHX paBeHcTBax npHBeJJ.eHHe nOJJ.06HbIX CJIaraeMbIX BbmOJI-
HeHO HeBepHo (npoBepb):
I) 3a-4a+ 7a=6a;
3) 4y-3y-y=0;
2 7 2 .
5 ) -x--x+x= --x,
333
ECJIH C 3aJJ.aHHeM CnpaBHJICH, TO BbIHCHHJI, 4TO B OJJ.HOM paBeHCTBe
oWH6Ka. B KaKOM? B 4eM OHa 3aKJIlO4aeTcH?
3. BbIpa>KeHHe MO>KeT cOJJ.ep>KaTb HeCKOJIbKO pa3JIH4HblX rpynn
nOJJ.06HbIX CJIaraeMblX, HanpHMep, 5a+6a+7b+2-2b-3a-6.
B TaKOM CJIY4ae nOJJ.06HbIe CJIaraeMbIe pa3Hwx rpynn LleJIeco06pa3Ho
nOJJ.4epKHYTb no-pa3HoMY, HanpHMep:
5a + 6a + 7b +2- 2b - 3a -6=8a+5b-4,
TaK KaK 5a+6a-3a=8a, 7b-2b=5b H 2-6= -4.
2) -5x+3x-7x+x=8x;
4) -1,2a-0,8a+4a=2a;
6) IOk+k-2k-5k=4k?
6.
1024. PacKpoH cK06KH:
I) - 5 (3x + 7 - t)
(1,2a+3,4b +2).3
-(4a-5b-3)
3) a(3x-y+z)
2a (x-3y+2z)
-6 (
x-+y++z )
1025. Bbll/HCJIH (y C THO), HCnOJIb3YH
yMHO>KeHHH:
7.(-92)=7.(-90-2)= -630-14= -644.
I) 5 . ( - 95) 2) - 9. 10 1 3) I 7 . 102
24.(-99) 67.1002 -67.1002
9.(-98) 19.(-103) -24.(-99)
1026*. Bbl4HCJIH KaK MO>KHO np0lll,e:
I) 54.36-42.54+6.74;
2) 478.62+ 13.478-75.678.
2) 0,7 (x+0,5y-l)
-(8x+y-9)
(2x-4t+zH -2)
4) -m (2n-3x+ I)
(a-3b+5c).( -2m)
15 ( J... a +
b - J... c )
235
pacnpeJJ.eJIHTeJIbHbIH 3aKOH
4. HaHJJ.H oWH6KH B cJIeJJ.YIOIlI,HX paBeHcTBax:
I) 2a-3b+4a-b=6a-4b;
2) m-5a+3-m+2=5-5a;
3) -2,lx-3y-0,9x+3,2y=3x+0,2y;
4) 2-x+y-3x-4y-5= -4x-3y-3;
5) 3a+2x-3a+4=6a+2x+4;
3 2 I 1 1 1.
6) "'4x-TY-2x+""6Y="'4x-2Y'
7) 2x-(x-2y)-2y+2x-x-2y-2y=2x-4y;
8) 2-3 (2x-y)+2 (x+3)-2-6x+3y+2x+5= -4x+3y+ 7,
6.11. npHBe.ll.eHHe nO.ll.06HblX CJ1araeMbiX
ECJIH C 3aJJ.aHHeM CnpaBHJICH, TO 06HapY>Ki1JI OWH6KH B Tpex pa-
BeHCTBax. B KaKHX? B 4eM 3aKJlI04alOTCH 3TH OWH6KH?
MaTepHaJI :noro naparpa4>a H3Y4H caMOCTOHTeJIbHO.
1. PacnpeJJ.eJIHTeJIbHbIH 3aKOH YMHO>KeHHH n03BOJIHeT BbIHOCHTb
06WHH MHO>KHTeJIb 3a cK06KH (CM.
6.10), HanpHMep.
3a-6a+2a-6a=(3-6+2-6) a= -7a.
B BbIpa>KeHHH 3a-6a+2a-6a CJIaraeMble 3a, -6a, 2a, -6a
Ha3bIBalOTCH nOJl,06HblMH CJ1araeMblMH. nOJJ.06HbIe CJIaraeMbIe JIH60
He OTJIHl/aIOTCH OJJ.HH OT JJ.pyroro (HanpHMep, -6a H -6a), JIH60
OTJIH4alOTCH TOJIbKO K034>4>HLlHeHTaMH (HanpHMep, 3a H 2a). CJIO-
>KeHHe nOJJ.06HbIX CJIaraeMblX Ha3bIBaeTCH npHBeJl,eHHeM nOJl,06HblX
CJ1araeMbiX.
4T06bI npHBecTH nOJJ.06HbIe CJIaraeMbIe, HaJJ.O CJIO>KHTb HX K034>-
4>HLlHeHTbI H nOJIY4eHHbIH pe3yJIbTaT YMHO>KHTb Ha 061l1,YlO 6YK-
BeHHYIO 4aCTb.
202
A.
1027. npHBeJJ.H (y C THO) nOJJ.06HbIe CJIaraeMwe:
I) 7 x + 6x 2) - 5m - 7 n 3) 4a - 5a + 7 a
5y-3y -9a+2a 6x-3x-5x+x
5y-8y 6x-lOx t-4t+t+2t
1028. npHBeJJ.H nOJJ.06HbIe CJIaraeMble:
I) 5a-6a-7b+b 2) 4a+5b-6a-4b+10
x-y+3x-2y-4x 8-a+ 7 +5a-IOa
5t-2x+3x-5t-x 6x-7 +8y-3+4y
1029. YnpocTH BbIpa>KeHHe H BbI4HCJIH ero 3Ha4eHHe:
I) 3x-8x, eCJIH x= -4; x=O,7;
203
I
2) -6a+9a, eCJIH a= -15; a =""6;
3) -8y-6y, eCJIH y=2,4; y= -
.
1030. OTpe30K KL=8n CM, a OTpe30K LM=3n CM (pHC. 6.9).
COCTaBb Bblpa>KeHHe JJ.JIH BbI4HCJIeHHH JJ.J1HHbI OTpe3Ka KM,
BbI4HCJIH ero 3Ha4eHHe, eCJIH: I) n = 6; 2) n = 1,4.
1031. OTpe30K AB=4a CM, a OTpe30K BC=7a CM (pHC. 6.10).
I) HaHJJ.H OTpe30K AC, 2) Ha CKOJIbKO oTpe30K BC JJ.JIHHHee
oTpe3Ka AB?
1032. PacKpoH cK06KH H npHBeJJ.H nOJJ.o6Hble CJIaraeMble:
I) 4 (x-3)-2x 2) -(c+5)+2c
-2 (3a+ 1)+6a 3 (6-m)-1O
5 (2-m)-10 -( -4x-3)-4x
3) 2 (4-y)+2y 4) -2 (x-5)-1
-3 (2x-4)+8 (2a-I).( -2)+2
6 (-2a-3)+ 18, -(a-5)+2a
6.
K
.
PHC. 6.10
A
.
1033. YnpocTH BbIpa>KeHHe:
I) 0,3x-0,4x+x 2)
2,6c-5,ld-0,3d
- 7 ,5a - 2,5b + 4a
3) 4m-6,4-5,6m-I,9 4)
11,2u-2v+7,2v-12u
-4,3x+0,8-2,7x-l,5
L
.
d
20
50
c 3b
4b
CD
M
.
PHC. 6.9
o
B
.
3c
b
c
.
d
CD
o
PHC. 6,11
3) 5 (x-I)+2 (x+3)-7x 4) 2 (a+3)-(a-3)+ I
x-I-2 (x+2)-1 3 (a+b-I)-3a-3b+3
(2a+ 1).2-a+5 5 (1-2a)-3 (a+ I)
1035. YnpocTH Bblpa>KeHHe 3 (2-c)-4 (c+3) H BbI4HCJIH ero 3Ha-
4eHHe, eCJIH c= - 3,
1036. HaHJJ.H nepHMeTpw 4>Hryp, H306pa>KeHHbIX Ha pHcYHKe 6.11.
1037. PewH ypaBHeHHe:
I) 2x+3-x=5
5-2y+3y-7= -I
7a-9-6a+4= -4
2) 2 (x-I)-x=O
3 (a+5)-2a=-1
8 (z-5)-7z+3= -10
6.12. J].eJ1eHHe
2 1 1
-a+l-a--a
333
1 3
--x-x+-x
4 4
2 3 1
--b--c--b
5710
5 I 3 1
-a-I-a+-x--x
6 344
4 1
--I 5n-0 6--n + 2n
5 ' , 2
2 2
""'5 a +T b - b + 0 ,8a
MaTepHaJI 3Toro naparpa4>a H3Y4H caMOCTOHTeJIbHO,
1. .LI.eJIeHHe - 3TO JJ.eHcTBHe, B KOTOpOM no JJ.aHHoMY npoH3-
BeJJ.eHHIO H OJJ.HOMY H3 MHO>KHTeJIeH HaXOJJ.HT JJ.pyroH MHO>KHTeJIb.
Pa3JJ.eJIHTb 4HCJIO a Ha 4HCJIO b - 3Ha4HT HaHTH. TaKoe 4HCJIO x,
4T06bl
bx=a.
HanpHMep, HaHTH 4aCTHoe (- 6): 3 - 3Ha4HT HaHTH TaKoe 4HCJIO
x, 4T06bl 3x= -6. 04eBHJJ.Ho, x= -2, TaK KaK 3.( -2)= -6. 3Ha-
l/HT, -6:3= -2. TaKHM >Ke 06pa30M,
-6:( -3)=2, TaK KaK -3.2= -6,
6:(-3)=-2, TaK KaK -2.(-3)=6.
2. KaKHe paBeHcTBa
MOlll,blO YMHO>KeHHH):
I) -10:(-5)=2;
4) -2:(-1)=-1;
HeT (npoBepb
C no-
BepHble, KaKHe
1034. PacKpoH cK06KH H npHBeJJ.H nOJJ.06HbIe CJIaraeMble:
I) 6x-2 (3x-l) 2) 3 (x+2)-x+2
3 (a-5)+ 15-2a 5 (t-7)-6t+35
-4 (2b+ 1)-2b+3 2a-5 (a+3)-a
204
2) -6:t -2)=3; 3) -10:(-0,1)= 100;
5) -6:(-0,5)= 12; 6) -24:(-4)=8?
205
ECJIH C 3aJJ.aHHeM CnpaBHJICH, TO 06HaPY>KHJI OWH6KY B JJ.BYX pa-
BeHCTBaX. B KaKHX? B 4eM 3aKJII04aeTCH oWH6Ka?
3. 113 npeJJ.blJJ.ywHX npHMepoB BbITeKaeT:
qacTHoe OT iJelleHWl iJBYX oTpu'laTellbHblX 'Iucell eCTb 'IUCIlO nOllo-
3ICUTellbHoe. qT06bl HauTu MOiJYllb 'IaCTHOZO, HY3ICHO MO
Yllb iJellUMo-
zo pll3iJelluTb Ha MOiJYllb iJellUTellJl.
4. CJIeJJ.YIOIlI,He paBeHcTBa npoBepb YMHO>KeHHeM KaKHe paBeHcT-
Ba BepHbIe, KaKHe HeT:
I) 10:(-2)= -5; 2) -9:3= -3; 3) 12:(-4)=3;
4) -7: I = -7; 5) 18:(-9)= -4; 6) -0,2:0,1 = -2?
ECJIH C 3aJJ.aHHeM CnpaBHJICH, TO HaWeJI OWH6KY B JJ.BYX paBeH-
CTBaX. B KaKHX? B 4eM 3aKJlI04aeTCH oWH6Ka?
5. 113 npeJJ.bIJJ.YIlI,HX npHMepoB BbITeKaeT:
qacTHoe OT iJelleHWl iJBYX 'Iucell C pa3HblMU 3HaICQMU eCTb 'IUCIlO
oTpu'laTellbHoe. QTo6bl HauTu MOiJYllb 'IaCTHOZO, HY3ICHO MOiJYllb iJe-
IlUMOZO pa3iJelluTb Ha MOiJYllb iJelluTellR.
6. 060CHYH, 4TO:
I) 'IaCTHOe OT iJelleHWl HYIlR Ha /I.Io6oe OTIlU'IHOe OT HYIlJl
'IUCIlO paBHo HY/I.IO; 2) iJelluTb Ha HYllb Hellb3R.
A.
1038. BbI4HCJIH (y C THO), OTBeTbI npOBepb C nOMOWblO YMHO>KeHHH:
I) -5:(-1) 2) 18:(-6)
-6:( -2) 25:( -5)
0:( -3) -12:( -4)
3) -12:6 4) -0,2:(-0,1)
15:( -5) -4,2:2
16:( -8) -0,9:3
1039. BbI4HCJIH:
I) 125:(-5)
- 0,45: 0,9
36,6:( -1,2)
1040. BbI4HCJIH:
I) -+:( -+)
2 . 4
-J'""'5
1041. BbI4HCJIH:
I) -32:(-4)-6
-6.( -3)+85:( -5)
(24-36):(-6).3
15-(4+8:2)-6
2) -500:( -25)
3,2: ( - 20)
-213,2: 10,4
3) 4:(-8)
-10,2:( - 30)
-15,54:( - 42)
3)
. :( -
)
I . ( 1 )
2) J' -"3
. 1
-4.2""'5
. :( - 1 14 )
2) -6,3:3+0,1
-1-1:( -0,5)
-50: 10+8.( -4+9)
8.(3-9)-15:( -10)
206
1042. BbI4HCJIH:
I) -5+:(-7)-2
-
-2
'
7 7' 7
-8:( -2
)++
2) (3
-
):(-29)
(2+-8+ ):3
1 3 ( 1 )
-12'":""'5- 6 -12
1043. 3anOJIHH Ta6JIHLlY:
-3 -1 1 0 1,5 6
a 3
a:( -3)
-6:a
1044. Bbl4HCJIH 3Ha4eHHe BbIpa>KeHHH 9:a-l,2, eCJIH a= -6;
a= -0,3; a=2.
1045. PewH ypaBHeHHe:
I) 2x=-4
5x = - 2
2 4
gX= -15
2) -3x= 15
3x = - 18
1 7
-1""'5 x =210
3) -O,2x= I
- 3x= - 9
-O,06x= -1,2
6.
1046. BbI4HCJIH:
5 25 ( 1 ) 2) -3
'( -I 54)
1), :- 20 24"
-2,06:( - 1 7 0 ) -1,25:3 1
-1,05:( - ;0) 4,25:( -f)
1047. BbmOJIHH JJ.eHcTBHH:
1) 300:(50-100)+500:(400-500)-100;
2) -39:(19-32)-4.(18+36:(-9»;
3) (27-24:(8-11».(-9+8:(27-35»;
4) (2,02-16,61 :5,5):2,5-1,6;
5) (5,44+(-8,16:4)-(-1,48-1,12»):(-1,5),
1048. BbI4HCJIH:
-84.(3.5-50). 2 98.( -15)
I) 12.(-7) , ) -14.(20-4.10)
3) -=-0,78:( -2
)
-3 .
:1,4
-3+:( -16)
207
1049. BbInOJIHH JJ.eHcTBHH:
I) 3,4-19,5-0,3.
'(3+-2+) ;
1 5
2) -7:14+ 2 . 0 ,095+2,172.1,2;
3
3) 0,4-(9".(4,22-28,07:3,5).
1050. Bbl4HCJIH 3Ha4eHHe BbIpa>KeHHH k-3 eCJIH k= -I;
1 I-k'
k= -"'4; k=2,5; k=3.
1051. BbI4HCJIH 3Ha4eHHH BbIpa>KeHHH la:bl H lal: Ibl, eCJIH: I) a=
= -42 H b=7; 2) a=60 H b= -12; 3) a= -0,836 H
b=O,95; 4) a=56,1 H b= -85,
fl pH Me p l. PeWHM ypaBHeHHe 3x-6=5x.
npH6aBHM K 06eHM 4aCTHM ypaBHeHHH 4HCJIO 6. flOJIY4HM:
3x-6+6=5x+6, HJIH 3x=5x+6.
BHJJ.HM, 4TO CJIaraeMOe -6 nepeMeCTHJIOCb H3 JIeBOH 4aCTH JJ.aH-
HOro ypaBHeHHH B npaBYIO C npoTHBOnOJlO>KHblM 3HaKOM.
.LJ.aJIee H3 06eHX 4aCTeH nOJIY4eHHoro ypaBHeHHH Bbl4TeM 5x.
nOJIY4HM:
3x-5x=5x+6-5x, HJIH 3x-5x=6,
1052. PewH ypaBHeHHe:
I) 0,4x+ 12,03= -0,13
l...!...x-52.= -6...!...
482
O,12+0,8x= -0,08
Tenepb CJIaraeMOe 5x nepeMeCTHJIOCb H3 npaBoH 4aCTH B JIeBYIO C
npoTHBOnOJlO>KHblM 3HaKOM_ B JIeBOH 4aCTH ypaBHeHHH npHBeJJ.eM
nOJJ.06HbIe CJIaraeMble, nOJIY4HM:
-2x=6, oTKYJJ.a x=6:(-2), HJIH x=-3,
2) 2 (3y-5)= -22
5+2 (2a+3)=39
-3(5-b)+7=-17
TaK KaK HeH3BecTHbIH MHO>KHTeJIb paBeH 4acTHoMY OT JJ.eJIeHHH npo-
H3BeJJ.eHHH Ha BTOpOH MHO>KHTeJIb. I1TaK, KopeHb JJ.aHHoro ypaBHeHHH
paBeH -3,
113 paCCMOTpeHHoro npHMepa CJJ.eJIaeM 04eHb Ba>KHbIH BbIBOJJ.:
C.IlaZaeMble M03ICHO nepeHOCUTb U3 OiJHOU 'taCTU ypaBHeHWl B iJpyzylO,
U3MeHJUl npu aTOM UX 3HaICU Ha npOTUBOnO.l103ICHble.
PeWHM 60JIee CJIO>KHOe ypaBHeHHe.
1053. BbIl/HCJIH H 06'bHCHH (y C THO):
I) 5.(37:5) 2) (1,2: 16). 16 3) 14.(91: 14)
47.( -10,3:47) m .(6: m) a-(b:a)
1054. Bbl4HCJIH H 06'bHCHH (YCTHO):
l) (3.5):5 2) (- 2 .4): ( - 2) 3) (-5,9.4):5,9
(3,2.4,1):4,1 (3a.5.):5 10x:x
n p H M e p 2.
3 (x+3)=5-2x
3x+9=5-2x
3x+2x=5-9
6.13. PeweHHe ypaBHeHHH
(paCKpbIJIH cK06KH)
(nepeHeCJIH CJIaraeMOe -2x
H3 npaBoH qaCTH B JIeBYIO,
a CJIaraeMOe 9 H3 JIeBOH
B npaBYIO, nOMeHHB HX
3HaKH)
(npHBeJIH nOJJ.06HbIe CJIa-
raeMwe)
x= -4:5, (HaWJIH HeH3BecTHblH MHO-
x= -0,8 >KHTeJIb)
l..h06w npoBepHTb nOJIY4eHHbIH OTBeT, HY>KHO HaHJJ.eHHoe 3Ha4eHHe
KOpHH nOJJ.CTaBHTb B 3aJJ.aHHoe ypaBHeHHe H y6eJJ.HTbcH B paBeHcTBe
06eHx 4aCTeH ypaBHeHHH. B nOCJIeJJ.HeM npHMepe nOJIY4HM:
JIeBaH l/aCTb: 3 (-0,8+3)=3-2,2=6,6;
npaBaH 4aCTb: 5-2-(-0,8)=5+ 1,6=6,6.
YpaBHeHHe peweHo BepHo.
OCHoBbIBaHcb Ha npHBeJJ.eHHbIx npHMepax, COCTaBHM 06wylO cxe-
MY peweHHH ypaBHeHHH, H3Y4aeMblX B 6 KJIaCce:
5x= -4
1055*. PewH ypaBHeHHe:
I) 12: Ixl-6,06= -0,06; 2) -
Ix I +2,5= -20.
.LJ.0 CHX nop MbI peWaJIH ypaBHeHHH, HCnOJIb3YH 3aBHCHMOCTH
Me>KJJ.y B3aHMo06paTHblMH JJ.eHcTBHHMH: CJIO>KeHHe - BbI4HTaHHe, YM-
HO>KeHHe - JJ.eJIeHHe. Tenepb MbI n03HaKOMHJIHCb C oTpHLlaTeJIbHblMH
4HCJIaMH, H XOJl. peweHHH ypaBHeHHH MO>KHO 3Ha4HTeJIbHO ynpocTHTb.
MbI 3HaeM, LITO eCJIH K JJ.BYM paBHbIM l/HCJIaM npH6aBHTb HJIH OT
3THX 4HCeJI OTHHTb OJJ.HO H TO >Ke 4HCJIO, TO nOJIY4HM onHTb paBHbIe
4HCJIa. 3Ha4HT, H K 06eHM 4aCTHM ypaBHeHHH MO>KHO npH6aBHTb H OT
HHX MO>KHO OTHHTb OJJ.HO H TO >Ke 4HCJIO (3TO JJ.OKa3bIBaeTCH B
CTapWHX KJIaCcax).
208
209
1060. PewH ypaBHeHHe:
I) 3(x-2)=4x
5 (z+3)= 10
3 (2x-7)=9
3) 2 (2-y)=y-5
-(4x-2)= -6
-(3x-4)=3x-8
I) no B03MO>KHOCTH ynpocTHM ypaBHeHHe (pacKpoeM cK06KH,
npHBeJJ.eM nOJJ.06Hble CJIaraeMbIe);
2) nepeHeceM CJIaraeMble, cOJJ.ep>KaWHe HeH3BeCTHoe, B OJJ.HY
4aCTb ypaBHeHHH (06bI4HO B JIeBYIO), a OCTaJIbHbIe CJIaraeMble B
JJ.pyrylO 4aCTb ypaBHeHHH, H3MeHHB npH 3TOM HX 3HaKH Ha npoTHBo-
nOJIO>KHbIe;
3) npHBeJJ.eM nOJJ.06Hble CJIaraeMbIe;
4) HaHJJ.eM KopeHb ypaBHeHHH.
B CJIY4ae Heo6xoJJ.HMOCTH BbmOJIHHM npoBepKY.
1056. np04TH TeKCT naparpa4>a, pa36epHcb B npHMepax H CKa>KH,
4TO HY>KHO CJJ.eJIaTb, 4T06bI nepeHecTH CJIaraeMble H3 OJJ.HOH
4aCTH ypaBHeHHH B JJ.pyrylO,
1057, nepeHecH CJIaraeMbIe, cOJJ.ep>KaWHe HeH3BeCTHoe, B JIeBYIO
4aCTb ypaBHeHHH, a Bce OCTaJIbHbIe CJIaraeMbIe B npaBYIO.
HaHJJ.H KopeHb ypaBHeHHH:
I) 2x + 4 = 6 2) 3x + 7 = x
5-3y= -7 -5m-24=m
-12+5a=8 -16-m= -2m
3y-3=5-y -7-4x= -7x+5
1058. PewH ypaBHeHHe:
I) 2x+3=x-6
5-3y=4-2y
6/-1 =3t+ 7
3) 9x=8x-6-x
-4x+8-7=x-1
6-4y-1 =y+3
A.
1061. PewH ypaBHeHHe:
I) 0,8x-3,5= -1,2x+0,5
1,3 (t-0,6)= 1,8/
8,6x - 3,7 = 7 ,6x - 5
1,lx+ I =x-2
1,2 (x-5)=0,2x+6
2) 6(z-I)=18
3 (x+6)=2 (x-3)
9 (x- 3)= 5 (x+ 5)
4) 3(x-5)=x+3
-2 (x+3)=2x-1
-(2x+ 1)= I-x
2) +x+6=2
1
T(4x-2)=-7
x+3=J....x
6 6
J.... x -2=x
3
x -5 =J.... x + J....
2 2
6.
1062. ECJIH 06e 4aCTH ypaBHeHHH cOJJ.ep>KaT OJJ.HHaKOBbIe CJIaraeMble,
TO 3TH CJIaraeMbIe MO>KHO BbI4epKHYTb. n04eMY?
1063. PewH ypaBHeHHe:
2) z+4-3=2z
7-3z+4z-9=0
IOx-3+5=x+3x
4) 5y=2y+9
6/=2/-12
5x= -x-13
I) 3x-4+2x=6+2x-4
50-7y-16=3y-16
2)
y-12= -5-+-12
2 3 2 I
""'5-'-""'5 z =""'5-w z + 3
2
0,7x+"3x-1 =O,7x
2(x+I+)-2=x-I
0,5 (2x+4)=x+3
-6a+ 16=4a-6a-24
5x-6+x=2 (x-I)
2 (x-6)-x=3x+4x
1064. PewH ypaBHeHHe:
I) 3(2x-I)+7=5(x-I)+7;
2) 3,1 (1-3/)+/=0,4 (/-14);
3) 0,8 (O,5-2v)=2v+0,4;
4) 0,2 (x-3)-1 =0,5 (x+3)-O,4;
5) -5 (z-7)=30-(2z+ 1);
6) -2 (x+5)+3=2-3 (x+ I);
7) 3 (2x-I)+6x= IOx-7;
8) 4n-2(n+7)=2n-2(n-I).
1059. npOJJ.YMaH 06wylO cxeMY peweHHH ypaBHeHHH (CM, TeKCT napa-
rpa4>a) H 06'bHCHH Ka>KJJ.bIH war peweHHH cJIeJJ.YlOllI,ero ypaB-
HeHHH:
3,5 (2x-I)+ 1,4=3x-4,9;
7x-3,5+ 1,4=3x-4,9;
7x-2,1 =3x-4,9;
7x-3x= -4,9+2,1;
4x= -2,8;
x= -2,8:4;
x= -0,7.
210
211
6.14. PeweHHe 3a.ll.alf c nOMOlU,blO ypaBHeHHH
n p H M e p I. B UlKOAbHOM caay Bcezo 35 f/JPYKTOBblX aepeBb-
eB - Jl6AOHb U ZpyUl, f/6AOHb Ha 3 60AbUle, '{.eM ZPYUl. CKOAbKO Jl6-
AOHb U CKOAbKO ZpyUl B UlKOAbHOM caay?
PewHM
ny 3aJJ.a4Y JJ.BYMH cnoco6aMH.
n e p B bl H C n 0 C 0 6. ECJIH 6bI H6JIOHb 6bIJIO CTOJIbKO. CKOJIbKO
rpyw, TO Bcero <j)PYKTOBbIX JJ.epeBbeB 6bIJIO 6bI Ha 3 MeHbwe, TO
eCTb 32. 3Ha4HT, YJJ.BoeHHoe 4HCJIO rpyw 6YJJ.eT 32 (pHC. 6.12).
OTclOJJ.a CJIeJJ.yeT, 4TO 4HCJIO rpyw 6bIO 32: 2 = 16, TaK KaK H6-
JIOHb 6bIJIO Ha 3 60JIbWe, TO HX 6bIJIO 16+3= 19.
o T BeT. B caJJ.y 6bIJIO 16 rpyw H 19 H6JIOHb,
npH peweHHH 3aJJ.a4 4aCTO XOpOWHM nOMornHHKOM 6blBaeT ypaB-
HeHHe. .D.JIH TOro l/T06bI pewHTb 3aJJ.a4Y C nOMOlll,blO ypaBHeHHH, CHa-
4aJIa no YCJIOBHIO 3aJJ.a4H HaJJ.O COCTaBHTb ypaBHeHHe. .D.JIH 3Toro co-
OTHoweHHH Me>KJJ.y BeJIH4HHaMH B 3aJJ.al/e He06xoJJ.HMO nepeBecTH
Ha MaTeMaTH4eCKHH H3bIK.
B TOp 0 it C n 0 C 0 6.
C 0 C TaB JI e H Hey p a B H e H H H. 0603Ha4HM 4HCJIO rpyw B
caJJ.Y 6YKBOH x, TorJJ.a 4HCJIO H6JIOHb 6YJJ.eT (x+ 3), 06rnee KOJIH-
4eCTBO JJ.epeBbeB x+(x+3) JJ.OJI>KHO paBHHTbcH 35. nOJIY4HM ypaB-
HeHHe
x+(x+3)=35,
Pewe H Hey p a B He H H H: x+(x+3)=35;
2x+3=35;
2x=35-3;
2x=32;
x=16.
I1TaK, B caJJ.y 6blJIO 16 rpyw. 5I6JIOHb 6bIJIO Ha 3 60JIbWe, TO eCTb
16+3= 19,
npoBepKa. CJIO>KHM 4HCJIO rpyw H H6JIOHb: 16+19=35.
o T BeT. B caJJ.y 6bIJIO 16 rpyw H 19 H6JIOHb.
3 R 6nOH\II
} 351\EPEBbEB
R6nOH\II
rpYW\ll
PHC. 6.12
212
n p H M e p 2. 3a KOHTPOAbHYIO pa60TY no MaTeMaTUKe U3 32 ylte-
HUKOB 14 nOAyltuAu o
eHKy «3». OCTaAbHue nOAyltuAu «4» UAU
«5», npUlteM IteTBepOK 6blAO B aBa pa3a 60AbUle, iteM nJlTepOK. CKOAb-
KO ylteHuKoB nOAyltuAU «5», a CKOAbKO «4»?
C 0 C TaB JI e H Hey p a B H e H H H. 0603Ha4HM l/HCJIO nHTepOK
6YKBOH x. 4HCJIO 4eTBepOK B JJ.Ba pa3a 60JIbWe, 3Ha4HT, 2x. Tpo-
eK 6bIJIO 14, 3Ha4HT, OLleHOK Bcero 6blJIO x+2x+ 14. 4HCJIO
OLleHOK JJ.OJI>KHO 6blTb paBHo 4HCJIY Y4eHHKoB (BeJJ.b Ka>KJJ.bIH Y4eHHK
nOJIY4HJI OJJ.HY oLleHKY). nOJIY4HM ypaBHeHHe x+2x+ 14 =32.
Pew e H Hey p a B H e H H H: x + 2x + 14 = 32;
3x+ 14=32;
3x=32-14;
3x = 18;
x=6
3Ha4HT, nHTepOK 6bIJIO 6. 4eTBepoK 6blJIO B JJ.Ba pa3a 60JIbWe, TO
eCTb 6.2= 12.
n po B e p K a. HaHJJ.eM 06rnee 4HCJIO OLleHOK: 6+ 12+ 14=32.
o T BeT. OLleHKY «5» nOJIY4HJIO 6 Y4eHHKoB, a «4» 12 Y4eHHKoB.
3 a M e 4 a H H e. nOCJIe peweHHH 3aJJ.a4H C nOMOlll,blO ypaBHeHHH
npOBepKY HY>KHO JJ.eJIaTb no YCJIOBHIO 3aJJ.a4H, HO He no COCTaBJIeHHOMY
ypaBHeHHIO. I1Hal/e MbI npOBepHJIH 6bI npaBHJIbHOCTb peweHHH ypaB-
HeHHH, a He npOBepHJIH 6bI npaBHJIbHOCTb ero COCTaBJIeHHH,
A.
1065. COCTaBb ypaBHeHHH H pewH HX:
I) 4HCJIO X CJIO>KHJIH C ero YJJ.BoeHHbIM 4HCJIOM H nOJIY4HJIH 39.
2) 113 4HCJIa 45 Bbl4JIH 4HCJIO x H nOJIY4HJIH YTpoeHHoe
4HCJIO x.
3) 113 4HCJIa 42 BbI4JIH 4HCJIO Y H nOJIY4eHHbIH pe3YJIbTaT
YBeJIH4HJIH B TPH pa3a. nOJIY4HJIH 96.
4) 4HCJIO m YMHO>KHJIH Ha 4, H3 pe3YJIbTaTa BbI4JIH 73 H
nOJIY4HJIH 183,
5) 4HCJIO x YMHO>KHJIH Ha 6, pe3YJIbTaT CJIO>KHJIH C CYMMOH
4HCeJI x H 9, nOJIY4HJIH 44.
1066. CYMMa JJ.BYX 4HCeJI paBHa 165, npH4eM nepBoe 4HCJIO 60JIbWe
BToporo Ha 21. HaHJJ.H 3TH 4HCJIa.
1067. Macca JJ.BYX KOHTeHHepoB 102 Kr. npH4eM Macca OJJ.Horo H3
HHX B JJ.Ba pa3a 60JIbWe MaCCbl JJ.pyroro. BbI4HCJIH Maccy
Ka>KJJ.oro KOHTeHHepa.
213
1068. B03paCT nanbl H MaMbl BMeCTe 75 JIeT, npH4eM MaMa Ha
5 JIeT MOJIO>Ke nanbI. OnpeJJ.eJIH B03pacT nanbl H MaMbI.
1069. OJJ.Ha CTopoHa TpeyrOJIbHHKa 20 CM, BTopaH - B TPH pa3a
60JIbWe TpeTbeH. Bbl4HCJIH JJ.Be JJ.pyrHe CTOpOHbl TpeyrOJIb-
HHKa, eCJIH ero nepHMeTp 52 CM.
1070. WHpHHa npHMoyroJIbHOrO Y4acTKa 3eMJIH Ha 15 M Kop04e
ero JJ.JIHHbI, a nepHMeTp ero 126 M. Bbl4HCJIH ero CTOpOHbI.
1071. OJJ.Ha CTopoHa npHMoyroJIbHHKa B TPH pa3a 60JlbWe JJ.pyroH, ne-
pHMeTp ero 52 CM. Bbl4HCJIH CTOpOHbl npHMoyrOJIbHHKa.
1072. CYMMa rpaJJ.ycHblX Mep JJ.BYX yrJIOB paBHa 123°. OJJ.HH yroJI B
JJ.Ba pa3a 60JIbWe JJ.pyroro, HaHJJ.H rpaJJ.ycHBle Mepbl 3THX yrJIOB.
1073. 113 4HCJIa 47 Bbl4JIH HeKoTopoe 4HCJIO. nOJlY4eHHYlO pa3HocTb
YMHO>KHJIH Ha 2 H H3 pe3YJIbTaTa Bbl4JIH 15, nOJlY4HJIH 45.
KaKoe 4HCJIO Bbl4JIH H3 47?
1074. 3aJJ.YMaHHoe 4HCJIO YMeHbWHJIH B JJ.Ba pa3a H pe3YJIbTaT CJIO-
>KHJIH C 4HCJIOM 19. nOJlY4HJIH 37. OnpeJJ.eJIH 3aJJ.YMaHHoe 4HCJIO.
6.
1075. CYMMa Tpex 4HCeJI paBHa 94. 113BecTHo, 4TO nepBoe 4HCJIO
Ha 18 MeHbwe BToporo, a TpeTbe 4HCJIO Ha 4 60JIbWe BTO-
poro. HaHJJ.H 3TH 4HCJIa.
1076. KOJIH Ha 2 rOJJ.a cTapwe BOBbl, a J1eHa MOJIO>Ke KOJIH Ha 3 ro-
JJ.a. B03pacT KOJIH, BOBbl H J1eHbl BMeCTe COCTaBJIHeT 37 JIeT.
CKOJIbKO JIeT BOBe?
1077. B03pacT OTLla, JJ.04epH H CblHa BMeCTe COCTaBJlHeT 47 JIeT.
OTeLl cTapwe CbIHa B 5 pa3, a ceCTpa MOJIO>Ke 6paTa Ha 2 roJJ.a.
CKOJIbKO JIeT CbIHY?
1078. OJJ.Ha CTopoHa TpeyrOJIbHHKa B JJ.Ba pa3a 60JIbWe BTOpOH, a
TpeTbH cTopoHa Ha 5 CM MeHbwe BTOpOH. Bbl4HCJIH CTOpOHbl
TpeyrOJIbHHKa, eCJIH ero nepHMeTp 85 CM.
1079. Cawa noweJI Ha 03epo, KOTopoe HaXOJJ.HJIOCb OT JJ.OMa Ha
paCCTOHHHH 4,2 KM. BHa4aJIe x 4 OH weJI co CKOpOCTblO
3
-, a 3aTeM elll,e x 4 CO CKOpOCTblO 4 -. CKOJlbKO B p e-
lJ lJ
MeHH Cawa 6blJI B nYTH?
1080. nBa TypHcTa BblWJIH oJJ.HoBpeMeHHo HaBcTpe4Y JJ.pyr JJ.pyry H3
nYHKToB, paccToHHHe Me>KJJ.Y KOTOpblMH 18 KM. CKOpOCTb OJJ.-
4 KM KM
HOro -, a BToporo 5 -. 4epe3 CKOJIbKO 4aCOB OHH BCT p e-
lJ lJ
THTCH?
214
1081. 113 OJJ.Horo ropoJJ.a B JJ.pyroH naCCa>KHpCKHH noe3JJ. npHe3>KaeT
Ha 45 MHH 6b1CTpee TOBapHoro. Bbl4HCJIH paccToHHHe Me>KJJ.Y
KM
ropoJJ.aMH, eCJIH CKOpOCTb naCCa>KHpCKOro noe3JJ.a 48
, a TO-
KM
BapHoro 36 -.
lJ
1082. Ha XXII OJIHMnHHCKHx Hrpax B MocKBe 60JIbWe Bcero MeJJ.aJIeH
3aBOeBaJIH cnopTcMeHbl COBeTcKoro COI03a, Ha BTOpOM MeCTe
no KOJIH4ecTBY MeJJ.aJIeH 6bIJIH cnopTcMeHbl fepMaHcKoH LleMo-
KpaTH4ecKoH Pecny6JIHKH, CKOJIbKO MeJJ.aJIeH 3aBOeBaJIH cnopT-
CMeHbl Ka>KJJ.oH cTpaHbl, eCJIH cnopTcMeHbl 3THX CTpaH nOJIY4H-
JIH Bcero 321 MeJJ.aJIb, npH4eM cnopTCMeHbl f LlP nOJIY4HJIH Ha 69
MeJJ.aJIeH MeHbwe, 4eM cnopTcMeHbl CCCP?
1083. 4HCJIO 30JIOTblX MeJJ.aJIeH, 3aBoeBaHHblX cnopTcMeHaMH CCCP
Ha XXII OJIHMnHHCKHx Hrpax B MocKBe, 6blJIO Ha II 60JIbWe
4HCJIa cepe6pHHblx H Ha 34 60JlbWe 4HCJIa 6pOH30BblX Me-
JJ.aJIeH. CKOJIbKO 30JIOTbIX, cepe6pHHblX, 6pOH30BblX MeJJ.aJIeH 3a-
BOeBaJIH cnopTcMeHbl CCCP Ha 3THX Hrpax? 4TO Hy>KHO elll,e
3HaTb, 4T06bl OTBeTHTb Ha 3TOT Bonpoc? He06xoJJ.HMble JJ.aH-
Hble nOJIY4Hwb npH peweHHH npeJJ.blJJ.Ylll,eH 3aJJ.a4H.
1084. LlBe peMoHTHble MaCTepcKHe B Te4eHHe HeJJ.eJIH JJ.OJl>KHbl OT-
peMoHTHpoBaTb no nJIaHY 18 MOTOpOB. nepBaH MaCTepcKaH
BbInOJIHHJIa nJIaH Ha 120%, a BTopaH - Ha 125%, n03ToMY B
Te4eHHe HeJJ.eJIH oTpeMoHTHpOBaJIH 22 MOTopa. KaKoH nJIaH no
peMoHTY MOTOpOB Ha HeJJ.eJIIO HMeJIa Ka>KJJ.aH MaCTepcKaH?
1085. Ha OJJ.HOH 4alll,e BeCOB 7 OJJ.HHaKOBbiX naKeTOB lIaH H nHTHJJ.e-
cHTHrpaMMoBaH rHpH. Ha JJ.pyroH 4awe BeCOB JJ.Be JJ.BYXCOT-
rpaMMoBble rHpH. Becbl HaXOJJ.HTCH B paBHoBecHH, CKOJIbKO
BeCHT OJJ.Ha na4Ka 4aH?
1086. (C Tap H H H a H H H JJ. H H C K a H 3 a JJ. a 4 a.) ECJIH 3aJJ.Y-
MaHHoe lIHCJIO YMHO>KHTb Ha 5, H3 nOJIY4eHHoro npoH3BeJJ.eHHH
Bbl4eCTb ero TpeTb, OCTaTOK pa3JJ.eJIHTb Ha 10 H K pe3YJIbTaTY
1 I 1
npH6aBHTb nOCJIeJJ.OBaTeJIbHO 2' 3" H"'"'4 nepBOHa4aJIbHOrO 4HC-
JIa, TO nOJlY4HM 68. 4eMY paBHo 3aJJ.YMaHHoe 4HCJIO?
J],JlH nOBTopeHHH
1087. Bbl4HCJIH (y C THO):
1) -10+(-18)
-45+(-56)
-32+( -28)
2) -10+8
25+(-32)
85+( - 15)
215
3) 5-25
II -12
-7+(-9)
1088. Bbl4HCJIH (y C THO) :
1) -5+7-12+6
-4-3+ 15-8+3
30-18-12+5-8
1089. BbInOJIHH JJ.eHcTBHH:
I) - 3,89 + ( - l,ll)
-6,98+( -0,22)
- -.!.+ ( _..!. )
5 15
1090. Bbl4HCJIH (y C THO):
I) -7.(-9)
-12.0,1
-
.(-+)
4) -0,4:(-4)
0,18:( - 0,6)
-rr:( -+ )
1091.
BbInOJIHH JJ.eHcTBHH:
I) -16.(-15)
-1,3.2,8
J..2.. 4..!.
11 5
4) 2,5:(-15)
- 1,8: ( - 0,4)
-3:( -1+)
1092.
4) 2-(-3)
-0-( -7)
-5-( -12)
2) 28-32-16-9+ 16
-3+9- 11 + 16-9
-25+ 15+ 10+ 13-6
2) -0,87+1,2
4,9+( -9,8)
++( -
)
2) -30.0,6
-4,5.( - 2)
--.!..2
9
5) -0,1:(-10)
0,12:( -6)
l,l:(-II)
2) 4,08.(-4,5)
-0,12.( -1,25)
-2
.( - 1
)
5)
:( -0,75)
: .( - 1
)
-8:(-1+)
BbInOJIHH JJ.eHcTBHH:
I) -2.(-5)+20:(-2)-10.(- 1)-9;
2) (-650+350):( -50)+35:( -20+25);
3) 6.(-15-5)-9:(-20+ 11);
4) -6,15.(-1,02)+7,8:(-0,26);
5) (-43,56: 0,66+50):( -0,2);
1
6) -39'""4:(-3).(-16);
216
3) 1,07-59,8
-9,76-15,9
I
-( -+)
3) -28:(-7)
-54:( -2)
7
-8: 4
6) 7. ( - 1,2)
0.8.( -7)
-1,2.( -3)
3) 12:(-24)
-1,1 :0,22
8 . 4
9'""'5
6) 3,5:( -
)
- 1,2 -( -
)
0,5.( -I +)
7) 12.( -
)-15:
;
7
8) 8:7+ 12,5: 10-0,07.
1093. BbInOJIHH JJ.eHCTBHH:
-125.2.(-12) 9.(-14).(-2)
1) 10.(-6) ; 2) -4.(-7) .
1094. Bbl4HCJIH 3Ha4eHHe x no <l>opMYJIe x=(2a-b).c, eCJIH:
1) a= -1,5, b= -10, c= -0,1;
2) a= -0,1, b=0,8, c= 151;
3) a=6,75, b=2,5, c= -19,1;
1 1
4) a=2' b=3' c= -0,3.
1095. Bbl4HCJIH 3Ha4eHHe Bblpa>KeHHH 21 a 1- 31 b I, eCJIH: I) a = - 3,4
H b=2,7; 2) a= 1,6 H b= -0,9.
1096. Bbl4HCJIH 18% OT 3Ha4eHHH Bblpa>KeHHH (
-+).50,
1097. HaHJJ.H l/HCJIO, 24% KOToporo paBHbl 3Ha4eHHIO Bblpa>KeHHH
4,578: 3,27 + 3,4.
1098. HaHJJ.H 4HCJIO, KOTopoe Ha 10% MeHbwe 3Ha4eHHH Bblpa>KeHHH
32.25-3400: 17.
1099*. 4HCJIO YMeHbWHJIH Ha 50%. Ha CKOJIbKO npoLleHToB He06-
XOJJ.HMO YBeJIH4HTb nOJIY4eHHoe 4HCJIO, 4T06bl nOJIY4HTb nepBo-
Ha4aJIbHoe?
6.15. MHKpoKanbKynslTop H pau,HOHanbHble IIHcn8
Tbl, HaBepHoe. YMeeWb CKJIaJJ.bIBaTb, BbI4HTaTb, YMHO>KaTb H JJ.e-
JIHTb nOJlO>KHTeJIbHble 4HCJIa Ha MHKpOKaJIbKYJIHTope. A KaK 6blTb, ec-
JIH l/HCJIa oTpHLlaTeJIbHble? LleHcTBHH C OTpHLlaTeJIbHblMH l/HCJIaMH BbI-
nOJIHHIOTCH TaK >Ke, KaK H C nOJIO>KHTeJIbHbIMH. Bonpoc B TOM, KaK
BBeCTH B MHKpOKaJIbKYJIHTOp OTpHLlaTeJIbHble 4HCJIa. Y 60JlbWHHCTBa
MHKpOKa JIbKYJI HTOpOB JJ.JIH 3Toro HMeeTCH CneLlHaJIbHaH KJIaBHWa
I I-I I HJIH I + I -1 - KJIaBHWa nepeMeHbI 3HaKa. 4T06bl BBeCTH
OTpHLlaTeJIbHOe 4HCJIO, HY>KHO BBeCTH BHa4aJIe MOJJ.yJIb 3Toro 4HCJIa,
a 3aTeM Ha>KaTb Ha KJIaBHWY nepeMeHbl 3HaKa. Ha 3KpaHe nepeJJ.
JJ.aHHbIM 4HCJIOM nOHBHTCH 3HaK «-». HanpHMep, BBOJJ. 4HCJIa - 32,07
6YJJ.eT npoHcxoJJ.HTb no CJIeJJ.YlOllI,eH cxeMe:
3 2 0 0 7 1 I-I I
n p H Me p l. -32,7 -42,1 = -74,8.
CxeMa: 32 0 7 1 I-I 18 42 [J I E]
217
TaK KaK CYMMa JJ.BYX OTpHuaTeJIbHblX 4HCeJI - 4HCJIO, npOTHBO-
nOJIO>KHOe CYMMe MOJJ.YJIeH 3THX 4HCeJI, TO npeJJ.blJJ.YIlI,HH pe3YJIbTaT
MO>KHO 6blJIO nOJIY4HTb H no- JJ.PyroMY:
3 2 D 7 EJ 4 20 1 Ell I-I I
npHMep 2. 0,87.(-25)=-21.75,
C x e M a:
0 87 0 25 1 I-I IEJ
HJIH
0 8 70 2 5 EJI I-I I
n p H M e p 3. -562: 1,12
-501,79,
C x e M a:
5 6 2 1 I-I I EJ I 0 I 2 EJ
HJIH
5 6 2 EJ I 0 I 2 EJI I-I I
n p H M e p 4. 17 - 32.0,35 = 5,8.
C x e M a:
3 2
D 3 5 1 I-I I EJ I 7 EJ
A.
1100. npoBepb paBeHcTBa C nOMOlll,blO MHKpOKaJIbKYJIHTOpa (HaHJJ.H 6
HenpaBHJIbHblX oTBeToB):
1) -5,47 +3,23= -2,24 2) -4,36.(-6,5)=28,34
-0,48+3,5=3,98 -0,08.420= -33,6
-497,3- 108,6= -605,9 5,4.( -0,58)=31,32
-9,07-0,096= -8,974 -1,05.(-8,6)=9,03
3) 45,63:(-0,9)=50,7 4) 186,4-352,64= 166,24
-17,01:( -21)=0,81 -32,472:2,64= 12,3
-6,345:7,05= -0,9 -19,7.( -4,28)=84,316
3,248: ( - 0,08) = - 40,6 - 0,806 + 15,02 = 15,826
1101. Bbll/HCJIH C nOMOlll,blO MHKpOKaJIbKYJIHTOpa (oTBeTbl oKpyrJIH JJ.O
JJ.ecHTblx) :
l) 2,718:(-1,8) 2) -4,17-12,083 3) -7,83:(-0,34)
-0,893+2,17 -9,9.(-1,47) -0,17:0,29
-3,47.5,95 -132,192: 15,3 160:( -(15)
218
6.
1102. Bbl4HCJIH C nOMOlll,blO MHKpOKaJIbKYJIHTopa:
I) 1,45+ 117,45:( -13,5)
-1,89.( -0,51)-5,103:0,3
2) -86: 17,2-6,5+ 15,69
-0,888:( -0,12).( - 5,25)
.D.JlH caMonpOBepKH
1103. 06'bHCHH Ha npHMepax, KaK:
I) CKJIaJJ.bIBaIOTCH JJ.Ba paUHOHaJIbHblX 4HCJIa;
2) Bbl4HTalOTCH JJ.Ba paUHOHaJIbHblX l/HCJIa;
3) YMHO>KaIOTCH JJ.Ba paUHOHaJIbHblX 4HCJIa;
4) JJ.eJIHTCH JJ.Ba paUHOHaJIbHblX 4HCJIa.
1104. 06'bHCHH Ha npHMepax:
I) nepeMeCTHTeJIbHblH H c04eTaTeJIbHblH 3aKOHbl CJIO>KeHHH;
2) nepeMeCTHTeJIbHblH, c04eTaTeJIbHblH H pacnpeJJ.eJIHTeJIbHblH
3aKOHbl YMHO>KeHHH.
1105. Bbl4HCJIH (y C THO):
I) 29+(-29)
100 -( - 64)
0,8+( -4,2)
3 : -( -2+)
2) -6+ 14
0-30
-1,8-0,6
-2
6
1106. Bbl4HCJIH (YCTHO):
I) -9+12-7-12+6
14-8-6+8-7
-6+4.( -3)
3) 4,3.(-0,1)
- 1 7 2 '( -+)
-4,9:0,7
10:( -+)
2) -6.(-8).5
0,5.( -7).0,4
-18:( -2)-5
1107.
BbInOJIHH JJ.eHcTBHH:
I) 392 + ( - 697)
-62,09-5,11
-3
- ( -2.i.. )
5 15
2
-2- + 525
3 '
BbInOJIHH JJ.eHcnmH:
I) -3,96+(-8,7)-7,04-(-30,7);
1 2
2) -63"+8,75-53"+ 1.25;
3) 12.(-0,75)+ 15:0,75;
2) 4,35.(-2,4)
-15,6.( -0,07)
6.( -1+)
1
-15.1-
, 3
1108.
3) -24,6:(-1,2)
- 9664: 3,2
5+:( -I ;0 )
0,6:1+
4) (-1+:3++)'(-
)'
219
1109.
1110.
1111.
1112.
1113.
11 14.
1 1 1 5.
1116.
11 17.
1118.
Bbl4HCJIH 3Ha4eHHe Bblpa>KeHHH (3b - 2): b, eCJIH b = - 4;
b=0,5; b=9.
KaKHe CJIaraeMble Ha3blBalOTCH nOJJ.06HbIMH? npHBeJJ.H npHMep.
npHBeJJ.H nOJJ.06Hble CJIaraeMble:
I) 4c-3c-5c+6c; 2)
3) -I,5a+4,7b-8,3a+ lOa; 4)
3x-7y+4y-6x-3x;
2 2 5
-"3 x +5+""6 x - I - x .
Ha KaKOM 3aKOHe YMHO>KeHHH OCHOBaHO paCKpwTHe CK060K?
PaCKpOH CK06KH H npHBeJJ.H nOJJ.06Hble CJIaraeMble:
I) -5(3x+y)-6y; 2) 7 (3a-b)-20a+7b;
3) -(12n-5m)+ 12n; 4) 5x-6y-5 (x-y):
PewH ypaBHeHHe:
I) 5x-7=x+9; 2) O,77+0,3x=0,5-0,6x;
3) 2 (5+3x)= -2; 4) 3 (y-4)=4 (3y+6).
CYMMa JJ.BYX 4HCeJI paBHa 157, npH4eM OJJ.HO 4HCJIO 60JIbWe
JJ.pyroro Ha 28. OnpeJJ.eJIH 3TH 4HCJIa.
B Tpex uexax 3aBOJJ.a Bcero 685 pa604Hx. Bo BTOpOM uexe
pa60 l IHx B TPH pa3a 60JIbWe, 4eM B nepBOM, a B TpeTbeM - Ha
15 pa604Hx MeHbwe, 4eM BO BTOpOM uexe. CKOJIbKO pa604Hx
B Ka>KJJ.OM uexe?
113 JJ.BYX ropoJJ.oB, paccToHHHe Me>KJJ.Y KOTOpblMH 150 KM,
oJJ.HoBpeMeHHo HaBcTpe4Y JJ.pyr JJ.pyry BbleXaJIH JJ.Ba aBToM06H-
JIH. CKOpOCTb OJJ.Horo aBTOM06HJIH 65 KM/4, a BToporo 60 KM/4.
4epe3 CKOJIbKO 4aCOB OHH BCTpeTHJIHcb?
Bbl4HCJIH paCCTOHHHe Me>KJJ.Y T04KaMH K ( - 4,8) H L (1,3)
4HCJIOBOH OCH.
OTBeTbi
fnABA I. 122. 1, 2, 4, 7, 8, 14, 28, 56; I, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72.
124. 1,2,3,6,9, 18. 126. 26. 52, 78. 128.6984: 18=388. 131. HeT. 133.42=2.3.7;
76=2.2.19; 88=2.2.2.11; 495=3.3.5.11. 135. I) 6; 2) 21; 3) 18; 4) 72;
5) 42; 6) 200. 136. 29 KOMaHJl, 5 MaJlblJHKOB, 3 JleBOlJKH. 137. 120.
24325 645 27 28
fnABA2.260' 5 '9'S'3'9. 26 1.1) 120 H 120 ; 2) 48 H 48 '
50 51 11 2 11
3) 150 If 150 ' 262. PmKb, npoco, nweHlfIl,a, rpe'lHxa, OBec. 264. 1) 24 ' 1 21 ' 4 20 ;
13 25 32 19 5 3 1 13 31 1 3 15
2) 9 27 ' 4 36 ' 7 45 ; 3) 48 ' 24 ' 1 4 ; 4) 3 12 ' 14 ' 1 60 ' 265. 1) 33"; 2) "5; 3) 8 16 .
2 13 4 4 1 9 8 7 1
266.29,1 18 .267. 3"5+3"5+x; 8 1O KM, 8 1O KM . 268.12 9 , 269. 20 .270.7 10 T.
3
271. B TpeTbeM Ha 20 n 60nbwe. 272. 1) 2; 2) 0,1.
220
3 12 21 1
fnABA 3.377. 0,63; 3,2; 2,65; 0,42; 4,25. 378. 3 5 , 25 ' 2 200 ' 25 ' 379. 0,41;
9 I 3 2 1 1 7
1,89. 382.1) 32 ,2 4 , 1 5 ; 2) 123'3' 1 3 ; 3) 2; 8,1; 2s
7 1 17 1 1 3
383. 1) 1 9; 2) 2 3 ; 3) 2 18 . 384. 22 2 KM. 385. 1) 3 2 p., 2) Ha 10 p. JleWeBJle.
128
386. 1) x+ 2 ; 2) 3-7a; 3) 5x+4. 388.1) 42; 1,4; 2) 60; 15; 3) 23,8; 0,343;
4) 96; 14,58. 389. 5 M 2 . 390. 162 KM.
1 1 5 I I 3 12 7 3
fnABA4. 663'8,3 2 '6,3 3 .666.1) 2 4 ,1 5 ' 25 ;2) 24 ,10' 50 ;
13 1 3 2 2 3 4 3 KM
3) 15' 210' 1 5. 667. 7' 668. 3' 1, 5' 669. 1 5 p. 670. Y TaHH 3 5 q'
1 5
y £leTH 3 q' 15 pa3a. 672. I) 50; 2) 4 6 ; 3) 20; 4) 4,9. 673. 1) 12,5%;
2) 112%.674. 1) 16%; 2) 75%, 675. 70%. 676. 160. 677. 37,5%. 678. 45 KM.
2 1 3 1
682. 1) 5; 2) 1 3 ; 3) 30; 4) 3"5' 5) 8' 684. 1188 KM. 685.6 'I. 686. 6400 M 2 .
689. 47 M, 177 M 2 .
fnABA 5.859. A (-4,5), B (2,5), C (-2,3), D (-0,6), £ (5).860. 1) -5; -3,7;
-3,4; -0,5; 0; 2.29; 2,9; 2) -2,9; 3.7; 5; -2,29; 0.5; 3,4; 0; 3) 2,9; 3,7; 5; 2,29;
0,5; 3,4; O. 861. -5, -4, -3, -2, -1,0, 1,2..862. 5; -1,8; 0; -4,9. 863. 3,6;
125
5;3,2.864.1) 4.9; -15;2) 5; -2 9 ;3) 10,2,13,7;4) 8;0.865.1) -3,8<0;
2) 7>0; 3) x";;;O; 4) y
O. 871. A (-2; 1), B(I,5; -4), C(O; 2,5), D(3,5; 2).
£(-1.5; 0). F(-3.5; -1.5), G(O; -1), H(2; 0). 872. II, IV, I, III. 873. (-2; 1);
(-1,4; 0,7); (0; 0); (1; -0,5); (2; 1).874. (-1,5; -4); (0; -1); (0,5; 0); (1; 1); (2; 3).
877. 1) _5°, _2°,4°,0°; 2) B 2 lJ, B 8 'I H B 22 'I, B 10 'I H B 21 'I; 3) B 0 'I _6°;
4) B 14'15°; 5) nOBblWaJlaCb C 6 'I JlO 14 'lOT _2° JlO 1\0 H nOHH>KaJlaCb C 14'1 JlO 16 'I
C 5° JlO 3°. 879. 1) 120 KM; 2) 0,5 'I, 1 '1,2,5 '1,5 lJ; 3) 50 KM, 80 KM; 4) 80 KM, 1 'I.
fnABA 6.1107.1) -305; -67,2; -If; 2 1
; 2) -10.44; 1.092; -6
;
27 2
-2; 3) 20,5; -3020; -5; 50 ' 1108. 1) 11; 2) -2; 3) - II; 4) 9.1109. 3,5; -1;
5 3
-6 x- 5
-2; 4) -4.
7
29. 1111. I) 2c; 2) -6x-3y; 3) O,2a+4,7b; 4)
1113. 1) -15x-lly; 2) a; 3) 5m; 4) - y. 1114. 1) 4; 2) -0,3; 3)
1115.64,5 H 92,5. 1116. 100,300 H 285. 1117. 1.2 'I. 1118.6,1.
A6cLlMcca 152
6eCKOHelJHaH JleCHTHlJHaH Jlp06b 50
83aMMHO oopaTHbie lJMCJla 72
B3aHMHo npocTble lJHCJla 19
BbllJMTaHHe Jlp06eH 38
rpaljJHK Jl.BH>KeHHH 163
rpaljJHK TeMnepaTypbl 161
.ll.eJJeHHe Jl.pOOeH 73
lleJJMTeJlH lJMCJla 8
llecHTHlJHoe npH6J1H>KeHHe 53
llHaMeTp OKPY>KHOCTH (Kpyra) 95
llHaMeTp wapa (cljJepbl) 107
llJlHHa OKPY>KHOCTH 96
1l0nOJlHHTeJJbHbIH MHO>KMTeJlb 32
llP06Hoe Bblpa>KeHHe 26
llP06b lJHCJla 63
EJlHMMlJHaH Jlp06b 54
KOHelJHaH JleCHTHlJHaH Jlp06b 50
KOOpJlMHaTa TOlJKM 130
KOOpJlMHaTHaH nJlOCKOCTb 152
KOOpJlMHaTHaH npHMaH 130
KOOpJlMHaTHaH lJeTBepTb 153
K031jJljJHLlMeHT 197
KpaHHHe lJJIeHbI nponopLlHM 110
KpaTHbie lJHCJla 8
Kpyr 95
KpyroBaH JlMarpaMMa 105
KpyroBoH ceKTop 103
MacwTa6 118
MHKpoKaJlbKYJlHToP 217
MOJl.YJlb lJHCJla 135
HaH60JlbWHH OOIUHH Jl.eJJHTeJJb 18
HaHMeHbwee 06IUee KpaTHoe 20
HaXO>KJleHHe JlP06H OT lJHCJla 63
HaXO>KJleHHe npoLleHToB lJHCJla 68
HaXO>KJl.eHHe lJMCJla no ero Jl.P06H 78
HaXO>KJleHHe lJMCJla no ero npoLleHTaM 82
HalJaJlo KOOpJlHHaT 152
HecoKpaTMMaH Jlp06b 30
06paTHaH nponopLlHOHaJlbHOCTb 122
06b1KHOBeHHaH Jlp06b 25
OKPY>KHOCTb 94
npe.ll.MeTHblH YKa3aTenb
OpJlHHaTa 152
OCHoBHoe CBOHCTBO JlP06M 27
OCb a6cLlHCC 152
OCb OpJlMHaT 152
OTHoweHHe 110
OTpHLlaTeJJbHoe lJHCJlO 128
napaJlJleJlbHble npHMble 148
flepeMeCTMTeJlbHblH 3aKOH CJlO>KeHHH 181
flepeMeCTHTeJlbHblH 3aKOH YMHO>KeHHH 58,
195
flepeceKalOIUHecH npHMble 142
flepHOJl Jl.eCHTHlJHOH JlP06H 50
flepHOJl.HlJeCKaH JleCHTMlJHaH Jl.p06b 50
flepneHJlHKYJlHp 146
flepneHJlMKYJlHpHble npHMble 145
flJlOIUaJlb Kpyra 99
floJl06Hble CJlaraeMbie 202
flOJlHblH yroJl 104
flOJlO>KMTeJJbHoe lJHCJlO 128
flpM3HaKH JleJJHMOCTH 9
flponopLlHH 109
flpocToe lJHCJlO 13
flpocTble MHO>KMTeJlM 16
flpoTMBOnOJlO>KHble lJHCJla 133
flpHMaH nponopLlHOHaJlbHOCTb 114
PaJlHYc OKPY>KHOCTH (Kpyra) 95
PaJlMYc wapa (cljJepbl) 107
PacKpblTMe cK060K 200
PaCnpeJleJJHTeJJbHblH 3aKOH 58, 200
PaLlHOHaJlbHOe lJHCJlO 133
PeweTo 3paTocljJeHa 14
CJlO>KeHHe Jlp06eH 38
CocTaBHoe lJMCJlO 13
COlJeTaTeJJbHblH 3aKOH CJlO>KeHHH 181
COlJeTaTeJlbHblH 3aKOH YM.HO>KeHHH 58. 195
CpeJlHHe lJJleHbI nponopLlHH 110
CljJepa 107
YMHO>KeHHe Jlp06eH 56
u'eJloe lJHCJlO 133
lJacTHoe B npOLleHTax 85
4HCJlOBaH OCb 130
Wap 107
OrnaBneHHe
MOH Jlpyr! . . . _ . .
I. .ll.eJJMMOCTb HaTypaJlbHblx lJMCen
floBTopeHMe .....
1.1. lleJlHTeJJH H KpaTHble lJHCJla
1.2. flpH3HaKH JleJlHMOCTH . .
flpH3HaKH JleJlMMOCTH Ha 10, 5 H 2
flpH3HaKH JleJlMMOCTH Ha 9 M 3 . .
CaMocToHTeJJbHaH pa60Ta 1 cflpoCTble H COCTaBHble lJHCJla»
1.3. Ta6J1HLla npocTblx lJHCeJJ
HCTOpHlJeCKHe CBeJl.eHHH . . . . . . . . . _ _ _ . . . _
1.4. Pa3J10>KeHHe COCTaBHblX lJHCeJl Ha npocTble MHO>KMTeJlH
1.5. HaH60J1bWHH 06IUMH Jl.eJlHTeJJb
1.6. HaHMeHbwee 06IUee KpaTHoe
llJlH caMonpoBepKH
2. 06w.Me cBoiicTBa 06b1KHOBeHHblx Jl.p06eii. CnO>KeHMe M BbI'IMTaHMe
2.1. 06b1KHOBeHHaH Jlp06b KaK 'IaCTHOe OT JleJJeHHH
2.2. OCHoBHoe CBOHCTBO JlP06H
2.3. COKpaIUeHHe Jlp06eH . .
2.4. flpHBeJleHHe Jlp06eH K 06IUeMY 3HaMeHaTeJJIO
2.5. CpaBHeHHe 06b1KHOBeHHblx Jlp06eH .
2.6. CJlO>KeHHe H BbllJHTaHHe JlpOOeH C pa3HblMH 3HaMeHaTeJlHMH
llJlH caMonpOBepKH .
3. npe06pa30BaHHe Jl.p06eii. YMHO>KeHHe 06b1KHOBeHHblx Jl.p06eii
CaMocToHTeJJbHaH pa60Ta 2 c06paw.eHHe oobiKHoBeHHblx Jlp06eH
TH'IHble»
3.1. flepHOJlH'IeCKaH JleCHTH'IHaH Jlp06b
3.2. llecHTHlJHOe npH6J1H>KeHHe 06b1KHOBeHHoH JlP06H
HCTOpHlJeCKHe CBeJleHHH . . . . .
3.3. YMHO>KeHHe 06bIKHoBeHHblx Jlp06eH
3.4. 3aJla'lH Ha YMHO>KeHHe Jlp06eH
3.5. HaXO>KJleHHe npoLleHToB OT JlaHHoro lJHCJla
llJlH caMonpoBepKH
4. .ll.eJJeHHe 06b1KHOBeHHblx Jl.p06eii. nponopLlHH
CaMocToHTeJJbHaH pa60Ta 3 cB3aHMHo oopaTHbie 'IHCJla»
4.1. lleJJeHHe 06b1KHOBeHHblx Jlp06eii . . .
4.2. 3aJla'lH Ha JleJJeHHe Jlp06eH . _ _
4.3. HaXO>KJleHHe 'IHCJla no ero npoLleHTaM
4.4. 4TO nOKa3b1BaeT lJaCTHOe JlBYX 'IMCeJJ
4.5. H3MeHeHHe BeJJH'IHHbI B npoLleHTax .
4.6. COBMecTHbie JleHCTBHH C JleCHTHlJHblMH H 06b1KHOBeHHblMH JlP06HMH
4.7. OKPY>KHOCTb H Kpyr
4.8. llJlHHa OKPY>KHOCTH
4.9. flJlOIUaJlb Kpyra . .
HCTOpH'IeCKHe CBeJleHHH
4.10. KpyroBOH ceKTOp
3
5
8
9
11
13
14
16
18
20
24
25
27
30
32
35
38
46
49
B JleCH-
50
53
54
56
62
68
70
72
73
77
82
84
88
90
94
96
99
101
103
223
4.11. KpyroBaH JlHarpaMMa
4.12. Wap
4.13. flponopLlHH
4.14. OCHOBHoe CBOHCTBO nponopLlHH
4.15. flpHMaH nponopLlHOHaJlbHOCTb BeJlHlJHH
4.16. MacwTa6
4.17. 06paTHaH nponopLlHOHaJlbHOCTb BeJJHlJHH
llnH caMonpoBepKH
5. nonO>KMTenbHble M oTpMLlaTenbHble lJMCna. npHMoyronbHaH CMCTeMa
HaT
5.1. OTpHLlaTeJJbHble lJHCJla
HCTOpHlJeCKHe CBeJleHMH
5.2. 4HCJlOBaH OCb .
5.3. flpoTHBOnOJlO>KHble lJMCJla
5.4. MOJl.YJlb lJHCJla
5.5. CpaBHeHMe lJHCeJl
5.6. flepeCelJeHHe npHMblx
5.7. flepneHJl.HKYJlHpHble npHMble
5.8. flapaJlJleJlbHble npHMble
5.9. KOOpJlHHaTHaH nJlOCKOCTb . .
CaMoCTOHTeJlbHaH pa60Ta 4 «floCTpoeHMe TOlJKH no ee KOOpJlHHaTaM»
5.10. rpaljJMKH
llJlH caMonpoBepKH
6. .ll.eiicTBHH C paLlHoHanbHblMM lJMCJlaMM
6.1. CnO>KeHMe Jl.BYX OTpHLlaTeJlbHblX lJHCeJl
6.2. CJlO>KeHHe Jl.BYX lJHCeJJ C pa3HblMH 3HaKaMH
6.3. 3aKoHbI CJlO>KeHMH. CJlO>KeHMe HeCKOJlbKHX lJHCeJl
6.4. BbllJHTaHHe
6.5. Bblpa>KeHHH. COJlep>KaIUHe CJlO>KeHMe If BbIlJMTaHMe
6.6. PaCCTOHHHe Me>KJl.Y Jl.BYMH TOlJKaMH KOOpJlMHaTHOH OCH
6.7. YMHO>KeHHe JlBYX lJHCeJl .
6.8. flepeMeCTHTeJlbHbIH H COlJeTaTeJJbHbIH 3aKOHbI YMHO>KeHMH
6.9. K03IjJIjJHLlHeHT
6.10. PaCnpeJl.eJJHTeJJbHblH 3aKOH YMHO>KeHHH. PacKpblTHe CKo60K
6.11. flpHBeJl.eHHe nOJl06HblX CJlaraeMbiX
6.12. lleJJeHHe
6.13. PeweHHe ypaBHeHHH . .
6.14. PeweHHe 3aJlalJ C nOMoIUblO ypaBHeHHH
6.15. MHKpOKaJlbKYJlHTOp H paLlHOHaJlbHble lJHCJla
llJlH caMonpoBepKH
OTBeTbI
flpeJl.MeTHbIH YKa3aTeJJb
105
107
109
112
114
118
122
124
KOOPJlM-
127
128
130
133
135
138
142
145
148
154
156
160
170
173
175
181
183 ' .
186
189
191
195
197
200
202
205
208
212
217
219
220
222
TA6J1HlJ,A npOCTbIX 1JHCEJI (.l{O 997)
PAIJ.110HAJlbHbIE 1JI1CJlA
2 97 227 367 509 661 829
3 101 229 373 521 673 839
.
5 103 233 379 523 677 853
7 107 239 383 541 683 857
11 109 241 389 547 691 859
13 113 251 397 557 701 863
17 127 257 401 563 709 877
OTPHQATEJlhHhlE nOJlO}KHTEJlhHhlE 19 131 263 409 569 719 881
PAQHOHAJlhHhlE PAQHOHAJlhHhlE 23 137 269 419 571 727 883
IJHCJlA IJHCJlA 29 139 271 421 577 733 887
31 149 277 431 587 739 907
37 151 281 433 593 743 911
41 157 283 439 599 751 919
QEJI hIE QEJlhlE 43 163 293 443 601 757 929
OTPHQATEJlhHhlE nOJlO}KHTEJlhHhlE 47 167 307 449 607 761 937
(HATYPAJlhHhlE) 53 173 311 457 613 769 941
IJHCJlA IJHCJlA 59 179 313 461 617 773 947
61 181 317 463 619 787 953
67 191 331 467 631 797 967
.lI.POI>HhlE .lI.POI>HhlE 71 193 337 479 641 809 971
OTPIIQATEJI hHhlE nOJlO}KHTEJIhHhi E 73 197 347 487 643 811 977
79 199 349 491 647 821 983
IJHCJlA IJHCJlA 83 211 353 499 653 823 991
89 223 359 503 659 827 997
--