Mathimatika A
Transcript of Mathimatika A
-
2008 41 & , . 210 3608065, Fax: 210 3608197www.volonaki.gr, e-mail: [email protected]
: : : , . 35, , . 26510 20069
, .
ISBN 978-960-381-391-0
-
- - . . - . . : 3 . : , , - , , -, ( ) , ( ). - , -, . , , .
, 20/06/2008
-
:
1: 1.1 - ............................ 161.2 , ................. 281.3 ........................................................................... 391.4 ....................................................................................... 491.5 . . . . . . ............................................................................. 57
2: 2.1 ................................................................................. 742.2 ..................................................................................... 82 2.3 ..................................................................................... 932.4 ............................................................ 1012.5 ...................................................................... 1132.6 ..................................................................................... 121
3: 3.1 ....................................................................................... 1363.2 ...... 1473.3 ............................................ 1563.4 ........................................................ 1573.5 ........................................................................................ 160
4: 4.1 : + x = , x = , x = , x = , : x = x : = ............................................................... 1784.2 .................................................................................. 187
5: 5.1 ......................................................................................................... 1985.2 ............................................................................... 208
10
-
6: 6.1 .................................................................. 2246.2 ..................................................................... 2296.3 ................................................. 2336.4 ......................................................... 2376.5 ................................................................................. 2446.6 ........................................................................... 247
7: 7.1 ( ) ............................................................................................... 2547.2 ............................ 2587.3 ............................................................................. 2647.4 .............................................................................. 2697.5 ............................................................................... 2737.6 .................................................................... 2797.7 .................................................................... 2837.8 ................................................... 2877.9 ................................................. 2917.10 ................................... 297
:
1: 1.1 ......................................................................................................... 3021.2 ....................................................................................................... 3091.3 , ............................................................ 3161.4 ....................................... 3211.5 , ....................... 3271.6 ..................................................................... 3311.7 ...................................... 3351.8 ......................................................................................... 3411.9 ........................................................................... 3491.10 .......................... 3531.11 .................................................................. 357
11
-
1.12 .................................................................................................. 3621.13 ........................................................................... 367
2: 2.1 ............................................................................... 3782.2 ......................................................................................... 3822.3 ........................................................... 3852.4 ............................................................................. 3912.5 ......................................................................................... 3932.6 .............................. 396
3: , 3.1 .......................................... 4063.2 .......................................... 4113.3 .......................................................................................... 4193.4 .................................................................. 424
& ............................. 437
12
-
-
-
15
1
1, 2, 3, 4, 5, 6, 7, 8,...99, 100
-
16 1
1, 2 , 3 , 4 , 5 , ... , 99 , 100 , 101, 102 ,... , 999 , 1000 ,...
.
()
(). 2 .
0 , 2 , 4 , 6 , 8 1, 3 , 5 ,
7 , 9 .
:0 , 2 , 4 ,, 10 ,... , 102 ,... , 998 , ... , 2004 ,...
:1, 3 , 5 , ... , 11,... , 205 ,... , 999 ,... , 2007 ,...
N 1N 1N + . 0 ( 1),.
N ,:
:
A 6N = (), 1 5N = ()
1.1
.
-
17 - -
1 7N + = ().A 55N = (), 1 54N = () 1 56N + = ().
0,1,2,3,4,5,6, 7, 8, 9. ()
.,
, ,
...:
254.897
7 ,9 ,8 ,4 ,5 ,2 .
..
:
> . < . = . ,
.
.:
100 89> , 1010 1000> , 500 500=
0 0 .
-
18 1
.
.
0 1 2 3 4
, ,
..,.
:
0 , 1, 2 , 3 , 4 . :
13.637 .
6
3 , 6
13.600 .
5 , 6 , 7 , 8 , 9 .
: 13.637 .
3
6 , 3 4
14.000 .
-
19 - -
9 ,0 1.:
202.987 .
9
1 0 2 3 .
203.000 .
,
,...
1. :. 455 , . 203 , . 860 , . 2007 .
,
2 .
. 455 227 2 1 + , 2 .
.203 101 2 1 + ,2 .
. 860 430 2 ,2 .
.2007 1003 2 1 + ,2 .
-
20 1
2. . 2 9 . . 2 9 .
: 1 = + .
( , ) :
1 = .
: = .
.:2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 .,:
9 2 1 8 = + = .. : 3 , 4 , 5 , 6 , 7 , 8 .
,:
9 2 1 6 = = .
3. :. 400 .4 000 , . 789 987 , . .10 100 .10 010 .
.
,
, .
. 400 4.000 ,:
400 4.000< ..:7 9< .:
789 987< .
-
21 - -
.
,8 0> ,:80.800 80.080> .
4.. 4.402, 4.204, .4 420 , .4 240 .. 50.500, 50.005, 50.050,
55.000.
(
) ( ) ,
.
.( 4 ).
, 4.402 4.420
4.204 4.240 .
,: 4.402 4.420< 4.204 4.240< .
:
4.420 4.402 4.240 4.204> > > ..,
:
50.005 50.050 50.500 55.000< < < .
5. 5 , 35 , 65 95 :
. , .
-
22 1
,,,10
1. 5 ,
5 5 (10 ).
,35 5 (40),655(70)
95 5 (100).
6. : . : 196959 , : .1 255 ,
: .4 344m , : 1020kg ,
2651095637 ..
;
.
.
,
, , ,
, .
.
,
.
.1.300 ,
2 5 .
4.300m ,
3 4 .
1.000kg ,0 2 .
-
23 - -
13.
1.. 205 .732 . 20.813.
2. ..
..
..
3.289 288 ,287 286, 290 291.
4.,:3.508 3.515 3.620 4.800 4.801> > > >
5.. 45 45= .38 36> . 456 465< .8.765 8.970< .90.876 86.945< .345 5.690 . ()
.
. ()
.10 ,.()
..
()
2..:22,567 , 1.453 , 1.921, 49.870 , 567.532 , 1.234.709 .
.,,
.
3..: 233, 8455 , 332 , 1234 , 700 , 12 , 0 , 32.459 .
.
.
4..5 500 .
-
27 - -
. 21 2007.
5.( , ,> = < ):
34 . . 43 457 ..0457
1.234 .. 1.3246.010 .. 6.0014.568 .. 4.659
5.944 .. 59.440
6..:3.124 , 1.243 , 4.312, 4.213 , 2.134 , 2.431, 1.234 .
.: 5.678, 6.786 , 8.576 ,
8.765 , 5.687 , 6.587 , 7.685 .
7. . 5 , 6 , 7 ..
.
.
..
8. 50
500 .
0 75 150 225 300 375 450 525
-
28 1
9.:
12 , 18 , 20 , 24 , 32 , 38 , 40 .
10.11.134 , 21.254 , 58.398 , 97.951..
..
..
11. : : 48.265kg , .: 98826754 ,.: 45554 ,
:384.394km,: 4.550.644 .
,.
12.83 3 8 , ,
830 ;
, 1.2
-
29,
(+ ). , ,:
+ = ( , ).
:
0 ,:
0 + = 0 .
:
,
,:
+ = + .
: ,
,:
( ) ( ) + + = + + .
( ).
-
30 1
= .:
= +
> .
:
0 ,:
0 = 0 .
,:
0 =
( ). , ,:
( , ).
: 1,,:
1 =
-
31,
1 . 0 , 0 ,:
0 0 =
:
,,:
= .
:
,,:
( ) ( ) = .
::
( ) + = + ( ) =
,:
( ) + + = + + .
.
:
-
32 1
,
,.
,
,.
1. 0 10 :
..
..
.
1, 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 .
.:1 2 3 4 5 6 7 8 9 45+ + + + + + + + = ..:1 2 3 4 5 6 7 8 9 362.880 = .
2. :
. ( )+ +35 12 8 . ( )+ +123 134 43 202 . ( ) + 44 35 4 12 . ( ) + 32 2 3 22 2 14
.
-
33,
.:
( )+ + = + =35 12 8 35 20 55 ..
:
( )123 134 43 202 123 91 202 214 202 416+ + = + + = + = .. ,
:
( )44 35 4 12 9 4 12 9 48 57 + = + = + = .. , ,
,
:
( ) ( ) ( )32 2 3 22 2 14 32 6 22 28 32 28 28 32 0 32 + = + = = = .
3. + = 25 = 5 :
. + + +10 5 . + + +2 7 . + +
.
.: 10 5 1 0 5 1 0 20 5 35 + ++ = + ++ = + + = .
.: 2 7 2 7 2 7 2 20 7 29++ += ++ + = + ++ = + + = .
.: 25 5 30 + + = ++ = + = .
4. , cm :
-
34 1
,
.
60cm
25cm
18cm
12cm
13cm
3cm
:
21 60cm 25cm 1.500cm = = .
2E
18cm 3cm
13cm ( )18 12 cm :
( ) ( ) ( )2
2E 3 18 3 2 13 18 12 3 54 2 13 6 48 2 78 48 156 318cm= + = + = + = + =
( ) ( ) ( )2
2E 3 18 3 2 13 18 12 3 54 2 13 6 48 2 78 48 156 318cm= + = + = + = + = .
E 1E 2E :2 2 2
1 2 1.500cm 318cm 1.182cm = = = .
5. . 25 , 20
15 . 3
4 . ;
-
35,
, ,
.
15 .9 2 3
25 20 ,:
N 9 2 3 7 3 4= = =,
:
2 25 3 20 4 15 5 (3 4) 2 25 3 20 4 15 5 7 50 60 60 35 205 = + + + + = + + + = + + + =
2 25 3 20 4 15 5 (3 4) 2 25 3 20 4 15 5 7 50 60 60 35 205 = + + + + = + + + = + + + = .
17.
1... ...0 . ... = .... ..
2..52 5.200 =100 .37 370 =10 . 490 4.900.000 =10.000
3. .5823 . 4 58 . .5 53 6 + 7.5 19 + 525 + 528 11.173 1.010 4.093
-
36 1
4. 1 2 3 4+ + + = 10 1 2 3 4 1 2 12 15+ + = + + = 1 2 3 4 2 12 + = + = 14 1 2 3 4 = 24
5..190 . 225 . 462 . 2.726 .60 18 2 +
. ( )52 11 9 +. 230 .9.700 .879.000
6. 10 100 ,.:
. ( )3 13 3 10 3 3 10 3 3 30 9 39 = + = + = + = .. ( )7 11 7 10 1 7 10 7 1 70 7 77 = + = + + + = .. ( )45 12 45 10 2 45 10 45 2 450 90 540 = + = + = + = .. ( )12 101 12 100 1 12 100 12 1 1.200 12 1.212 = + = + = + = .. ( )5 110 5 100 10 5 100 5 10 500 50 550 = + = + = + = .. ( )4 111 4 100 10 1 4 100 4 10 4 1 400 40 4 444 = + + = + + = + + = .. ( )34 99 34 100 1 34 100 34 1 3.400 34 3.366 = = = = .. ( )58 98 58 100 2 58 100 58 2 5.800 116 5.684 = = = = .
7. 1 ,
2 3 .:
1 2 14 = , 2 2 3 = , 3 2 3 = . :
( ) 31 2 3 2 14 2 3 2 3 2 14 3 3 2 20 40m = + + = + + = + + = = .
-
37,
8. . .
156 160 , 30 30 , 38 40 , 369 370 , 432 430 .:
160 30 40 370 430 1.030 1.000+ + + + = >,.
.: 156 30 40 370 430 1.025+ + + + = .
9.: 35 48 77 160+ + = ..
10.:( ) ( )120 135 25 38 107 112 19 23 318 261 57+ + + + + + = = .
11..:
200719832007 1983 24 = ..:
1983 25 1958 = .
12. 20 40 12 24 .12 20 12 .:
7 40 5 24 280 120 400 + = + = .:80 58 61 199 400+ + = < ,.
1.:
-
38 1
.82 ( )22 13 2 16 5 11+ + + ..45+ ( ) ( )45 21 87 2 14 2 3 4 22 ..( ) ( )14 2 7 2007 3 20 4 15 2004 + .
. ( )29 8 2 24 3 2 3 4 2 3 4 + + + .
2.:. 27 101.68 111.52 99.12 199. 2007 11
3. :.12 147 12 3 12 150 + . 24 16 24 18 24 32 +
5..(5 500 ),;
.
(
500 , 200 ),;
6.12 .7 17
5 ,
8 22 ..
-
39
7. .1 28 97 ,2 36
98 3 19 99 .;
8. 8 + = 9 9 + = 8 +2+4 . 9...
.
.
1.3
.:
7 7 49 = , 4 4 4 64 = , 3 3 3 3 81 = .,
, :
..... =
,
-
40 1
.
2 , 2 .
2 = .
3 , 3 .
3 = . :
1 , 4 5 .
1 :1 = .
1 1 1 . 1 1,.:
1 1 =
0 0 0 . 0 0 ,.:
0 0 = , ( )0
0 , 0 1
: 0 1 =
00 .
10
10
-
41
10 ,:
10 1 0 10 10 10 .. 10 1 0000 .0 = = :
010 1= , 110 10= , 210 10 10 100= = , 310 10 10 10 1.000= = , 410 10 10 10 10 10.000= = .
10
10 ,:
1 01 10= = , 1 110 10= = , 1 2100 10= = , 1
31.000 10= = .
10 .:
1 014 10 4 1 1 0 4 10= + = + .2 1 0785 7 100 8 10 5 1 7 10 8 10 5 10= + + = + + .
3 2 1 03.452 3 1.000 4 100 5 10 2 1 3 10 4 10 5 10 2 10= + + + = + + + .
10 .
,
.
,
.
-
42 1
1. . : 33 , 44 , 55 .. : 7 7 7 7 ,
8 8 8 8 8 , 9 9 9 9 9 9 .
.
.: 33 3 3 3 9= = , 44 4 4 4 4 16= = , 55 5 5 5 5 5 3.125= =
.: 47 7 7 7 7 2.401 = = , 58 8 8 8 8 8 32.768 = = , 69 9 9 9 9 9 9 531.441 = =
2. :
.
.
1 4 6 102 1 24 16= 26 36= 210 100=3 1 34 64= 36 216= 310 1.000=
3. .
;
. 2 25 +4 ( )24+5 , 3 35 +4 ( )35+4
. 2 25 -4 ( )25-4 , 3 35 -4 ( )35-4
-
43
. 2 24 5 ( ) 25 4 , 3 35 4 ( ) 35 4
:
( ) + > + , ( )- - + .
. 2 25 4 25 16 9 = = ( )2 25 4 1 1 = = .
: ( )22 25 4 5 4 >
3 35 4 125 64 61 = = ( )3 35 4 1 1 = = .
: ( )33 35 4 5 4 >
:( ) < .
. 2 25 4 25 16 400 = = ( )2 25 4 20 400 = = .
: ( )22 25 4 5 4 =
3 35 4 125 64 8.000 = = ( )3 35 4 20 8.000 = = .
: ( )33 35 4 5 4 =
:( ) = .
-
44 1
4. 10 .. 234.450 , . 1.344.788 , . 45.700.201
, : 010 , 110 ,
210 .
. 5 4 3 2 1 0 5 4 3 2 1234.450 2 100.000 3 10.000 4 1.000 4 100 5 10 0 1 2 10 3 10 4 10 4 10 5 10 0 10 2 10 3 10 4 10 4 10 5 10= + + + + + = + + + + + = + + + + 5 4 3 2 1 0 5 4 3 2 1234.450 2 100.000 3 10.000 4 1.000 4 100 5 10 0 1 2 10 3 10 4 10 4 10 5 10 0 10 2 10 3 10 4 10 4 10 5 10= + + + + + = + + + + + = + + + +
. 6 5 4 3 2 1 01.344.788 1 1.000.000 3 100.000 4 10.000 4 1.000 7 100 8 10 8 1 1 10 3 10 4 10 4 10 7 10 8 10 8 10= + + + + + + = + + + + + + 6 5 4 3 2 1 01.344.788 1 1.000.000 3 100.000 4 10.000 4 1.000 7 100 8 10 8 1 1 10 3 10 4 10 4 10 7 10 8 10 8 10= + + + + + + = + + + + + +
. 45.700.201 4 10.000.000 5 1.000.000 7 100.000 0 10.000 0 1.000 2 100 0 10 1 1 = + + + + + + + =45.700.201 4 10.000.000 5 1.000.000 7 100.000 0 10.000 0 1.000 2 100 0 10 1 1 = + + + + + + + =
7 6 5 4 3 2 1 0 7 6 5 2 0 4 10 5 10 7 10 0 10 0 10 2 10 0 10 1 10 4 10 5 10 7 10 2 10 1 10= + + + + + + + = + + + + + 7 6 5 4 3 2 1 0 7 6 5 2 0 4 10 5 10 7 10 0 10 0 10 2 10 0 10 1 10 4 10 5 10 7 10 2 10 1 10= + + + + + + + = + + + + +
5. :
. ( )2 26+ 3 4+4 10 . . ( )22 222-4 + 7-2 . :
- .
. ( ) ( )2 22 2 2 26 3 4 4 10 6 2 4 10 6 6 10 6 36 100 6 3.600 3.606 + + = + + = + = + = + =
( ) ( )2 22 2 2 26 3 4 4 10 6 2 4 10 6 6 10 6 36 100 6 3.600 3.606 + + = + + = + = + = + = .
. ( ) ( )2 22 2 2 2 222 4 7 2 22 4 7 4 22 4 3 22 16 9 1 5 + = + = + = + = .
-
45
22.
1. 8 9 10 11 12 13 14 15 16 17 18 19 20 252 64 81 100 121 144 169 196 225 256 289 324 361 400 6253 512 729 1.000 1.331 1.728 2.197 2.744 3.375 4.096 4.913 5.832 6.859 8.000 15.625
2.. 65 5 5 5 5 5 5 = . 6 38 8 8 8 8 8 6 6 6 8 6 =
. 61 1 1 1 1 1 1 = . 4 = . 3x x x x =
. 4 32 2 2 2 2 = .
3. 12 2= . 62 2 2 2 2 2 2 64= = .2 2 2 2 4= = . 72 2 2 2 2 2 2 2 1 28= = .32 2 2 2 8= = . 82 2 2 2 2 2 2 2 2 256= = .42 2 2 2 2 1 6= = . 92 2 2 2 2 2 2 2 2 2 512= = .52 2 2 2 2 2 32= = . 102 2 2 2 2 2 2 2 2 2 2 1 .024= = .
4. 210 10 10 100= = . 220 20 20 400= = . 230 30 30 900= = .240 40 40 1.600= = . 250 50 50 2.500= = .
-
46 1
260 60 60 3.600= = .270 70 70 4.900= = . 280 80 80 6.400= = . 290 90 90 8.100= = .
5. 310 10 10 10 1.000= = . 320 20 20 20 8.000= = . 330 30 30 30 27.000= = .340 40 40 40 64.000= = . 350 50 50 50 125.000= = .
6.. 23 5 3 25 75 = = . . 23 5 2 3 25 2 75 2 77 + = + = + = .
. 2 2 3 5 2 3 25 4 75 4 79 + = + = + = .
. 23 5 2 3 5 4 1 5 4 1 9 + = + = + = .
. ( )2 23 5 2 3 7 3 49 1 47 + = = = .
7.. 2 3 3 43 3 2 2 9 27 8 16 60+ + + = + + + = .
.( )4 213 2 5 3 1 4.641 5 9 1 4.641 45 1 4.686 + = + = + = .
8. .( )2 26 5 1 1 121+ = = , 2 26 5 36 25 61 + = + =
( )2 2 26 5 6 5+ > +
.( )2 23 6 9 81+ = = , 2 23 6 9 36 45+ = + =
( )2 2 23 6 3 6+ > +
: ( )2 2 2 + > + .
9.. 4 + + = . . 3 = .
. x x x x 4x+ + + = . . 4x x x x x = .
-
47
10..34.720 :34.720 3 10.000 4 1.000 7 100 2 10 0 1 = + + + + =
4 3 2 1 0 4 3 2 1 3 10 4 10 7 10 2 10 0 10 3 10 4 10 7 10 2 10= + + + + = + + + ..123.654 :
123.654 1 100.000 2 10.000 3 1.000 6 100 5 10 4 1 = + + + + + =5 4 3 2 1 01 10 2 10 3 10 6 10 5 10 4 10= + + + + + .
.890.650 :
890.650 8 100.000 9 10.000 0 1.000 6 100 5 10 0 1 = + + + + + =5 4 3 2 1 0 5 4 2 18 10 9 10 0 10 6 10 5 10 0 10 8 10 9 10 6 10 5 10= + + + + + = + + + .
11. ( ) ( )1 2 3 4 + + = 3 7 = 21( )1 2 3 4 + = ( )1 2 12 1 14 + = = 14
( )1 2 3 4 + = ( )2 3 4 5 4+ = = 20( )1 2 3 4 + + = 1 5 4 1 20+ = + = 21
12. 2 2 2 + = 2 4+ = 63 3 3+ = 3 9+ = 124 4 4 4+ = 4 64+ = 685 5 5 5 5+ + = 5 25 25+ + = 555 5 5 5 5 + = 25 125+ = 150
1..( )1 2 3 4 1 3+ + = . ( )1 2 3 4 1 4 + = .1 2 3 4 1 4+ + = .( )1 2 3 4 36+ =
-
48 1
2.,:
.51 .75 .126
20 13 18 26 21 28 18 36 72
15 17 19 27 25 23 24 78 24
16 21 14 22 29 24 84 12 30 .
1. : 43 , 54 , 65 , 1110 .
2. :2 2 2 2 2 2 2 2 2 , 3 3 3 3 3 3 3 , 5 5 5 5 5 5 , 10 10 10 10 .
3.: 2 4 6 8 12 14 18
23
4.: 1 2 3 4 5 6 7 14 13 12 11 10 9 8
2 2 +
( )2 +
-
49
5.: 7 6 5 4 3 2 1 14 13 12 11 10 9 8
2 2
( )2
6.10 :2.007 , 88.106 , 277.890 , 2.342.909 , 12.456.987 ,
7.007.007.007
7.:
. 3 2 20072 4 4 2 6 1+ + , . ( )32 2 3 0 20074 5 2 2 4 5 10 0+ + + 22( )32 2 3 0 20074 5 2 2 4 5 10 0+ + + . ( )2 2 44 4 4 4 4 2 4 4 4 + + ,.( ) ( ) ( )2 22 1 3 3 23 5 3 9 2 2 3 2 3 +
1.4
+ . (:)
: =+ :
-
50 1 -
= +
.
.
:
= . = .: 0 .
= . = .: < .
1 = : :1 1 = 0 = : 0 : 0 = = = : : : 1 = =
,
:
: = = . .
, . .
.
,
.
-
51
1. .
. 24:5 , . 125:9 , . 450:15 .
,
.
.245 .1259 .45015
- 20 4 - 117 13 - 450 30 1 8 0
::
24 5 4 1= + 125 9 13 8= + 0 = .: 450 15 30=
2. .
. 37=3 12+1 , . 223=8 28 1 , . 156=14 10+16 .
: = + ,
-
52 1 -
. 223
8 28 1..
14 = 10 , 16 = :16 14> 16 10> . < .
3. :.5 4 2 .
.122 24 2 .
.129 6 21 .
.
..: 5 = , 4 = , 2 = : 5 4 2 20 2 22= + = + = + = .
..: 122 = , 24 = , 2 = :
= + = ( ) ( ): 122 2 : 24 120 : 24 5 = = = = ...: 129 = , 6 = , 21= :
= + 1 29 6 21 129 126 3 = = = = .
4. .. :3 . :5 . :8
!
< , :
. 0= , 1, 2 , =3
-
53
. 0= , 1, 2 , 3 , 4 , =5. 0= , 1, 2 , 3 , 4 , 5 , 6 , 7 , =8
5. . ; .1/1/2007,1/1/2008;
.
. 365 7 ,
365 : 7 .
:
365 364 1 7 52 1= + = + 365 : 7 52 1.
, 152 1.
.52 01/01/07,52 7 364 = ,31/12/07.
,01/01/08,.
6. :
. ( ) ( )2 32 33: 5-2 +16: 29-3 . ( ) ( )2 3 4 2 236+14:2-5 : 3 -3 :3 +144:12 :
-
.
. ( ) ( ) ( ) ( )2 3 2 32 3 2 33: 5 2 16 : 29 3 3 : 5 4 16 : 29 27 3 :1 16 : 2 3 :1 16 :8 3 2 5 + = + = + = + = + = ( ) ( ) ( ) ( )
2 3 2 32 3 2 33: 5 2 16 : 29 3 3 : 5 4 16 : 29 27 3 :1 16 : 2 3 :1 16 :8 3 2 5 + = + = + = + = + = .
-
54 1 -
.( ) ( ) ( ) ( )2 3 4 2 2 2 236 14 : 2 5 : 3 3 :3 144 :12 36 14 : 2 25 : 27 81 :3 144 :12 + + = + + =( ) ( ) ( ) ( )2 3 4 2 2 2 236 14 : 2 5 : 3 3 :3 144 :12 36 14 : 2 25 : 27 81 :3 144 :12 + + = + + =
( ) ( ) ( )2 236 7 25 : 27 81 :9 144 :12 1 8 : 27 9 144 :12 1 8 :18 144 :144 1 1 2 + + = + = + = + =( ) ( ) ( )2 236 7 25 : 27 81 :9 144 :12 1 8 : 27 9 144 :12 1 8 :18 144 :144 1 1 2 + + = + = + = + = .
26.
1..(): 4.002 69 58= . 4.002 69 552 58 345 0
.():1.445 17 85= . 1.445 17 85 85 136 0
.():925 37 25= . 925 37 85 25 74 0
.():3.231 213 17= . 3.321 213 1431 17 213 0
-
55
.():35.280 2.940 12= . 35.280 2.940 5880 12 2940 0
.():5.082 77 66= . 5.082 77 462 66 462 0
2..1 65 :5 1 3= ..1 30 :3 1 0= .. 46.592 :52 896= 52 .
3..125 35 3 20= + . 125 = , 35 = , 3 = 20 = . 3 20 .
. 762 38 19 40= + . 38 = 19 , 40 = :40 38> 40 19> .
.1.500 42 35 30= + . 1.500 = , 42 = 35 , 35 = 42 30 = .
. 300 18 16 12= + . 300 = , 18 = 16,16 = 18 12 = .
4., 8 = , :
0 = , 1, 2 , 3 , 4 , 5 , 6 , 7 .
5. 9 = , 73 = , 4 = ,: 9 73 4 657 4 661= + = + = + = .
-
56 1 -
6. : 247 7 35 2= + , 247 35 2 .
7 , 247 .
1. .
. 23:3 , .145 :12 , .1550 : 25 , . 2.345 : 44 .
2.: 24 55 465 924 221 730 4 6 13 99 17 19 10 3 8 12 5 52 0 2 4
3..
.58 4 14 2= + , .88 4 20 8= + ,.109 2 55 1= , .144 6 24= ,.94 10 8 14= + .
4. . 8 20 .,.
.
11 4 .,.
-
57 /---
5...416 52
;
. 32 1 , ;
.
,;
;
6.:
. ( )2 2 288 :11 4 : 2 169 :13 12 + 68( )2 2 288 :11 4 : 2 169 :13 12 + . ( ) ( )24 3 2 2 4 55 :5 : 180 :5 10 2 + .( ) ( )5 2 2501 64 : 2 :5 2.000 : 4 5 26( ) ( )5 2 2501 64 : 2 :5 2.000 : 4 5 . ( ) ( ) ( )2 2 2 2 232 3 144 : 48 : 12 : 6 5 3
- - - 1.5 .
-
58 1 -
. :0 , 1 , 2 , 3 ,. .
:
(). , . , .
.
0 , 0 0 = .
, (, ),
.
()
.: = ,
.
:
-
59 /---
1 . 1 .
,: 2 , 3 , 5 , 7 , 11, 13 , 17 , 19 , 23, 29 , 31, 37 , 39 ,
41, 43, 47 , 51,,1,.
1 ,.,: 4 , 6 , 8 , 9 , 10 , 12 , 14 , 15 , 16 , 18 , 20 , 21, 22 , 24 ...,1,.
1,,.
.
1, :1 = .
, (, ), .
( ), 1 = , .
.
-
60 1 -
,: 44 2 2 2= = , 6 2 3= , 88 2 2 2 2= = , 29 3 3 3= = , 10 2 5= ,.,.
.
,:
( 2 ).
2 .,
( 3 ). 3 , ( 5 ).
1 .
120 :
120 : 2 60= 1.60 : 2 30= 2.30 : 2 1 5= 3.
15 2 ,3 :
15 :3 5= 4.5 3 ,5 :
5 : 5 1= 5.1.:
3120 2 2 2 3 5 2 3 5= = .
-
61 /---
125 :125 2 , 3 ,5 :
125 :5 25= (1) 25 :5 5= (2)5 :5 1= .:
3125 5 5 5 5= =
.
, 80 , 40 , 120 :
480 2 5= , 340 2 5= , 3 120 2 3 5= .
3 2 , 5 .:
( ) 380,40,120 2 5 8 5 40 = = = .
.
,:
( ) 480,40,120 2 3 5 16 3 5 240 = = = .
2 , 3 , 5 , 9 , 10 25 .
-
62 1 -
2 : 2 , .
, 2 , 4 , 26 , 48 , 244 2 .
3 : 3 3 .
,117 , 585 3 1 1 7 9+ + = , 5 8 5 18+ + =9 18 3 .
9 : 9 9 .
, 567 , 1.998 9 5 6 7 18+ + = , 1 9 9 8 27+ + + = 18 27 9 .
5 : 5 0 5 .
,50 , 105 , 255 , 500 5 .
10 , 100 , .1 000 : 10 , 100 , 1.000 0 , 00 , 000 .
, 30 , 400 5.000 10 , 100 1.000 .
4 25 : 4 25 , 4 25 .
, 77.816 4
16 , 4 , 4.975 25 75 ,25 .
-
63 /---
1. : 19 , 20 , 21 , 22 , 23 24 .
.
:
2 2 .
19 119 ,.
20 1, 2 , 4 , 5 , 10 20 , .
21 1, 3 , 7 21 ,. 22 1, 2 , 11, 22 ,.
23 1 23 ,.
24 1, 2 , 3 , 4 , 6 , 8 , 12 ,24 ,.
2. : 38 , 155 , 71 , 108 .
,
, 2 , 3 , 5 . , .
.
-
64 1 -
38 :
38 : 2 1 9= (1)19 :19 1= (2) :38 2 19= .55 :2 3 55 ,
5 :
55 :5 1 1= (1)19 :19 1= (2) :38 2 19= .71 :71 ,.
108 :
108 : 2 54= (1)54 : 2 27= (2)27 :3 9= (3)9 :3 3= (4)
3:3 1= (5) : 2 3108 2 2 3 3 3 2 3= = .
3. 160 , 200 256 .
(),
().
5160 2 5= , 3 2200 2 5= , 8256 2= 2 5 :
( ) 8 2160,200,256 2 5 256 25 6.400 = = = .
2 :
( ) 3160,200,256 2 8 = = .
-
65 /---
4. :
. 2_6 9 .
. 4 _ _ 25 3 .
.
.9
9 .1
9 .1
9 : 2 1 6 1 8+ + = . 216 ..9
25 ,
2 5 ( 25 ),5 0 (50 ),7 5 ( 75 ).
3 ,
3 .5
0 :4 5 0 9+ + = . 4 2 5 1 1+ + = 4 7 5 1 6+ + = .1 1 , 1 6 3 . 450 .
5. .
. 19
29 , .
.
2,.:20,22,24,26,28.
-
66 1 -
,
.
., 24 3 ( )2 4 6+ = . 24 .
30.
1..5 8 80 . ( )5,8 40 = .. ( ), = , .. 1
.
1 ,.1.
. ( ), 1 = , .
2. 9 :
.684 ( )6 8 4 1 8+ + = . .9.504 ( )9 5 0 4 1 8+ + + = 9.594 ( )9 5 9 4 27+ + + = ..6.012 ( )6 0 1 2 9+ + + = .
3. . ( )3,5 15 = . ( )11,6 66 =
-
67 /---
. ( )5,10 10 = . ( )3,2,5 30 = . ( )3,6,9 18 = . ( )8,12,15 120 =
4. ( )2,3,5 30 = . 30 , 2031 .
5. ( )3,5,7 1 05 = ..
6.:10 2 5= 212 2 3= . ( ) 210,12 2 3 5 60 = = 60 , 10 .
60 :10 1 5 = 60 :12 1 5 = .
7. . ( )5,8 1 = . ( )16,24 8 = . ( )30,15 15 = . ( )10,30,60 10 = . ( )22,32,50 2 =
8. ( ), 24 = 24 .24 1, 2 , 3 , 4 , 6 , 8 12 .
9. 10 1, 2 , 5 , 10 . 111, 11.
12 1, 2 , 3 , 4 , 6 , 12 .
13 1, 13 .
14 1, 2 , 7 , 14 .
-
68 1 -
15 1, 2 , 3 , 5 , 15 .
16 1, 2 , 4 , 8 , 16 .
17 1, 17 .
18 1, 2 , 3 , 9 , 18 .
19 1, 19 .
20 1, 2 , 4 , 5 , 10 , 20 .
11, 13 , 17 , 19
10 , 12 , 14 , 15 , 16 , 18 , 20 .
10. 2 .
11. 28 1, 2 , 4 , 7 , 14 , 28 . 82 1, 2 , 41, 82 .
95 1, 5 , 19 , 95 .
105 1, 3 , 5 , 7 , 15 , 21, 35 , 105 .
124 1, 2 , 4 , 31, 62 , 124 .
345 1, 3 , 5 , 15 , 69 , 115 , 345 .
1.232 1, 2 , 4 , 8 , 14 , 16 , 22 , 28 , 44 , 56 , 77 , 88,
112 , 308 , 616 , 1.232 .
3.999 1, 3 ,1.333 , 3.999 .
12.. 78 : 2 39= 39 :3 1 3= 13:13 1= .78 2 3 13= .
.348 : 2 1 74= 174 : 2 87=
87 :3 29=
-
69 /---
29 : 29 1= .2348 2 3 29= .
.1.210 : 2 605=
605 :5 1 21=
121:11 1 1=
11:11 1= .21.210 2 5 11= .
. 2.344 : 2 1 .172=
1.172 : 2 586=
586 : 2 293=
293: 293 1= . 32.344 2 293= .
1. 42 47 .
2. .
3. :. 25 , 45 , 65 . 40 , 50 , 60 .120,144 , 180.
4.:
.1.2 _ 7 9 .
. 1._ 5 _ 5 3 1.100.
5. 32 ,48
-
70 1 -
72
.
;
6. 8 , 30160 .,;
7. ( ), , , 16 = , , , 70 1, , , .
1
32.
1. ,( )100 30 10 70 10 60 = = ( )100 30 10 1 00 20 80 = = .2. .11 ,10 .3. .4. , 52 2 2 2 2 2 32= = .5. .
-
71 1
6. , 6 = .7. .8. .9. , 20 12 : 4 20 3 1 7 = = .10. .11. .12. .13. ., 2 . , 2 ,.14. .15. .38 3 8 1 1+ = 3 .16. .
17. . ( )35,210 35 = .18. . ( )2,24 24 = .19. .15 420 320 :15 28= .20. .21. .22. ,10 .23. .
1
1
.
;
-
72 1 -
. , ;. ;
.
;
2 :
. ( )4 2 2124 :31 2 : 4 2160 :18 8 + .( ) ( )2 3 2 3 2 81:3 88 :8 : 2 170 :34 26 3+ + :( ) ( )2 3 2 3 2 81:3 88 :8 : 2 170 :34 26 3+ + 9( ) ( )2 3 2 3 2 81:3 88 :8 : 2 170 :34 26 3+ +
3 .
7 16 .,.
.
24 7 .,.
4
.:432 ,
1250,1.890 ..
-
73
2
21
48 6
1 32 6
343 5
2 82
-
74 2 -
. .
:
1
, :
1 =
.
( ) . ( ) : 0 . .
, , :
1 =
2.1
-
75
1:
1 =
,
:
1 >
.
, :
: =
1. 25
.
,
1
.
-
76 2 -
60 . 15
5 15
. :
1 60 12 5 5
= = 1 60 12 5 5
= = 1 60 12 5 5
= = .
25
15
2 , :
2 1 2 2 12 24 5 5 = = = 2 1 2 2 12 24
5 5 = = = 2 1 2 2 12 24
5 5 = = =
2 1 2 2 12 24 5 5 = = = .
2. 58
10 ,
.
, 1
.
58
, 5 8 , 10 ,
18
, 1 , :
18
5 8
= : 5 1 0 : 5 2= = .
88
, 188
:
8 8
188
= 8 2 16= = .
-
77
3. 715
21 ,
.
7
15 21 1
15 :
115
7
15= : 7 21: 7 3= = .
, 1515
, :
1515
11 5
15= 1 5 3 45= = .
36.
1. . , .
. : 1 = , 1 =
,
0 0=
.
. ,
.
2. 3
4,
23
, 79
1820
1,
109
.
-
78 2 -
3. 4 28 428
.
4. 15
15 , 155
:
155 5 14= 70= .
5. 24
, 23
, 49
, 68
, 13
, 2540
.
6. 27
4= , 17
: 27
: 2 4 : 2 2= = .
, 77
, 1 77
2 7 1 4= =
..
7. . 100 1001.000
= . . 250 2501.000
= .
. 500 5001.000
= . . 600 6001.000
= .
4. . 30 , 15 1530
.
-
79
. 6 30 180 = , 15
15
180 .
. 365 , 15 15365
.
5. , 55
90 , 15
90 185= .
25
125 2 18 36= = .
90 36 54 = .
6. 3
8 12 1
8 12 : 3 4= .
88
: 88
1 88
=
8 4 32= = .
7. 1111
33cm , 111
33:11 3cm= . 311
3 3 9cm = .
33cm 9cm , :
2 33 2 9 66 18 84cm = + = + = .
-
80 2 -
8. . 1010
5cm , 1
10
1 55 cm
10 10 = 8
10 :
1 58 8 cm 4cm
10 10 = = .
. 55
5cm , 15
1
5 1 cm5 = 6
5 :
16 6 1cm 6cm
5 = = .
1. . 26
, . 14
, . 1016
, . 3
5.
2.
14
3
16
18
524
1. 11 4 .
.
.
-
81
.
.
2. 27
140
3
28 ,
.
3. 59
.
85 ,
.
4. , .
240 . 512
, 22
16 .
. ;
. ;
. ;
5. 89
,
72 .
6. 500
.
-
82 2 -
7. ,
, 18
.
21.500 :
. ;
. ;
. 1
10
;
8. 1.000
1
10 .
;
2.2
,
, , ,
, :
=
-
83
:
= ,
( ) .
.
, :
=
: 1 3 1 3
4 3 4 12
= =
.
, :
:
:
=
.
: 4 4 : 2 2
8 8 : 2 4= = .
.
-
84 2 -
. 1.
, .
, 2436
12 : 24 24 :12 236 36 :12 3
= = ( ).
,
:
: 15
, 45
, 175
.
, :
: 14
, 15
, 158
.
:
.
-
85
.
.
, 56
78
. ( )6,8 24 = .
, 24 : 6 4= 24 :8 3= .
56
4 78
3 . : 5 4 206 4 24 =
7 3 21
8 3 24
=
2024
2124
56
78
.
1. 3
5
100 .
.
100 5 : 100 : 5 20=
3
5 20 :
3 20 60 5 20 100 =
-
86 2 -
60
100 100
3
5.
2. 1848
.
.
: ( )18,48 6 = . 1848
:
18 18 : 6 3 48 48 : 6 8
= = .
3
8
1848
.
3. 1248
, 1452
3
12.
.
.
( ) .
1248
1452
:
12 52 624 = 14 48 672 624 =
, 1248
1452
.
-
87
1248
3
12 :
12 12 144 = 3 48 144 =
, 1248
3
12 .
1452
3
12 :
14 12 1 68 = 3 52 1 56 1 68 =
, 1248
3
12 .
4. 56
119
.
,
.
( )6,9 18 = , 18 : 6 3= 18 : 9 2= .
56
3 119
2 . :
5 3 15
6 3 18
=
11 2 22 9 2 18 =
1518
, 2218
56
119
.
-
88 2 -
40 .
1. . ,
.
. =
: =
. .
.
.
.
.
. , .
2. . : 2 27 54 = , 3 18 54 = , 23
, 1828
.
. :3 2 6 = , 4 1 4 6 = , 34
, 12
.
. : 7 40 280 = , 8 30 240 280 = , 78
, 30
40
.
. :13 28 364 = , 14 26 264 = , 1314
, 2628
.
-
89
3. . :100 : 4 25= , : 3 3 25 75 4 4 25 100
= =
.
. :100 : 5 20= , : 8 8 20 160 5 5 20 100
= =
.
. :100 : 20 5= , : 4 4 5 20 20 20 5 100
= =
.
. :100 : 2 50= , : 5 5 50 250 2 2 50 100
= =
.
. 75 100 6075
( )60,75 15 = : 60 60 :15 475 75 :15 5
= = .
45
.
:100 : 5 20= , : 4 60 4 20 80 5 75 5 20 100
= = =
.
4. . ( )6,10 2 = , : 10 10 : 2 5 6 6 : 2 3= = .
. ( )30,50 1 0 = , : 50 50 :10 5 30 30 :10 3
= = .
. ( )18,27 9 = , : 18 18 : 9 227 27 : 9 3
= = .
5. . : 6 : 3 2= , : 2 2 2 43 3 2 6
= =
.
. : 15 : 3 5= , 2 2 5 103 3 5 15
= =
.
-
90 2 -
6. . ( )33 3 11 .. ( )3 9 : 3 . . ( )70 14 5 .
. 4836
3
2, ( )32 3 8 .
7. . ( )25,30 5 = , : 25 25 : 5 3 30 30 : 5 5
= = .
. ( )9,12 3 = , : 12 12 : 3 4 9 9 : 3 3= = .
. ( )35,56 8 = , : 35 35 :8 4 26 26 :8 7
= = .
8. . ( )30,32 2 1 = , 3230
.
. ( )14,15 1 = , 1415
.
. ( )16,51 1 = , 1651
.
. ( )26,50 2 1 = , 2650
.
9. . ( )5,9 45 = , 3 3 9 27 5 5 9 45
= =
7 7 5 35 9 9 5 45
= =
.
. ( )8,10 45 = , 7 7 5 35 8 8 5 40
= =
3 3 4 12
10 10 4 40= =
.
. ( )3,12 12 = , 11 1 1 4 44 3 3 4 12
= =
7 7 1 7
12 12 1 12= =
.
-
91
10. . .
. .
. . x .. . .
. .
. .
1.. .
. .
.
. .
. .
. .
1. 6
15
:
.30 . 75 . 105 .135 .
2. 13
, 3
6,
58
7
12
24 .
3. :
. 1442
, . 14496
, . 39
40, .
225315
.
-
92 2 -
4. .
. 13
12
169144
, . 149
5836
, . 8497
11761358
.
5.
: 1213 4
5=
125 55 11
=
6.
. . 2832
98112
.
.
7. :
. 23
74
, .5
12,
718
1124
.75
, 9
12,
2115
, 1730
.
8. 45
.
55 62 .
-
93
2.3
,
:
> >
.
, 4 7
15 15< 4 7< , 8 6
5 5> 8 6> .
,
:
> <
.
, 4 4
11 15> 11 15< , 8 8
5 7> 5 7< .
,
-
94 2 -
,
:
=
=
, > >
, >
.
, 45
56
2430
2530
24 25 30 30
<
4 55 6< .
,
:
3 13= .
,
:
, 2 13< 2 3< ( ).
,
:
, 5 14> 5 4> ( ).
-
95
1. :
. 59
79
. 811
813
.
.
.
. :
5 7< 5 79 9< .
. :
11 13< 8 811 13
> .
2. 920
2125
.
.
: ( )20, 25 100 =
: 9 9 5 45 20 20 5 100
= =
21 21 4 84 25 25 4 100
= =
-
96 2 -
45 84< : 45 84100 100
< 9 2120 25
< .
3. 4554
35
34.
, .
45 54< 45 154
< .
35 34> 35 134
> .
4. , ( )2
14411 +
.
.
2144 12 12 12= = .
( )2144 11= + ( )2212 11= + 12 11= + , : 1 = .
43 .
-
97
1. . .
. ,
1. ,
1> . ,
1< .
. >
> .
2. . 3 5
7 7< .
. 5 9< 3 35 9> .
. ( )5, 12 60 = , :
4 4 12 48 5 5 12 60
= =
8 8 5 40
12 12 5 60= =
48 60> , : 48 4060 60
> 4 85 12> .
3. 10 11 12 13 14< < < <
: 31 31 31 31 31 10 11 12 13 14
> > > > .
4. . 5 8< 5 18< . . 9 10< 9 1
10< . . 12 11>
-
98 2 -
12 111
>
. 16 16= 16 116
= . 109 120< 109 1120
< .
5. ( )5,10,15 30 = :
3 3 6 18 5 5 6 30
= =
, 8 8 2 16
15 15 2 30
= =
, 5 5 15
10 10 3 30
3= =
,
20 20 2 40 15 15 2 30
= =
, 7 7 6 42 5 5 6 30
= =
.
: 15 16 18 40 42 30 30 30 30 30
< < < < 5 8 3 20 7 10 15 5 15 5
< < < < .
6. . : 3 5 6
3 3 3< < , , 51 2
3< < .
. : 6 7 82 2 2< < , , 73 4
2< < .
. : 60 63 65
5 5 5< < , , 6312 13
5< < .
. : 120 125 130
10 10 10< < , , 12512 13
10< < .
7. 12
, 14
, 3
4,
45
, 9
10 1<
.
( )2,4,5,10 20 = 1 102 20= , 1 5
4 20= , 3 15
4 20= , 4 16
5 20= 9 18
10 20=
-
99
:5 10 15 16 1820 20 20 20 20
< < < < 1 1 3 4 9 0 14 2 4 5 10
< < < < < < .
8. . 0 1 , 1 2 5 , :
:15
, :45
, .65
, :95
, : 115
. 0 1 , 1 2 3 , :
:13
, :23
, .43
, :73
.
9. ( )2,3,4,9,15,16 720 = :
540720
= , 528720
= , 540720
= , 495720
= , 360720
= , 560720
= , 480720
= ,
360
720 = .
: > = > > > > = .
1. :
. 7
15,
915
, 3
15,13
15 .
75
, 76
, 73
,79
. 56
, 1112
, 1718
,89
.
-
100 2 -
2. :
. 3
9,
59
, 89
,119
. . 112
, 114
, 1111
,1112
. . 53
, 3
5,
610
,9
15.
3. :
. 188189
189188
. 265256
256265
. 450150
62
.
4. :
.15
, 12
, 1
10 .
23
, 3
2,
13
.
5. :
. 1213
1413
. 143
103
. 215
295
.
6. 3
3
382
. :
. 1..
1.
7. . 24 140 5.000 220 7.500
-
101
.
;
8. 45
.
:
. 1.. 1.. 1
.
. 1 .
2.4
.
.
-
102 2 -
++ =
.
,
.
<
+ < < +
.
+
.
( )
(
).
:
+ =
:
=
, *= +
-
103
, > .
.
.
=
.
, .
,
.
1. : .15 29 31
8 8 8+ + , . 3 5 3
4 6 2+ + .
.
,
.
-
104 2 -
. , :
15 29 31 15 29 31 75 8 8 8 8 8
+ ++ + = = .
. , :
( )2,4,6 1 2 = , : 3 9 4 12= , 5 10
6 12= , 3 18
2 12= .
: 3 5 3 9 10 18 9 10 18 37 4 6 2 12 12 12 12 12
+ ++ + = + + = = .
2. : . 41 29 117 7 7 , . 5 1 2
3 6 9 .
.
,
.
. , :
41 29 11 41 29 11 1 7 7 7 7 7
= = .
. , :
( )3,6,9 1 8 = , : 5 303 18= , 1 3
6 18= , 2 4
9 18= .
: 5 1 2 30 3 4 30 3 4 23 3 6 9 18 18 18 12 12
= = = .
-
105
3. 3710
.
.
: 3 7 10 3 70 3 737 10 10 10 10
+ += = = .
4. 227
.
: 22 3 7 1= + , : 22 1 37 7=
5. 3 14 28 8+ + .
,
.
46
46 .
-
106 2 -
1. . 5 2 5 2 7 3 3 3 3
++ = = .
. 11 2 11 2 13 1 13 13 13 13
++ = = = .
. 4 2 4 2 3 4 6 4 6 10 9 3 9 3 3 9 9 9 9
++ = + = + = =
.
. 8 2 8 2 4 8 8 8 8 16 4
12 3 12 3 4 12 12 12 12 3
++ = + = + = = =
.
. 17 3 17 3 3 4 51 12 51 12 63 21 20 15 20 3 15 4 60 60 60 60 20
++ = + = + = = =
.
.15 5 15 5 3 15 15 15 15 30 5 12 4 12 4 3 12 12 12 12 2
++ = + = + = = =
.
2. . 3 1 3 1 2 1 2 2 2 2
= = = .
. 8 3 8 3 5 9 9 9 9
= = .
. 10 3 10 3 2 10 6 10 6 4 1 8 4 8 4 2 8 8 8 8 2
= = = = =
.
. 4 2 4 3 2 12 2 12 2 10 9 27 9 3 27 27 27 27 27
= = = =
.
. 7 5 7 8 5 3 56 15 56 15 41 3 8 3 8 8 3 24 24 24 24
= = = =
.
. 33 3 3 11 3 7 33 21 33 21 12 7 11 7 11 11 7 77 77 77 77
= = = =
.
3. .5 5 3 5 3 8 5 24 5 24 5 29
3 3 8 8 1 8 1 8 8 8 8 8 8
+= + = + = + = + = =
.
.1 1 4 1 4 10 1 40 1 40 1 414 4
10 10 1 10 1 10 10 10 10 10 10 += + = + = + = + = =
.
-
107
. 1 1 2 1 2 9 1 18 1 18 1 192 2 9 9 1 9 1 9 9 9 9 9 9
+= + = + = + = + = =
.
4. . 15 4 3 3 1 2 3= + = + , : 15/4 15 12 3 12 3 3 3 3 34 4 4 4 4 4
+= = + = + = .
. 5 2 2 1 4 1= + = + , : 5 4 1 4 1 1 1 2 22 2 2 2 2 2
+= = + = + = .
. 38 3 12 2 36 2= + = + . : 38 36 2 36 2 2 2 3 312 12 12 12 12 12
+= = + = + = .
5. . 3 3 2 3 2 8 3 16 3 16 192 8 8 1 8 1 8 8 8 8 8
++ = + = + = + = =
.
. 12 12 1 12 15 1 12 15 12 15 27 91 15 15 1 15 15 1 15 15 15 15 5
++ = + = + = + = = =
.
.16 3 16 3 2 5 20 16 6 100 16 6 100 122 615 20 10 20 10 2 1 20 20 20 20 20 100 10
+ ++ + = + + = + + = = =
.
6. . 1 1 2 1 2 5 1
3 2 3 2 3 3 5 5 1 5 1 5 5
= + = + = +
10 1 11 3 5 11 15 11 3 3
5 5 1 5 5 5 5+ = = = = =
15 11 4 5 5= = .
-
108 2 -
. 1 1 1 1 4 3 1 2 2 14 2 4 2 3 2 3 2 1 3 3 1 2 2
= + + = + +
12 1 4 1 13 5 3 3 2 2 3 2
= + + = =
13 2 5 3 26 15 26 15 11 6 5 6 5 5 51 13 2 2 3 6 6 6 6 6 6 6 6 6
+= = = = = = + = + =
.
. 2 4 2 4 1 3 2 4 3 2 41 13 5 3 5 1 3 3 5 3 3 5
= + = + = +
3 2 4 5 4 5 5 4 3 25 12
3 5 3 5 3 5 5 3 15 15
+ = = = = =
25 12 13
15 15= = .
7. 2 18
2/5 1. 1 25
20 000. . 20 000. , 55
.
15
25
:
15
20 000 5 4 000= = . : . .
25
2 4 000 8 000= = = . . 1.
-
109
2 18
8 1 78 8 8
=
1 . 1 8 000. , 18
78
:
18
1 8 000 8 1 000= = . : . .
78
1 7 1 000 7 000= = = . . 2 .
2 : 7 000 720 000 20
=..
.
: 20 000 8 000 7 000 5 000 =. . . . .
3 : 5 000 120 000 4
=. .
.
8. : 5 3 5 8 3 9 40 27 40 27 13
9 8 9 8 8 9 72 72 72 72 = = = =
.
9. ,
. :
( )3 5 10 15 30 =, , , ,
2 2 1 1 2 6 2 2 1 10 1 3 12 4 10 3 12 4 10 3 29
5 15 3 10 5 6 15 2 3 10 10 3 30 30 30 30 30 30
+ + ++ + + = + + + = + + + = =
2 2 1 1 2 6 2 2 1 10 1 3 12 4 10 3 12 4 10 3 29
5 15 3 10 5 6 15 2 3 10 10 3 30 30 30 30 30 30
+ + ++ + + = + + + = + + + = =
.
, :
-
110 2 -
30 29 30 29 1
30 30 30 30
= = .
10. . 3 4 3 4 7 5 2 5 2 2 21 15 5 5 5 5 5 5 5 5
+ ++ = = = = + = + = .
. 1 4 1 4 5 5 4 203 3 3 3 3 4 12
+ + = = = =
.
. 1 1 1 6 1 5 6 5 6 5 115 6 5 6 6 5 30 30 30 30
++ = + = + = =
.
. 3 1 3 1 5 3 5 3 5 3 815 1 5 1 5 5 5 5 5 5
++ = + = + = + = =
.
. 1 2 1 3 2 5 3 10 3 10 13
5 3 5 3 3 5 15 15 15 15
++ = + = + = =
.
. 33 5 3 5 3 1
5 5 5 5+ = + = +
.
. 8 3 8 3 31
8 8 8 8 = =
.
-
111
1. 8 4 4 8 12 12 2 610 10 10 10 10 2 5
++ = = = =: :
.
5 4 5 4 9 519 9 9 9 5
++ = = = = .
45 19 45 19 64 32
90 90 90 90 45++ = = = .
16 8 16 8 24 212 12 12 12
++ = = = .
2.
+ 5/7 3/2 1 3/5
5/75/7 + 5/7 =
10/7
5/7 + 3/2 =
31/14
5/7 + 1 =
12/7
5/7 + 3/5 =
46/35
3/23/2 + 5/7 =
31/4
3/2 + 3/2
= 3
3/2 + 1 =
5/2
3/2 + 3/5 =
21/10
11+5/7 =
12/7
1 + 3/2 =
5/21 + 1 = 2
1 + 3/5 =
8/5
3/53/5 + 5/7 =
46/35
3/5 + 3/2 =
21/10
3/5 + 1 =
8/5
3/5 + 3/5 =
6/5
1. ,
, :
-
112 2 -
.14 742 42
+ . 5 7 612 12 12
+ + . 25 344 11
+ . 1 3 94 7 14+ + .
2. ,
, :
.17 98 8
. 32 23 415 15 15
. 8 25 6
. 5 1 44 8 6
.
3. :
.25 2 1 3 94
3 3 6 5 15
+ + .
2 23
3
3 2 42 9 242 16
+
/ .
4. 13
15
15
, .
5.
97
.
6. ( ) ( ) :
. 445
. 1712
.1143
.72
. 2512
. 153
12
-
113
7. :
. 21 4 2 3
3 22 5 3 8
+ + 3
5 . 3
1 1 34 3 3
5 3 10+ .
8.
(), : 19100
,
150
, 25
15
. ;
2.5
:
=
.
-
114 2 -
:
=
.
1 .
:
. 1 = =
.
1
.
1 ,
0 .
: :
: =
.
-
115
.
1. :
. 3 , .15
, .27
, . 153
, .1 26 9+ .
.
.
. 3 13
.
. 15
5 .
. 27
72
.
. : 1 1 15 1 165 5 3 3 3 3 3= + = + = .
153
3
16.
. : 1 2 3 4 7 6 9 18 18 18+ = + = . 1 2
6 9+
187
.
-
116 2 -
2. :
. 1 1 3 1 222 3 6 6 3 + +
. 23 1 1 12 2 44 8 8 2
+
, ,
, , ,
.
. 1 1 3 1 2 1 2 1 1 2 1 3 2 1 3 2 3 4 72 2 2 2 3 3 6 6 3 3 6 6 6 3 3 6 18 3 6 18 18 18 18
+ + = + + = + = + = + =
1 1 3 1 2 1 2 1 1 2 1 3 2 1 3 2 3 4 72 2 2 2 3 3 6 6 3 3 6 6 6 3 3 6 18 3 6 18 18 18 18
+ + = + + = + = + = + = .
. 23 1 1 1 3 2 8 1 1 4 2 1 3 17 1 9 6 17 1 92 2 4 4 4 4 4 8 8 2 4 8 8 2 4 8 8 2 8 8 8 2
+ + + = + = + = + =
2 3 1 1 1 3 2 8 1 1 4 2 1 3 17 1 9 6 17 1 92 2 4 4 4 4 4 8 8 2 4 8 8 2 4 8 8 2 8 8 8 2
+ + + = + = + = + = 25 9 100 9 200 9 191 4 8 16 8 16 16 16 16
= = = = .
3. 180 . 160 . 29
;
,
.
29
:
-
117
2 2 180 360180 . . . 40 .
9 9 90 = = =
40 . :
40 1160 4
= .
49 .
1. .
.
.
1.
. , 1
,
: 1
, ,
.
. 1 .
2. . 3 3 3 9
3 4 4 4
= = . 10 7 10 707 514 14 14
= = =
. 4 4 2 82 22 2 2
= = = . 5 5 10 50 110 100 100 100 2
= = =
3. . 2 7 2 7 14 7 5 8 5 8 40 20
= = =
. 8 100 8 100 800 1 6
10 5 10 5 50 = = =
-
118 2 -
. 4 5 4 5 20 9 9 9 9 81
= =
. 3 2 3 2 6 1 2 15 2 15 30 5
= = =
4.
* 5/7 3/2 1 3/4
7/5 7/5 * 5/7 = 1 7/5 * 3/2 = 31/14 7/5 * 1 = 7/5 7/5 * 3/4 = 21/20
2/3 2/3 * 5/7 = 10/21 2/3 * 3/2 = 1 2/3 * 1 = 2/3 2/3 * 3/4 = 1/2
1 1 * 5/7 = 5/7 1 * 3/2 = 5/2 1 * 1 = 1 1 * 3/4 = 3/4
4/3 4/3 * 5/7 = 20/21 4/3 * 3/2 = 2 4/3 * 1 = 4/3 4/3 * 3/4 = 1
5. . 1 3 1 3 2 3 1 3 7 3 7 3 21 12 2 3 21 3 21 3 21 3 21 3 21 63 3
+ = + = = = = = .
. 1 1 1 1 4 5 1 2 2 1 21 5 21 5 105 100 5 100 5 14 2 4 2 1 0 5 2 5 2 5 2 5 2 5 2 10 10 10 10 2
+ + + = + + = = = = = = + = + =
1 1 1 1 4 5 1 2 2 1 21 5 21 5 105 100 5 100 5 14 2 4 2 1 0 5 2 5 2 5 2 5 2 5 2 10 10 10 10 2
+ + + = + + = = = = = = + = + = 1102
= .
. 1 1 3 8 1 25 250 125 1 24 1 124 1 1 1
3 10 3 10 10 10 31 318 8 8 8 8 4 4 4 4 4 4
+ + = + = = = = = = + = + =
1 1 3 8 1 25 250 125 1 24 1 124 1 1 13 10 3 10 10 10 31 31
8 8 8 8 8 4 4 4 4 4 4 + + = + = = = = = = + = + =
.
. 2 3 2 3 1 3 2 3 5 3 5 3 15 5 4 1 4 1 1 11 1 2 23 2 3 2 3 2 3 2 3 2 6 2 2 2 2 2 2
+ + = + = = = = = = = + = + =
2 3 2 3 1 3 2 3 5 3 5 3 15 5 4 1 4 1 1 11 1 2 23 2 3 2 3 2 3 2 3 2 6 2 2 2 2 2 2
+ + = + = = = = = = = + = + =
6. .74
.172
.85
.3 .8
739 . 1
-
119
7. 1 1 1 2 1 31 1 2 2 2 2
+= + = = .
23
, : 2 3 1 3 2 = .
8. . 6 3 1 6 3 24 3 27 5 5 4 5 20 20 20 20+ = + = + =
. 6 3 1 9 1 9 5 5 4 5 4 20
+ = =
. 6 3 1 3 1 3 5 5 4 5 4 20
= =
9. . 7 2 3 35 2 3 37 3 111 37 3 15 8 15 15 8 15 8 120 40
+ = + = = =
. 7 2 3 35 2 3 33 3 99 33 3 15 8 15 15 8 15 8 120 40
= = = =
. 7 2 3 7 6 280 6 274 137 * 3 15 8 3 120 120 120 120 60 = = = =
1. :
.2112
, . 1
2007, . 42 , . 13
4, . 2
23
3
2.
:
-
120 2 -
. 1 1 3 22 3 12 + . 7 7 110 5
15 30 5 +
. 3 1 1 3 12 28 3 4 2 2
+ +
3. 1 1.000 , :
. 3
4 1 . . 4
5 1 . . 5
8 2 .
. 9
10 5 .
4. 330ml :
. 22ml . 44ml . . 1101101ml . . 121ml . E. 440ml .
5. 45
. 41
100 ,
:
. 13
; . 25
;
. 79
;
6. 150 , 13
, 3
5 .
:
-
121
. .
. .
7. :
* 4/5 6/11 1 8/7
5/4
11/6
1
7/8
2.6
:
: = =
, ,
-
122 2 -
:
: =
,
.
, ,
.
:
=
:
: = =
-
123
1. :
.3 1
:2 3
.1
4 :6
, .8
: 27
, .1 1
5 :2 3
, .4 2
3 : 23 9
.
,
.
. 3 1 3 9
: 3 2 3 2 2
= = .
. 1
4 : 4 6 246= = .
. 8 8 1 8
: 2 7 7 2 14
= = .
. 1 1 5 2 1 1 11 1 11 33
5 : : : 3 2 3 2 3 2 3 2 2
+= = = = .
. 4 2 3 3 4 2 9 2 13 20 13 9 13 9 117
3 : 2 : : 3 9 3 9 3 9 3 20 3 20 60
+ + = = = = =
.
2. :
.
4327
. 435
.
1183
.
-
124 2 -
,
.
1 ,
.
. (1 ). :
4 4 7 28 143 2 3 2 6 37
= = =
( 2 ). :
4 4 2 4 7 4 7 28 143 : 2 3 7 3 2 3 2 6 37
= = = = =
. (1 ). :
44 4 5 201 3 3 1 3 35 5
= = =
( 2 ). :
-
125
44 4 3 4 5 4 5 201 : 3 3 1 5 1 3 1 3 35 5
= = = = =
. (1 ). :
11 1111 1 118 8
33 8 3 241
= = =
( 2 ). :
11 11 11 3 11 1 11 1 118 8 : 33 8 1 8 3 8 3 241
= = = = =
3. :
. +4 223 17
. +21224
3
,
.
-
126 2 -
. : 4 4 3 3 4 6 102 3 3 3 3 3 3
+ = + = + = .
: 2 2 1 7 2 91 1 7 7 7 7
+= + = = .
:
4 102 10 7 703 3 2 9 3 9 2717 7
+ = = =
.
. :
2 1 1 1 2 18 1 36 1 374 2 16 2 1 8 2 2 2 2 2 2 2
++ = + + = + = + = =
: 3
3 1
=
: 21 372 37 1 372 24
33 2 3 61
+ = = =
.
4. 12
710
.
;
;
.
-
127
:
7 10 120 119 1 17*7 1 1 112 : 1 2 7 7
10 7 7 7 7 7 7+ += = = = = + = .
, 8 7
1 17
.
51 .
1. .
.
. ,
.
2. .3 1 3 6 3
: 2 4 2 4 4 2
= = =
. 1 1 1 3
: 3 1 3 3 3 3
= = =
. 10 1 1 1 1 5 1
: : 5 100 5 10 5 10 10 2
= = = =
. 7 21 7 7 7 9 63
: : 33 27 3 9 3 7 21
= = = =
-
128 2 -
3. . 12 : 2 3 63= =
. 5 5 5:1 1 8 8 8
= =
. 1 2 2 1 5 5 1 5 1 52 : 4 : 4 : 4 2 2 2 2 4 2 4 8
+ = = = = =
. 1 1 4 10 1 3 3 1 41 10 41 3 1234 : 3 : : 10 3 10 3 10 3 10 10 100
+ += = = =
4. 1 1 1 3. : 3 2 3 2 2
= = 1 1 1 2. : 2 3 2 3 3
= =
3 2 1 2 3
= = , .
20 20 1 20 1. :10
6 6 10 60 3 = = = 20 6 60. 10 : 10 3
6 20 20 = = =
1
3 1 3
= = , .
5. 1 1 1 1 1 1 2 1 3 3
. : : : 2 : 8 3 2 8 3 8 3 8 2 16
= = = = .
1 1 1 1 1 3 1 3 6 3. : : 3 : : 2
8 3 2 8 2 8 2 8 8 4 = = = = =
.
.
6.
: 5/7 1/2 1 4/35/7 5/7 : 5/7 = 1 5/7 : 1/2 = 10/7 5/7 : 1 = 5/7 5/7 : 4/3 = 15/281/2 1/2 : 5/7 = 7/10 1/2 : 1/2 = 1 1/2 : 1 = 1/2 1/2 : 4/3 = 3/81 1 : 5/7 = 7/5 1 : 1/2 = 2 1 : 1 = 1 1 : 4/3 = 3/4
4/3 4/3 : 5/7 = 28/15 4/3 : 1/2 = 8/3 4/3 : 1= 4/3 4/3 : 4/3 = 1
-
129
7. . 3 4 3 10 30 3: 10 10 10 4 40 4
= = =
. 5 4 5 9 45 5: 9 9 9 4 36 4
= = =
. 45 15 1 15 1 9 9 3: : 90 9 2 9 2 15 30 10
= = = =
. 16 8 16 9 144: 63 9 3 8 24
= = =
8. .
33 5 158 4 8 4 32
5
= =
.
15 1515 1 15 53 3 44 3 4 12 4
1
= = = =
.
2020 20 4 801 1 65 5 1 5 54 4
= = = =
9. .
3 1 4 4 4 3 12 35 5 5 5 2 4 2 2 4 5 4 20 5 3 6 3 3 3
+ = = = = =+ +
.
4 2 8. 8 55 440 557 8 56
2 3 6 56 6 336 42.5 11 55
= = = =
.
2 4 2 3 6: 6 16 963 3 3 4 12 81 1 1 1 12 1 12 : 2
8 8 2 16
= = = = =
-
130 2 -
1. 2 1 1 3 2 6= +
v :
2 1 1 1 1 1 3 2 6 = + v v v
23
15
:2 1 2 3 5 15 = . :
2 1 1 1 1 1 1 1 3 1 4 2 3 5 2 5 6 5 10 30 30 30 30 15 = + = + = + = =
23
19
:2 1 2 3 9 27 = . :
2 1 1 1 1 1 1 1 3 1 4 2 3 9 2 9 6 9 18 54 54 54 54 27 = + = + = + = =
23
1
13 :
2 1 2 3 13 39 = . :
2 2 1 1 1 1 1 1 1 3 1 4 2 3 13 2 13 6 13 26 78 78 78 78 39 = + = + = + = =
1. :
. 5 1
: 3 4
.4
12 :7
, . 28
:87
, . 2 1
2 :3 3
, . 3 21 5
3 : 49 12
.
-
131
2. :
.
125 218
. 734
.
41194
.
3. :
.
2 25 7
33
8
+ .
2
3
14 3
42
37
+
+ .
2
4 2 5 9
145
4. :
. 3 2 1 2
: 4 24 5 2 3
+ .
22
3
1 1 1 2: 2 : 4 :
2 7 3 3
. 73 26 2 1 1 7
: 17 7 4 3 9
. 2 1 3
2 : : :3 2 2
5. :
: 3/5 3/4 1 8/9
3/5
3/4
1
8/9
-
132 2 -
6.
147 4
. 11 20
,
.
7. 23
:
. 49
.423
.
8.
1.400900 . 1 2
3. 47 ,
:
. .
. .
2
2
1
. , , , , : =
-
133 2
. ;
. ;
. ;
2 :
.
21 2 1 5: 2 :
32 3 57
+
. 1 1 22 2 12 3 39
+
3
, , : 23
+ = 12 4
= , :
. : 1 3 = + +3
2 2 = 2+83,
. 1 2 .
4
12 35
.
-
134 2 -
. ;
. 925
, ;
-
135
3
1,734
0,001
45,06
9,00210
26
-
136 3
-
10 , .. 110 10= , 2100 10= ...
: 10
,
.
.
.
.. 73,324 :
73
324
3.1
-
137 - - -
.
:
.
2 .
3 .
, 73,324 : 7 3 ( ) 3 2 ( ) 4
, .. 3 3,00=
, .. 102,2100 102,21=
,
. ..:
215100
2100 10= .
}}
-
138 3
. :
2
2
215 2, 151 00
=
10 ,
. ..
75,825
10 (3
):
33
3
75.820 75.82075, 82510 1 000
= =
10
.
: ,
: :
3
2
3
2
-
139 - - -
.
0 : ..
32 32 : 5 6,45= =
,
. ..
1010 : 3 3,333
=
2020 : 3 6,673
=
.
,
.
..:
14,35 9,48> , 14,35 (14) 9,48 (9) .
} ( )
-
140 3
, . ..:
32,12 32,21< , 1 32,12 (1) 32,21 (2) .
.
5 . ..:
11,255
. 11,255 11,300 11,3= =
: 22,826 22,800 22,8= =
1. :
: 2510
: 42
100 :
181000
, 0 .
: 110 10=
1 : 25 2,510
=
-
141 - - -
: 100, 2 :
42 0,42100
=
: 1000, 3 :
18 0,0181000
=
2. : A: 3 :8 : 32 : 44
:
.
.
: 33:88
= . 38
:
0,375 .
B: 3232 : 4444
= 3244
:
0,72727..... 0,73 ( )
0,727 ( )
-
142 3
3. : : 6,78 : 18,5 : 0,007
, .
: 6,78 678 100 :
6786,78100
=
: 18,5 185 10 :
18518,510
=
: 0,007 7 1000 :
70,0071000
=
4. 2,015 : . : .
.
: 1 2,015 0 : 2,015 2,000 2= =
: 3 2,015 5 : 2,015 2,020 2,02= =
-
143 - - -
(. 59)
1.) : 44 : 55
=
) : 99 :16
16=
) : 2525 : 7979
=
2. 221
2 : 21 .
163
19 : 3 .
77
105 77 :105 .
3.) 7 :16 :
7064
60 48
120 112
80 80
0
16
0,4375
77 :16 0,4375
16= =
7 0,4316 : .
7 0,43716 : .
) 21:17 :
-
144 3
21 40 60 90 16 ...
17
1,2352...
21
21:17 1,2352.....17
= =
21 1,2417 : .
21 1,23517 : .
) 20 : 95 :
200140 500 25 ...
95
0,2105
20
20 : 95 0,2105.....95
= =
20 0,2195 : .
20 0,21195 :
4. : 58 5,810
= : 3 0,03100
= : 5.025 50,25100
= : 1.024 1,0241.000
=
5. : 353,510
= : 4.52545,25100
= : 3.0043,0041000
=
-
145 - - -
6. ) 5,8909 : : 0 , : 9 .) 98,0005 : : 0 , : 5 .) 456,8756 : : 5 , : 6 .
7.) 45,345 54,413< : 3 4< .) 980,19 899,01< 980 899< .) 7,534 7,5340= , 0 .
8. :: 9.876,008 9.876,0 9.876= = : 67,8956 67,9000 67,9= =: 0,001 0= : 8,239 8,200 8,2= = : 23,7048 23,7000 23,7= =
:
: 9.876,008 9.876,010 9.876,01= = : 67,8956 67,9000 67,9= =: 0,001 0= : 8,239 8,240 8,24= = : 23,7048 23,7000 23,7= =
:
: 9.876,008 9.876,008= : 67,8956 67,8960 67,896= =: 0,001 0,001= : 8,239 8,239= : 23,7048 23,7050 23,705= =
9. : 3 3,4 4< < ().
) 4 4,5 5< < ) 2 2,3 3< < ) 2 2,8 3< < ) 4 4,7 5< >> >> : 4 2 20,4 10 4.000cm cm = .>> >> >> : 6 2 20,4 10 400.000mm mm = .>> >> >> : 3 40, 4 :10 4 :10 0,0004 = = .
3. 50 m , 3m , 3dm , 3cm , 3mm .
1.000 ( 310 ) 1.000 ( 310 ).
3 3 310 1 =m dm 31 11000 = =m cm .
50 : 450 1.000 50.000 5 10m m m = = .>> >> >> : 3 2 2 2 250 10 0,05 5 10m m m = = .>> >> >> : 350dm .
>> >> >> : 3 3 4 3 450 1.000 50.000 5 10 ( 5 10 )cm cm cm m = = = .>> >> >> : 3 3 7 350 1.000.000 50.000.000 5 10mm mm mm = = .
-
167
4. .
.
90 . : 90 min : 90 60 5.400s s = .>> >> >>: 90 : 60 1,5h h= .>> >> >>: 90 : (60 24) 0,0625 = .
5. : 0,2m 30cm . : 24cm 30,004mm .
. .
. 0,2m 30cm . cm m 230 30 10 0,3cm m m= = . 0,3 0,2m m> 30 0,2cm m> .
. 24cm 30,004mm .
24cm 30,004mm !
(.67)
1. . 23 23 10 230dm cm= = . . 33,1 3,1 10 0,0031m km km= = . . 245,83 45,83 10 0,4583cm m m= = . . 667,2 67,2 10 67.200.000km m m= = . . 195,5 95,5 10 9,55mm cm cm= = .
-
168 3
2. 33,1 3,1 1000 3.100 3,1 10a m mm mm mm= = = = . 34,2 4,2 1000 4.200 4,2 10m mm mm mm = = = = . 32,3 2,3 1000 2.300 2,3 10m mm mm mm = = = = .
3. 0,023 0,023 1.000 23km m m= = 456 456 :100 4,56cm m m= = 678 678 :10 67,8dm m m= =: 4,56 23 67,8 986m m m m< < < 456 0,023 678 986cm km dm m< < < .
4. 23a cm= , 45cm =
: 223 45 1.035E cm cm cm = = = . 2 2 21.035 1.035 100 103.500cm mm mm= = .
5. ) 2 2 7 256 56.000.000 5,6 10km m m= = .) 2 20,987 0,987 1.000 987 m m = = .) 2 2350 350 1.000 350.000 m m = = .
6. : 2E a= , 2 2 2210 44.100E m m= =
244.100 44.100 :1.000 44,1E m = = = .
7. : 25 7,2 36m m m = = .>> >> : 2 2 2' (40 ) (0,4 ) 0,16E cm m m= = = . :
22
56 225' 0,16mEN E m= = = 350 .
-
169
8. 3 3 315 15 100 15.000dm cm cm= = : 3 3 3 3 315 29 15.000 29 15.029dm cm cm cm cm+ = + =
6 3 315.029 :10 0,015029m m= 3 3 315.029 10 15.029.000mm mm = .9. 3a m= ,
2m = , 5m = : 33 2 5 30V a m m m m = = = .
: 3 3 3 3 33 30 90 90 10 90.000 90.000m m dm dm = = = = .
1 4 : 4 90.000 360.000 = .
10. 5 : 20 : 17 20minh + . 17 20min 8 10min 9 10minh h h+ + = + .
11. . 4 4 60min 240minh = = : 4 52min 240 52min 292min 292 60 17.520h s s+ = + = = = .
. 3 3 60min 180minh = = :3 12min 180min 12min 192min 192 60 11.520h s s+ = + = = = .
. 5 5 60min 300minh = = 30 30 : 60min 0,5mins = = :5 20min 30 300min 20min 0,5min 320,5min 320,5 60 19.230h s s s+ + = + + = = =
5 20min 30 300min 20min 0,5min 320,5min 320,5 60 19.230h s s s+ + = + + = = =
. 45 45 : 60min 0,75mins = = :56min 45 56min 0,75min 56,75min 56,75 60 3.405s s s+ = + = = = .
12. . 601 1 60min min 6min10 10 10h = = =
-
170 3
. 601 1 60min min 12min5 5 5
h = = =
. 601 1 60min min 10min6 6 6
h = = =
13. . 3 3.000kg g= : 3 600 3.600 3.600 :1.000 3,6kg g g kg kg+ = = = . 3,6kg 3 1kg , 1 500g 2 50g .
. 2 2 1000 2.000kg g g= = : 2 450 2.450 2,45kg g g kg+ = = . 2,45kg 2 1kg 9 50g .
14. . 5kg , 3kg 1kg 9kg .
. 5kg , 3kg 1kg 10kg .
15. . 5 2 2 1 0,5 .. 2,8 >> 1 2 , 1 0,5 3 0,1 .
. 2, 4 >> 1 2 4 0,1 .
16. 1T 1.200 3T
-
171
33 1.200 3.600 3,6m = = . 33,6 2,5 1V m m m = = 3 23,6 2,5m m = .
: 3
23,6 1,44 1442,5
m m cmm = = .
1cm 3.600 :144 25cm cm= .
17. 80 0,8cm m= , :31, 2 0,8 0,8 0,768 768m m m m = = .
1,2 120m cm= , 1cm 768 6,4120 = .
. 10cm 10 6,4 64 = . 1min 8 , 64 64 :8 8min= .
. 768 768 :8 96min= .
. 1 30min2 h = 30min 30 8 240 = 240 : 6,4 37,5cm= . 37,5cm .
18. 1 : 1 15min 60min 15min 75minh + = + = .2 >> : 1 45min 60min 45min 105minh + = + = .
. 75 575 :105 105 7= = . 105 7105 : 75 75 5= =
1. :
-
172 3
km m dm cm mm
1,44
52,2
250
125
36
2. :
3m 3dm 3cm 3mm m
0,02
4,5
62,2
44
250 650
3. 0,08a m=
. 2dm 3dm . . 2cm 3cm . . 2mm 3mm .
4. , 0,5a km= = 400m =6000dm.
5. .
1,2a m= , 200cm = , 8dm = 0,62 .
-
173 3
6. ) 120.000 21m 160
.
) 25m , .
7. 08 : 00 15 : 00 . , . 30 1 .
8. 0,45T . 80kg 65kg .
3
1. .
2.
3.: 1: 1 = =
4.
5.
-
174 3
6. 7 2139 9 =
7.
5 131 138 83 3 24
= =
8.
9. , 1.
10.
11.
12.
13. 10.
14.
15. 1.050 : 3.100 0,34 2.593: 4.650 0,56= . 1.050 2.5933.100 4.650
32x y =
2. ) 25 .
) 7 2 .
) .
) 88 .
3. : = + + + (1 1 1 1) = + + + 4 = .: 2 = = .
4. ) 2x x x+ =
) (1 1 1) 3 + + = + + =
-
183
) 3 52 (3 52) 55 + = + =
) 2 3 2 (2 1) (3 2) 3 5 + + + = + + + = +
) 4 8 3 (4 8) 3 12 3 9x x x x x x x x + = + = =
) 7 4 10 (7 4) 10 11 10 + = + = =
5. ( ) ( )x y z x y z = .
2 2 3 6 29 9 5 45 15
z = = =
6. 0x = 2 0+ = x 0 . 3x = 2 3+ = 1 = x 3 . 1x = 2 1+ = x 1 .
7. x 12 : 13 25x + = 12 13 25+ = 25 25= 12 13 25x + = .
8. 2 4x = 61 4y+ = 318 10 = 89 1 = 893 86 = 7
9. ) 4,9 15,83x + = 15,83 4,9x = 10,93x =
-
184 4
) 40,4 93,19x+ = 93,19 40,4x = 52,79x =
) 53,404 4,19x = 53,404 4,19x = 49,214x =
) 38 7,1x = 38 7,1x = 30,9x =
10. ) 3 12
20x= 123:
20x = 123:
20x = 203
12x = 5x =
) 5 157 x= 5 15 :
7x= 515 :
7x = 715
5x = 21x =
) 3540 8
x= 35 :840
x= 35 840
x = 7x =
) 49 4
5 5x= + 49 4
5 5x = 45
5x = 9x =
11. ) 3 1 7
4 2 4x + + = 3 7 1
4 4 2x + = 3 5
4 4x + = 3 5x + = 5 3x =
2x =
) 5 38 16 4
x+ = 10 1216 16
x+ = 10 12x+ = 12 10x = 2x =
) 3 2 15 10
x ++ + 6 2 110x+ + = 8 10
10 10x+ = 8 10x+ = 10 8x =
2x =
12. ) 3 4v + = 4 3v = 1v =
-
185
) 2 8x = 8 2x = + 10x =
) 4 1 3 19t + + = + 5 22t + = 22 5t = 17t =
) 6 5x = 6 5x = 1x =
13. x :
2145
x + = 21 45
x = 21 205
x = 15
x =
x 15
:
1 2145 5+ = 1 20 21
5 5 5+ = 21 21
5 5= 1
5.
14. x :5 313x + = 313 5x = 308x =
x 308 :308 5 313+ = 313 313= 308 .
15. ) 1 4cm , 2 8cm , 3 12cm , 4 16cm . 4 5 20cm .
) x 4 x .
) 4 128x = 128 : 4x = 32x = 32 128 .
-
186 4
1. 12
, 1, 2 , 3 :
) 4 5 19x + = ) 2 2 10 4x x x x+ + = ) 4 2 8x x+ = ) 5 6 6x x+ =
2. :
) 1 .
) 2 2 .
) 12
3. :
) 5 4 18x + = ) 1 1 12 2
x x+ = ) 2( 4) 4 2x x+ =
4. :
) 2 5 23x + = ) 15 5 6 4x x x = + ) 20 10 10x x+ = +
) 3 2 175
3 3x x+ + = + ) 24 8 3
2 x+ =
+ )
4
0,852
x =
5. x .) 20 : 5x = ) 4x + =
-
187
6. . 1 2 .
) .
) .
) 10 .
) 30 ;
4.2
-
.
.
. :
-
188 4
.
.
.
.
.
1. 4 29 . .
.
1. v , 1v + . : 1 2 1v v v+ + = + . : 2(2 1) 4 2v v+ = + .>> >> >> >> 4 4 2 4 4 1v v+ = + . : 4 1 29v + = 4 29 1v = 4 28v = 28 : 4v =
7v = . 7v = 1 7 1 8v + = + = . : 2 (7 8) 1 29 + = 2 15 1 29 = 30 1 29 = 29=29 .
2. 52 4 . ;
-
189
x 4 x 52 . : 4 52x = 52 : 4x = 13x = . 13 .
: 4 13 52 = 52 52= .
( 78)
1. vvvv , v ( 10v < ). vvvv 9 9 :
9v v v v + + + = 4 9v = (9 ) : 4v = . 1 = , 9 : 4 2,25v = = v = . 2 = , 2 9 : 4 18 : 4 4,5v = = = >> >> >> . 3 = , 3 9 : 4 27 : 4 6,75v = = = >> >> >> . 4 = , 4 9 : 4 9v = = . 5 = , 5 9 : 4 11,25 10v = = > . v 9 9999 9 9 9 9 4 9 36+ + + = = 9 .
2. x : 2 608
x =
1 604
x = 160 :4
x = 60 4x = 240x = . 240
.
: 2 240 608 = 1 240 60
4 = 240 60
4= 60 60= .
3. v , 1v +
-
190 4
2v + , : 1 2 1.533v v v+ + + + = 3 3 1533v + = 3 1533 3v = 3 1530v = 1530 : 3v = 510v = . 510 , 511, 512 .
: 510 511 512 1533+ + = 1533 1533= .
4. v , v , 75 3v . 9 9 , :7 5 3 9v + + + = 15 9v + = 9 15v = . 1 = , 9 15v = . 2 = , 2 9 15 18 15 3v = = = . 3 = , 3 9 15 27 15 12 10v = = = > . 3v = 7533 .: 7533 , 7 5 3 3 18 2 9+ + + = = , 9 .
5. x , 100 x . : 3 x . : 1 (100 ) 100x x = . : 3 100 220x x + = 2 100 220x + = 2 220 100x = 2 120x = 120 : 2x = 60x = . 60 100 60 40 = .
: 3 60 1 40 220 + = 180 40 220+ = 220 220= .
6. x : 18 25x = 25 18x = + 43x = . 43 .
: 43 18 25 = 25 25= .
7. x 1 ( 600)x + .
-
191
y 2 ( 15.600)y . 3 15.000 1
2 : 600 15.000x + = 15.600 15.000y = :600 15.000x + = 15.000 600x = 14.400x = 15.600 15.000y =
15.000 15.600y = + 30.600y = . 14.400 30.600 .
: 14.400 600 15.000+ = 15.000 15.000=
30.600 15.600 15.000 = 15.000 15.000= .
8. ) 47 73+ = 73 44 = 26 = 2 = 6 = .
) 8 5 =
8
5
3 = 3 8 35 = 3 35 8 = + 3 43 = 4 = .
9. x 7 y ( 18)x 7 y :
18 7x y = . 90 100x< < 95x = : 95 18 7 y = 77 7y= 77 : 7y = 11y = . 95
11 . : 95 18 7 11 = 77 77= .
10. x x : 0,75 100x = 100 : 0,75x = 133,3333...x =
) x = 133x = 133 0,75 99,75 = .
-
192 4
) 100 99,75 0,25 = .
11. 3 75
18 18,754 4
km km km= = .
1 1 7
3 3,52 2= = km/
2 >> >> 3 11
2 2,754 4= = km/.
x :3,5 2,75 18,75x x + = 6, 25 18,75x = 18,75 : 6,25x = 3x = . 3 .: 3,5 3 2,75 3 18,75 + = 10,5 8,25 18,75+ =
18,75 18,75= .
12. x 115 15
xx =
.
1 65 5 5
x xx x x+ = + = .
y : 615 5x xy=
6
:15 5x xy = 5
15 6xy
x= 50
90
xyx
= 590
y = 118
y =
1
18 .
13. x , x 7 . 7 , 14 , 21 , 28 .
1x + 9 , 18 , 27 .
-
193
35x = 35 7 5= 1 36 9 4x + = = . 35 36 . 98x = 98 7 14= 1 99 9 11x + = = .2 98 ( 99 ).
1. 65 32 . ;
2. 1 2. ;
3. 13
2 . .
4. 1kg . ;
5. 20 , . 0,2 0,15
3,6 .
6. 120 .. . ;
7. 3. .
-
194 4
8. 17 . 77 2 .
4
4
1
:
) 1
2 3 ( 5)2
x x+ = + 12 3 ( 5)2
x x+ = +
) 2 3 41
2 3x x x+ = +
) 2610
24 : 8xx
+ = 1024 : 8xx
+ =
2
) 2 :
2 2( 1) 3 1x x x x+ = + + .
) : 1 5 2 232
x x = .
-
195 4
3
.
4
1 35 . 3 .
-
196 4
-
197
5
% % % % % %
-
198 5
5.1
%
100 .
1.000 .
% , , %100 = .
100
.
, 1.000 = .
% :
-
199
% 100 .
:100 .
:1.000 .
:
100 ( 100 ).
100 % .
1.000 ( 1.000 ) 1.000
.
2
:
100
%100
= =
( 1.000
=
).
-
200 5
: . 100
% (
1.000
. ).
% :
%
100 =
1. . 925
. 7
13
100 , : (
) .
. 9 9 4 3625 25 4 100
= =
9 36%25
=
. 13 100 713
100
. :
-
201
77 :13 0,53846 0,538
13= =
0,538 100
: 53,8
0,538 53,8%100
= = 7 53,8%13
= .
2. : . 32% . 75 . 2,5%
, .
. 32% 32 8
100 25=
. 75 75 3
1.000 40=
. 2,5% 2,5 2,5 10 25 1
100 100 10 1.000 40= = =
3. 25% 40 .
x x % = , .
x . 25% x 40 :
25% x 40 = 25 x 40100
= 25 x 40100 = 40 100x 160
25
= =
: 25% 160 25 160 25 4.000160 40100 100 100
= = =
-
202 5
4. 120) 25% ) 30% .
.
. 25% 120 : 25 1 120
25% 120 120 120 30100 4 4
= = = =
120 25% 120 30 150+ = .
: x
( )x 120 25%120 120 0,25 120 120 1 0,25 120 1,25= + = + = + =
125
120100 x 125% 120=
. 30% 120 : 3030% 120 120 36100
= =
120 30% 120 36 84 = .
: 30
x 120 30%120 120 120100
= =
( ) 70120 0,3 120 1 0,3 120 0,7 120 120100
= = = =
x 70% 120= .
( 81)
1. 100 :
-
203
. 1 1 20 20
20%5 5 20 100
= = =
. 3 3 50 150
150%2 2 50 100
= = =
. 1 1 25 25
25%4 4 100
= = =25
. 3 3 25 75
75%4 4 25 100
= = =
. 3 3 20 60 60%5 5 20 100
= = =
2. 100 .
. 52
0,52 52%100
= = . 3413,41 341%100
= = . 190,19 19%100
= =
. 30,03 3%
100= = . 70,07 7%
100= =
3. . 15 315%100 20
= = . 77%100
= . 48 1248%100 25
= =
. 50 1
50%100 2
= =
4. . 10 30.00010% 3.000 3.000 300100 100
= = = . . 1h 60min= , 45% 1h :
45 2.70045%1h 45%60 min 60min min 27 min
100 100= = = = .
. 31 1.000m 1.000cm= = 20% 1 :
3 3 3 320 10.00020%1 20%1.000cm 1.000cm cm 100cm100 100
= = = = .
-
204 5
. 50% 500g : 50 25.000
50%500g 500g g 250g100 100
= = = .
. 1kg 1.000g 25%= 1kg :
25 25.000
25% 1kg 25% 1.000g 1.000g g 250g100 100
= = = = .
5. % :
) 50 5
100 1000 100 = = 5 = 50 5% 1.000 .
) 1 365= : 30 8,220,0822100 365 100
= = = 8, 22 =
30 8,22%= .
) 50 2
100 2.500 100
= = 2 = 50 2%
2.500 .
) 1 1dm= 1m 10dm= 10m 10 10 100dm= = .
3 3100 100 = = 3dm 3% 100dm .
6. 22% 0,61 ,
22 22 0,61
22%0,61 0,61 0,1342100 100
= = =
7. ) : R 50km 2.900km 3.450km 6.400km= + + = . ) R :
50 0,780,0078 0,78% R
6.400 100= = .
-
205
>> >> >> >> >> >> >> :
2.900 45,310,4531 45,31% R
6.400 100= = .
) >> >> >> >> >> >> >> :
3.450 53,910,5391 53,91% R
6.400 100= = .