[a Gumnasiou]Mathimatika-Theoria-Askiseis Kefalaio 2

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Transcript of [a Gumnasiou]Mathimatika-Theoria-Askiseis Kefalaio 2

2

.

. .

; : . . 7. , . . . x . 5x . . : 2 + 7 5x, 2 + 7 ( )

. . .

24.

(=) (=) .

1. . 2. . 3. . 4. .

: . . . . . , 8 . . 4x , x , , 4. . 2x , x , .

x . 8 , x , 2 -

25.

E . = . 2 : . x + 2 = 7 . 2x = 6 . 3 > x + 1 . 7x + 3 = 24 . 2, 7. . 6. . 3 , . . 3, 24. : x + 5 = 8 2x = 6 . x 0, 1, 3 x . : 3 + 5 = 8 23 = 6 . x = 3 . , .

: 1 + 5 = 6 , 1, 1 22 = 4 , 2, 2 24 : 4 = 6 , 3, 3 29 7 = 22 , 4, 4 . . x : 6 = 2 : . 3, . 12, . 4, . 6 . 5x = 5 : . 5, . 0, . 1, . 2

-

26.

. 12 : 6 = 2 . 51 = 5 13 6 . . x . , : x = 613 x = 78 . 78 . : 40 x . 4x = 40 . x = 10 , 410 = 40 .

200 . , 20 . x . 20x = 200 . x = 10 , 2010 = 200 .

1. :i. x + 3 = 7 : . 1, . 2, . 3, . 4 ii. 3 : . x + 2 = 7 , . 3x = 9 , . 25 : x = 3 , . x + 1 = 0

2. .

-

27.

3. :. 5x = 0 , 0. . 5 : x = 1 , 1. . 5 x = 1 , 0.

4. 0, 1, 2 :. x + 1 = 2 . 3x = 0 . 6x = 14 2 . 6 : x = 3

5.

45. .

6.

1 3. 2 .

7. 20 . , ;

8. 75.. 5 , .

9. 180 . ; ; 10. 150. . 3 ; ( (S) = () (t))

-

28.

11. : 2 = 5

+ 3 = 4 . 3, 4 7 ;

12. :3 + 2 = 14

13.

+ 3 = + = 5 . 1 2 3 4 , ;

-

29.

1 , ;

2 x + 3 = 7 , , ;

3 ;

1 .

2 () : . 2 + 1 = 7 . x < x + 2 . 3 < 2 + 1 . x < y +

-

3

, ... .

; ; . .

; , , .

+ = + . .. 50 + 34 = 84 34 + 50 = 84 . + + = ( + ) + = + ( + ) . .. 15 + 32 + 9 = (15 + 32 ) + 9 = 47 + 9 = 56

32.

15 + 32 + 9 = 15 + (32 + 9 ) = 15 + 41 = 56+0 = . .. 17 + 0 = 17 ; , (), (), (), . :

= + =

, . :

125 39 = 86 = 125 > = 39

.

; 10 100 1000 ; 0,1 0,01 0,001 .; :

= + + + ... +

- -

33.

, , . 10, 100, 1000 ..., , , ... . 10, 100, 1000 ..., . . , .

, , . , , . .. 5 5 , . .. 57 57 , .

0,1 0,01 0,001 . . ; / , , : - -

34.

= . .. 415 = 60

154 = 60

= ( ) = ( ) . .. 3710 = (37 )10 = 2110 = 210

3710 = 3(710 ) = 370 = 2101 = 1 . 0 = 0 0, .

1. . 2. : = + = 3. . 4. . 5. . 6. , , . .. 3x = 3x , = 7. . .. 35 35

- -

35.

: . 673 + 89 . 92 + 39 . 1325, 6 + 7, 2 . 92 + 39 = 131 . 0, 062 + 6, 32

. 6, 32 + 29, 2 . 9, 72 + 2, 3 . 6, 32 + 29, 2 = 35, 52 . 9, 72 + 2,3 = 12, 02

. 673 + 89 = 762 . 1325, 6 + 7, 2 = 1332,8 . 0, 062 + 6,32 = 6,382 : . 632 + 6, 32 + 0, 632 + 63, 2 . 0, 937 + 65 + 8, 9 + 72, 25 . 632, 0 6,32 . 723 0, 72 .

. 723 + 0, 72 + 723, 9 + 7, 23

0,937 65 8,9 +72, 25 147, 087

0, 632 + 63, 2 702,152

723,9 + 7, 23 1454,85

: . 159.500 + 997.000 + 3.000 . 4.400.000 + 630.000 + 5.600.000 . 997.000 + 3.000 = 1.000.000 : 1.000.000 + 159.500 = 1.159.500 . :

1.159.500 + 997.000 + 3.000 = 159.500 + (997.000 + 3.000 ) = 159.500 + 1.000.000 = 1.159.500

- -

36.

. 4.400.000 + 630.000 + 5.600.000 =

( 4.400.000 + 5.600.000 ) + 630.000 =10.000.000 + 630.000 = 10.630.000 . .

7 32 5 + 43 9 6 7 5 12 5

.

84 470 + 1 0 07 7 03 7

3,2 8 + , 97 12,6 7

.

.

.

73285 + 437967 511252

.

84470 9992567 10077037

.

3, 278 + 9, 397 12, 675

. 97 , 632 + 5, 798103, 43 0

. , . , .

- -

37.

: . 63 - 12 . 6, 9 - 3, 2 . 6, 72 - 2, 39 . 5.232, 5 - 2.351, 6 . . 739, 2 - 67, 9 . . 620 - 5, 39 . 5.232,5 2.351, 6 . 739, 2 67,9 . 620, 00

63 12 51

. 6,9 3, 2

6, 72 2,39 4,33

5,39 614, 61

3, 7

2880,9

671,3

: . x + 9, 2 = 13, 25 . 7, 32 + x = 9, 76 . 218, 3 + x = 927, 5 . x = 9, 76 7,32 = 2, 44 . x = 927,5 218, 3 = 709, 2 . . x + 6, 35 = 9, 63 . x = 13, 25 9, 2 = 4, 05 . x = 9, 63 6, 35 = 3, 28

:

63, 25 28,3 34,95

47, 08 + 26, 20 73, 28

93, 25 7, 05 86, 20

- -

38.

: .

6 9, 2 8 2,6 9 6 ,0 8

.

8 ,6 2 7,7 8 7 4, 4 3 2 7 ,8 6 9 ,6 1 7 2, 3

1 6 3,5 0 8, 2 1 3 ,7 .

.

. 69, 7 28 - 2, 639

. 82, 632 - 7, 78 9

67, 089.

74,843.

163,50 - 28, 72 134,78

271,86 - 99, 63 172, 23

: . 18015, 2 . 697, 20, 3 . 6, 328, 2 . 1, 0562 . . 8267, 3 . 0, 017, 63

6,32 x 8, 2 1264 +5056 51,824

.

1801 x 5, 2 3602 9005 9365, 2

.

697, 2 x 0,3 20916 00000 209,16

.

1, 05 x 62 210 + 630 65,10

.

x

67,3 82 1346

+ 5384 5518, 6

. 0, 017, 63 = 0, 0763 0,1 0,01 0,001 . . . : . 629 - 12 + 52 . 87 + 9( 32 - 8 )

. 63 - 93 + 87 . 62, 3735 , .

- -

39.

. 629 12 + 52 = 558 12 + 10 = 546 + 10 = 556 . 87 + 9(32 8) = 87 + 924 = 56 + 216 = 272 . 63 93 + 87 = 63 27 + 56 = 36 + 56 = 92 . 62,3735 = ( 62,37 )(35) = 14, 2215 = 213,3 , 10, 100, 1000 ... 0,1, 0,01, 0,001 . . 83000x = 83 . 63, 821x = 638, 21 . 100x = 82, 5 . x0, 01 = 72, 25 . 52x = 5, 2 . x0,1 = 65 . 83.0000, 001 = 83 , x = 0, 001 0,001 3 . 83.000 , 83.000,0. 3 83.000,0 83 : . 63,82110 = 638, 21 x = 10 . 1000,825 = 82, 5 x = 0,825 . 72250, 01 = 72, 25 . 520,1 = 5, 2 . 6500,1 = 65

x = 7225 x = 0,1 x = 650

- -

40.

1. :. 63, 28 + 2,932 . 835 + 9, 28 . 1.085,9 283 . 863, 7 99, 632 . 0, 065 + 93,5 . 63,5 6,35

2. :. 8, 20,1 . 5,329 . 6382,5 . 6, 34100 . 80, 01 . 7, 20,1

3.

. .

6 4 ,5 7 + 3 2, ,9 6

.

,3 2 5 + 2 5 , 35 1 1 4 ,9.

.

6 ,3 7 17, 4 5 ,0 9

. 5 3 , 28 9, 7 5 1 4 ,7

x

3 78 6 96 8

.

x +

93 6 5

2

4. :. 1, 5 + 1, 25 + 3, 75 + 2, 5 . 8,3 + 2,8 + 7, 7 + 1, 2 . 30, 75 + 50 + 10, 25 . 4, 6 + 3,5 + 1, 4 + 6,5

5. :. 30 + x = 45 . 10 + x = 12,5 . 180 + x = 360 . x + 15, 3 = 16, 7

6. :. 40 (13 11) = . (12 + 13) 5 = . (60 15) + 7 = . (5 + 7 ) 3 =

- -

41.

7.

= 13, 5 , = 12 , = 7,5 , = 3 . . + . + +

8.

x + y = 2, x + = 5, y + = 3 . :. x + 3 + 2 + y + 9 + 3 . x + y + 3 + 1 + x

. x + 1 + y + 3 + x + 7 + . y + 3 + x + y 2

9. :

10. :. 1, 5 + 1, 5 + 1, 5 . x + x + x + x + x

11. :. 2 + (53) . (3 + 1)2 . ( 45) 2 . (3 + 5 2 6 )10

12. :. 68100 . 930,1 . 520000, 016,81009, 30,1

0, 681009300, 01

52000,1

520000, 001

13. : . 0, 7325 . 7, 32, 5 . 1304,8 . 1, 34,8

7325 = 1825 1348 = 624 . 7, 30, 025 . 13480

14. :. 5(3 2 ) . (5 + 3)2 . (53) + (32 ) . ( 6 + 5 ) 22 + (3 2 )

- -

42.

15. :. 4, 32, 5100 . 6, 31, 5102 . 63.0000,1100, 01 . 200300, 001

16. 17,2 10 .. . , ;

17. 13,7 , 30 , 14,5 , 20 . ;

- -

43.

1 ;

2 10, 100, 1000...;

3 ;

1 . 3210 = 320 3, 20,1 = 320,3210 = 3, 2 0, 032100 = 0, 32 320,10 = 3, 20

2 200 , 0,20 0,10 . 4 3 . ;

- -

4

; 6 8; , , . . 6 : 0, 6, 12, 18, 24, 30, 36, 42, 48, ... 8 : 0, 8, 16, 24, 32, 40, 48, 56, ... 6 8 : 0, 24, 48,... ...

T ; 0, 1, 2, 3, ... :

0 = 0, 1 = , 2 = 2, 3 = 3 , ...

; ; 3, 5 6.

46.

: ... 3 : 0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33,... 5 : 0, 5, 10, 15, 20, 25, 30, 35, 40,... 6 : 0, 6, 12, 18, 24, 30, 36, 42,... ... (3, 5, 6) = 30

; .... ,

, . .

2 , , 3

: 0 = 1 ( 0 )1 =

10; 10 , 1 . : 10 2 = 100105 = 100000

; ; 3 ( x + 7 )

- -

47.

:

( + ) = + + = ( + ) . :

3( x + 7 ) = 3x + 37

; :

( ) = = ( )

1. , . 2. ( ) ( ). 3. (...) . 4. ( > 1) .... = , , > 1 .

5. = 1 , : 1 . 6. , . 8(3 + 5) = 83 + 85 .

- -

48.

: 0, 1, 2, 3, ... 6. . ; . ( 6); . 0 6 1 6 6; () 0, 6, 12, 18, 24, ... . , . , .

. 4. . 4 6 12; . 4 6 12; ; . 4, 8, 12, 16, 20, ... . 12 . ,

. ; . ; , ; . , 0 1. . , . , 0, 0, ... . 1. 1 1 1 = 1 . () , ; . , ;

- -

49.

. ... ( ) . . . (8, 12) = ; 8 12 (8, 12). ; (8, 12) =24 8: 24, 48, 72, ... 12: 24, 48, 72 24: 24, 48, 72, 96, 120, ... ... 5, 6, 10 (5, 6, 10). 5, 6, 10 (5,6,10); 5: 30, 60, 90, ... 6: 30, 60, 90, ... 10: 30, 60, 90, ... 30: 30, 60, 90, ... ... . 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, ... 6, 12,18, 24, 30, 36, 42, 48, 54, 60, 66,... 30, 60, 90, 120 2 2 2 2 2 2, 6 2, 26. 26 2 . 2 6 .

. 135 11; . , 11 135. . , . 121 < 135 < 132

. 2 + 2 + 2 + 2 + 2 + 2; ; ;

- -

50.

. : 2 2 2 2 2 2; ; ; . 6 2 = 12, 6 . 26 = 64, 6 . 5 3; ; . 35; ; ; . 53; ; . 5 3 = 15 . 35 = 3 3 3 3 3, 3, 5 243 . 53 = 5 5 5, 5, 3 125. 2 2 3 2 , 2 3 . , 2 . 3 . . . . . 110 1010. . 1 10. . 1, 1, 1, 1, 110 10 . 1 = 1 1 ... 1 = 1 , 10 = 10 10 ... 10 = 10.000.000.000 10 10

- -

51.

. 1 = 1, , 10 = 100...0, 4105+6104+0103+3102+0102+2100 460302 460302 10.

: 4 105 + 6 104 + 3 102 + 2 ; 4 105 + 6 104 + 3 102 + 2 = 460302

10 3275 408906. 3275 = 3 103+ 2 102 + 7 101 + 5 408906 = 4 105 + 8 103 + 9 102 + 6

: ( 4 + 3) 2 = 4 2 + 3 2

( 4 + 3) 2 = .4 2 + 3 2 = . . . 25 4 8 9 2 + 5 6 = 60 . 3 15 + 5 5 14 + 5 8 = 20 . 25 4 8 9 + 2 + 5 6 = 4 ( 25 2 9 ) + 2 + 5 6 = = 4 ( 25 18) + 2 + 30 = = 4 7 + 2 + 30 = 60 . 3 35 + 5 + 5 8 5 14 = 5 (3 3 + 1 + 8 14 ) == 5 (9 + 1 + 8 14 ) = = 5 (18 14 ) = = 54 = 20

- -

52.

1. :

2. :

(6, 8);

3. (4, 6, 18). 4. . 6 10.. (6, 10)

5. 8, 10 15 . , .

- -

53.

6. 7 7 7 7. .

7. 8.

13 + 13 + 13 + 13 + 13 . : . 3 3 . 4 4 4 4 4 . 23 23 23 23 23 23 . 1 1 1 . . : . 23 . 32 2 . 11 . 122

9.

. 43 . 52

. 34 . 53

10. . . 36 . 125 . 49 . 169 . 110 . 32 . 10.000 . 200 . 400 . 81 . 128 . 144 . 196

11. .. 32 ..... 23 . 10 2 ..... 210 . 53 ..... 35 . 14 ..... 17 . 42 ..... 24 . 7 2 ..... 27

12. .. 22 + 32 ..... ( 2 + 3)2 . 32 ..... 3 + 3 . (7 3)2 .....7 2 32 . 16 32 ..... (16 3)2 . 03 + 43 ..... (0 + 4 )3 . 1 5 ..... 15

13. .. 32 ..... 3 2 . 7 2 ..... 7 2 . 2 2 ..... 2 2 . 72 2 + 69 2 ..... 27 2 + 962 . 52 ..... 5 2 . 13 + 53 + 33 ..... 153

14. : . 532 . 6329 . 50314 . 7 935 028

15.

:

- -

54.

. 3 102 + 4 10 + 2 . 6 104 + 0 103 + 2 102 + 5 10 + 1 7 6 5 . 8 10 + 0 10 + 0 10 + 2 104 + 3 103 + 7 102 + 0 10 + 3

16.

, . . 8 16 = 23 24 = ( 2 2 2 ) ( 2 2 2 2 ) = 27 . 9 81 . 27 27 . 25 125 . 3 . x 4 x 2 x

17. , , B = (53 ) , = ( 24 ) , = (7 2 ) , 2 3 2

. = (32 ) = 32 32 32 = (33) (33) (33) = 333333 = 363

18. :. 9 . + . 8 5 . + + . x x x 6 x x

19. 20. 21.

= 4, , = 3 2 + (3 ) 9 .2

= 3 = 4, , : = 2 + 2 + 2 . : 352 84 = 29568 . 29568, 3;

22. :. 4 (8 + 2 ) . ( 20 6 ) 7 . (3 + 9 ) 12 . 3 (5 + 6 + 7 ) . 5 (12 4 )

23. :. 7 6 + 13 6 . 4 6 + 4 7 + 4 8 . 8 9 + 8 7 . 9 7 + 9 11 + 9 . 12 16 + 12

24. :. 12 12 12 5 . 14 8 5 8 . 16 15 15

- -

55.

25.

:

( 2 + 7 ) (6 + 5)

26. . , . . .

- -

56.

1 ; ; *;

2. 2 + 2 + 2 + 2 + 2 + 2; ; ; . : 2 2 2 2 2 2; ; ;

1 . . 36 . 125 . 48 . 81 . 144 . 169 . 110 . 32 . 128 . 196 . 10.000 . 200 . 400

2 : . 3 42 + 2 ( 4 5 24 + 3) . 2 4 ( 4 2 36 : 6 ) (3 4 11) + 2 325

. 3 23 + 3 ( 62 1 1 8 ) . 3 23 ( 43 : 2 31) 19

3 + = 3, , : = 5 ( 2 + ) + 2 + 3 32

- -

5

T ; ; . , , : : = =

: = . , . :

: =1 1 = :1 = 0 : = 0, 0 = 0

,

; 24. . 24 : 1, 2, 3, 4, 6, 8, 12, 24

58.

; . 1. . 2, 5, 7, 11, 13 , 4, 6, 12 . ; . . ; ; 8 36. . : ... 8 : 1, 2, 4, 8 36 : 1, 2, 3, 4, 6, 9, 12, 18, 36 ... (8,36 ) = 4

(...)

... 1 .

2, 3, 5 9; 2, -

- - - - - -

59.

0, 2, 4, 6 8. 3 3. 5 0 5. 9 9.

. , 3 9, 12, 15 , . , . , 12 3 . , . 4 8. 16, 24, 32 . 8. . 2 4 8 4 + 8 = 12 8 4 = 4 .

; : . 30 . 100 . . .

- - - - - -

60.

.

; , () () , ( ) () : = + < . = 0 , = . 0 , .

; , 10 , , .

; , ( ).

; , , -

- - - - - -

61.

, ,

10, 100, 1000 ; 10 100 1000 ..., ... .

. .

1. . : : = = 2. , . 3. . 4. 3 9 3 9 . 5. 2 0, 2, 4, 6, 8 5 0 5. 6. : = + .

- - - - - -

62.

. . 91 : 7 . 120 : 5 . 136 : 17 . 3588 : 23 . 58117 : 89 . 34200 : 456 . 91 21

7 13

13 x 7 91

0. 136 0

. 120 20 0 . 3588 128 138 0

5 24

24 x 5 120

17 8

17 x 8 136

23 156

156 x 23 468 312 3588

. 58117 471 267 0

89 653

653 x 89 5877 5224 58117

. 34200 2280 0

456 75

456 x 75 2280 3192 34200

: . 4x = 92 . 11x = 165 . x : 3 = 11 . 23x = 115 . 4x = 92 x = 92 : 4

. 27 : x = 9 . 192 : x = 32 . 27 : x = 9

x = 23

. 11x = 165 x = 165 :11 x = 15

9x = 27 x = 27 : 9 x=3

- - - - - -

63.

. x : 3 = 11 x = 311

. 23x = 115 x = 115 : 23

. 192 : x = 32

x = 33

x =5

32x = 192 x = 192 : 32 x=6

: . , 7 14. . 231 . . 5 320 . . : 14 : 7 = 2 . 231: 3 = 77 . . 320 : 5 = 64 5 . 15 23 3790 . 80 , . 2380 = 1840 , 15 3790 1840 = 1950 . : 1950 :15 = 130 . 18. 6 30. ; 6 30 18 = 12 12 : 6 = 2 . 8, 24 56 ... . 8 : 1, 2, 4, 8 24 : 1, 2, 3, 4, 6, 8, 12, 24 56 : 1, 2, 4, 7, 8, 14, 28, 56 8, 24 56 : 1, 2, 4, 8 ... (8, 24, 56) = 8 ... 36. .

- - - - - -

64.

36 , 36, . 1, 2, 3, 4, 6, 9, 12, 18, 36 . 3432, 4581, 864, 156, 62775, 730 . . 2 . 5 . 3 . 9 . 3432, 864, 156, 730 2, 2, 4, 6 0. . 62775, 730 5, 5 0. . 3432, 4581, 864, 156 3, 3. . 4581, 864, 62775 9, 9. . . i. 52....3 ii. 2....61 3. . 6....4.... 2 9. . 52....3, 2....61 3 3 : i. 5223, 5253, 5283 ii. 2061, 2361, 2661, 2961 . 6....4.... 2 9 0, 2, 4, 6, 8 9, : 6840, 6642, 6444, 6246, 6048, 6948 . . 132 . 500 . 140 . 132 2 66 2

132 = 22 311

33 3 11 11 1

. 500 2 250 2 125 5 25 5 5 5 1

500 = 22 53

. 140 2 70 2

140 = 22 57

35 5 7 7 1

- - - - - -

65.

... ... 60 150. ... , . ... . :

60 2 30 2 15 3 5 5 1

100 2 50 2 25 5

60 = 22 35

55 1

100 = 22 52

... (60,100 ) = 22 352 = 300

... (60,100 ) = 22 5 = 20 . . 315 : 25 . 5124 : 15 . 134 : 8 . 134 8 54 16 6 :

16 x 8 128 + 6 134

. 315 25 65 12 15

:

25 x 12 50 + 25 300 + 15 315

. 5124 15 62 341 24 9

:

341 x 15 1705 341 5115 + 9 5124

- - - - - -

66.

. : 5 . . , , 5, 7. . : 5 : 0 1 2 3 4 . : = 57 + < 5

=0 =1 =2 =3 =4 . . 0, 805 : 35 . 8, 25 : 2 . 8, 25 3 22 2, 75 15 0 .

= 57 = 35 = 57 + 1 = 36 = 57 + 2 = 37 = 57 + 3 = 38 = 57 + 4 = 39

. 51, 84 : 9, 6

. 0, 6 : 0,12

. 0,805 35 105 0, 023 0

.

: . i. (64 + 32 ) : 4 . i. (64 + 32 ) : 4 = 96 : 4 = 24

ii. (64 : 4 ) + ( 32 : 4 ) . ; ii. (7, 56 + 9, 72 ) : 1, 2

. N . i. (73, 8 + 44, 4 ) : 6

ii. (64 : 4 ) + (32 : 4 ) = 16 + 8 = 24

(64 + 32 ) : 4 = (64 : 4 ) + (32 : 4 ) = 24 . i. (73,8 + 44, 4 ) : 6 = 118, 2 : 6 = 19, 7

(73,8 + 44, 4 ) : 6 = 73,8 : 6 + 44, 4 : 6 = 12,3 + 7, 4 = 19, 7 ii. (7,56 + 9, 72 ) :1, 2 = 17, 28 :1, 2 = 14, 4 - - - - - -

67.

(7,56 + 9, 72 ) :1, 2 = 7,56 :1, 2 + 9, 72 :1, 2 = 6,3 + 8,1 = 14, 4O 8 10,56. ; 8 10,56 10, 56 : 8 = 1, 32 . . 2,5 12, 1250 5,75. . 1 1 12 : 2, 5 = 4,8 2 1 5, 75 :1, 250 = 4, 6 2 . . . 62, 8 : 13 . 185, 6 : 33, 6 . 564 : 23 i. . ii. . iii. . . 564

23

i. 564 : 23 = 24,5 . ii. 564 : 23 = 24, 52 . iii. 564 : 23 = 24, 521 .

104 24,5217... 120 50 40 170 9

. 62,8

13 4,8307...

i. 62,8 :13 = 4,8 ii. 62,8 :13 = 4,83 iii. 62,8 :13 = 4,830 .

108 40 100 9

- - - - - -

68.

. i. 185, 6 : 33, 6 = 5,5 . ii. 185, 6 : 33, 6 = 5,52 . iii. 185, 6 : 33, 6 = 5, 523 .

1. .. 80 : 5 . 7125 :15 . 215 : 5 . 16235 : 34 . 405 : 45 . 88576 : 346

2. :. 5x = 70 . x : 6 = 13 . 15x = 120 . 56x = 168 . 64 : x = 4 . 1088 : x = 64

3. :. 513 = 65 . 124 = 48 . 716 = 112 . 2731 = 837 . 126 = 72 . 3645 = 1620

4. .. 2004 : 2004 . 0 :1324 . 5398 :1 . 1932 : 0

5. :. 1040 , 130 . . 72 . , 2 6. . 60 17 .

- - - - - -

69.

6. 18000 . 24000 20 . ;

7. 42. 4 70. ;

8. 3 5 765 , 6 14 2010 . .

9. 16, 20, 60 ... .

10. 20. .

11. : 3, 5, 9, 11, 12, 15. :. . . .

12. 33 , 48 , 63 . . , ;

13. 4816, 805, 3600, 28575, 933 .. 2 . 5 . 3 . 9

14. . i 6 _ 53 ii. 9 _1 3 . 8 _ 3 _ 5 9.

15. :. 9 . 18 + 21 . 15 3 , 3.

16. - - - - - -

70.

. 126

. 300

. 256

. 620

17. ... ... 36 124. 18. .. 314 : 5 . 278 : 23 . 12449 : 23 . 43396 : 452

19. . : 8 . , , 8, 5.

20. :. 30, 03 : 7 . 0, 406 : 29 . 18, 72 : 5, 2 . 3,864 : 0,56

21. :. 925 :100 . 31, 4 :1000 . 0, 04 :10 . 658,124 :10000

22. :. 47,16 : x = 5, 24 . 3, 2x = 30, 72 . 1, 2x = 73,8 . 69,16 : x = 7, 6

23. 6,88 1,72; 24. 5 6,7. 12 ; 25. 34 266,56. 36,72.

26. :. 22, 4 :10 + 32, 7 :100 + 1682 :1000 . ( 625 :1000 )10

- - - - - -

71.

1 ; ;

2 ;

3 2, 3, 5 9;

1 ... 18, 48 64.

2 . . i. 12....9 ii. 8....73 9 . 3....4.... 2 3.

3 6 .

4 . 698 : 34 . 94, 68 : 21 . 895, 23 : 52, 7 i. . ii. . iii. .

- - - - - -

6

; ; , . , , .

; :

42 + ( 30 : 2 - 322 ) + (62 : 9 )2 - (15 + 3 ) : 32

: 1. . 2. . 3. . : .42 + (30 : 2 - 322 ) + (62 : 9 )2 - (15 + 3) : 32

( )= 4 2 + (30 : 2 34 ) + (36 : 9 )2 (15 + 3 ) : 3 =2

( )

74.

= 4 2 + (15 12 ) + 42 (15 + 3) : 3 = ( )2

= 42 + 32 + 42 18 : 3 = ( ) = 16 + 9 + 42 18 : 3 = ( , ) = 16 + 9 + 8 6 = ( , ) 25 + 8 6 = 33 6 = 27 ; 3000000, 27000, 80900000, 740300000. ( ) 1 10, 10. :

3000000 = 3106 27000 = 2, 7104 80900000 = 8, 09107 740300000 = 7, 4031081. . 2. . 3. : . . , . , . . 4. : 10 1 10.

- .

75.

: . 184 - 315 + 913 . 78 : 3 + 342 - 36 : 3 . 12, 35 - 1, 52 : 4 + 31,19 . 184 315 + 913 = ( ) = 72 45 + 117 = ( ) = 27 + 117 = 144 . 78 : 3 + 342 36 : 3 = ( ) = 26 + 68 12 = ( ) = 94 12 = 82 . 12, 35 1, 52 : 4 + 31,19 = 61, 5 0, 38 + 3, 57 = 61,12 + 3, 57 = 64, 69 : 2 2 i. = 16 : 2 + 3 4 - 6 : 9 ii. = 39, 36 : 3, 2 + (5, 42 - 3, 22 ) : 4 i. = 16 : 2 + 32 4 62 : 9 = ( ) = 16 : 2 + 94 36 : 9 = ( , ) = 8 + 36 4 = ( ) = 44 4 = 402 2 ii. = 39,36 : 3, 2 + (5, 4 3, 2 ) : 4 =

( )

= 39,36 : 3, 2 + ( 29,16 10, 24 ) : 4 =( )

- .

76.= 39, 36 : 3, 2 + 18, 92 : 4 = ( ) = 12,3 + 4, 73 = 17, 03

: . 8 2 + (8 ) = 2,12

. (6 ) - 6 2 = 0, 32

. 8 2 + (8 ) = 82,12 + (82,1) = 82,12 + 16,82 = 84, 41 + 282, 24 =2 2

= 35, 28 + 282, 24 = 317,52. (6 ) 6 2 = (60,3) 60,32 = 1,82 60,32 = 3, 24 60, 09 = 3, 24 0,54 = 2, 72 2

= 17, = 9, = 2 = 3 : . ( : ) - ( - ) 2 2

. ( - ) : - ( ) : + ( - ) : 2 2

. ( : ) ( ) = (9 : 3) (92 17 ) 3 = 32 (18 17 ) 3 = 9 12 3 =2 2 2 2 2

= 9 13 = 9 3 = 6

. ( ) : ( ) : + ( ) : = (17 2 ) : 3 ( 23) : 9 + (17 9 ) : 2 =2 2 2 2

= 15 : 3 62 : 9 + 82 : 2 = 15 : 3 36 : 9 + 64 : 2 = 5 4 + 32 = 1 + 32 = 33 = 15 : 5 + 32 - 7 , = (7 - 5 ) + 35 - 32 , = (12 : 3 ) - 52 = 52 - ( 210 + 3 )3 2

(

)

: = ( + ) : 2 + ( : ) ( 2 )

, , , = 15 : 5 + (32 7 ) = +3 + (9 7 ) = 3 + 2 = 5 = (7 5 ) + 35 32 = 23 + 35 32 = 8 + 15 9 = 23 9 = 143

= (12 : 3) 52 = 42 10 = 16 10 = 6 2

= 52 ( 210 + 3) = 25 ( 20 + 3) = 25 23 = 2 , , , : = ( + ) : 2 + ( : ) ( 2 ) = (14 + 6 ) : 22 + (14 : 2 )6 (52 14 )2 =

20 : 22 + 76 ( 25 14 )2 = 20 : 22 + 76 112 = 20 : 4 + 76 112 =5 + 42 22 = 47 22 = 25 - .

77.0, 4 = 0,12 4, 8 : t = 3, 2 = ( y : - x8 ) - (t : x )22

y - 2,1 = 0, 3, x + 3, 7 = 4, 2, . = ( x 2 + y ) : 0, 5 + (t 2 + x ) : 4 - 2

x, y, , t. :x + 3, 7 = 4, 2 x = 4, 2 3, 7 x = 0,5

y 2,1 = 0,3 y = 0,3 + 2,1 y = 2, 4

0, 4 = 0,12 = 0,12 : 0, 4 = 0,3

4,8 : t = 3, 2 3, 2t = 4,8 t = 4,8 : 3, 2 t = 1,5

:

(0,5

= ( x 2 + y ) : 0,5 + ( t 2 + x ) : 4 - 22

B = ( y : x8 ) ( t : x )2 =2

+ 2, 4 ) : 0,5 + (1,52 + 0,5 ) : 4 0,32 =

= ( 2, 4 : 0,3 0,58 ) (1,5 : 0,5 ) 2 =2

(0, 25 + 2, 4 ) : 0,5 + ( 2, 25 + 0,5) : 4 0,32 =2, 65 : 0,5 + 2, 75 : 4 0, 09 = 5,3 + 0, 6875 0, 09 = 5,9875 0, 09 = 5,8975

= (8 4 ) 32 =2

= 42 6 = 16 6 = 10

- = 3, 2 + = 1, 4 . = 23 + 5 2 - (4 2 - 10 ) + (15 : 3 ) - 45

= 6 + 25 6 + 5 20 = 6 6 + 25 20 + 5 = 6 ( ) + ( 25 20 ) + 5 = = 6 ( ) + 5 + 5 = 6 ( ) + 5 ( + ) = 63, 2 + 51, 4 = 19, 2 + 7 = 26, 2 : . 45000000 . 9830000 . 27600000000 . 827300000 . 524000 . 3000000 . 45000000 = 4,5107 . 35900000 = 3,59107 . 3000000 = 3106 . 9830000 = 9,83106 . 827300000 = 8, 273108 . 13200000 = 1,32107

= 23 + 52 - ( 42 -10 ) + (15 : 3) - 45 = 23 + 52 (16 10 ) + 5 45 =

. 35900000 . 13200000

. 27600000000 = 2, 761010 . 524000 = 5, 24105

- .

78.

: . 2, 03106 . 8, 64107 . 4, 721109 . 5,1108

. 2, 03106 = 2030000 . 4, 721109 = 4721000000

. 8, 64107 = 86400000 . 5,1108 = 510000000

. . 3, 4106 + 5, 2106 . 6104 + 1, 4105 . 300105 . 520106

. 3, 4106 + 5, 2106 = 106 (3, 4 + 5, 2 ) = 8, 6106 . 6104 + 1, 4105 = 60000 + 140000 = 200000 = 2105 . 300105 = 300100000 = 30000000 = 3107 . 520106 = 5201000000 = 520000000 = 5, 2108

1. : = 127 + 98 315 = 35 : 5 + 125 90 :15 = 189 173 + 224 : 56

2. :. (14,32 5, 7 ) : ( 0, 75 + 0, 05) . (9,35 27, 4 ) + ( 0,56 : 8 + 1, 03) . 927 (1, 2 : 8 ) (12,88 : 5, 6 ) + 3(7, 22 5, 04 ) . ( 73, 2 : 3)4 (19, 08 :1,8 )3

3. : = 18 : 32 + 43 + 3( 4 + 2 2 ) + 25 :16 27 : 32

- .

79.

= ( 29 43) + (6 + 4 2 ) 3 (7 23)2

20

= (32 23 )

2004

+ ( 43 + 62 3) 63 4 (52 64 ) + 11

4. :. 0,8 + ( 6, 2 2 4,3) + 5, 23 5, 62 : 2 . (3 2,8 ) + 53 1, 22 52

. 4,3 + (6,92 4,3) + 7,12 2,1

(

)

. 5 4, 2 + 6,142 + 82 5, 223 + 6,12

5. :. 9x 2 + (9x ) x = 0, 72

. (15t ) 15t 2 t = 1, 52

. 2y 2 + 5y + 4 y = 0, 02

6.

= 5, = 2, = 4 = 9 :2

= ( ) + ( ) : ( 2 )

= ( 2 2 ) : + ( ) ( )2

3

7.

x = 2,1, y = 4, = 3, 2 t = 1, 6 .2 5

A = ( y x ) + ( : t ) ( ty ) :

B = ( x + ) y t 2 y 2 + y32

8.

x + 26 = 31, 17y = 51, 15 = 11 t : 2 = 3 .A = ( t : y ) + 2x 2 (8x ) : t 2 : y6

B = ( 2 ) y3 (3t : 2 ) + ( x + 2 )2

2

= 7 + t 2 : 9 (15y + ) + x 2

9.

+ 12,5 = 13,9, 22, 4 = 19,3, 14,8 = 37 1, 2 : = 6 : = 3 2 (5 4 ) + (8 : 2 )42

= ( ) + 8 4,8 : + 72

= ( + ) (5 4 ) + 95 3

10. :

- .

80.

. 821000000 . 14100000

. 5320000 . 6020000

. 7140000000 . 28000000

11. :. 3, 2106 . 2,5381011 . 1, 04108 . 8, 72105 . 5,32108 . 6104

12. :7, 4106 , 2103 , 2, 4106 , 3109 , 6, 2107

- .

81.

1 ;

2 ; ;

= 7 + 532 = 1232 = 129 = 108

1 :

= (62 29 ) + (108 :12 ) 43 22 2

= (34 43 )4 + 52 22 (90 : 32 )

2 : . 31200000 . 1110000000 . 456000 . 87130000

- .

2

- 7: - - - - - - - 8: - - - - 9: - - - 10: - - -

7

.

.

. ; ; () . ( ) = 1 . , , , ...... . 1, 2, 3, 4, ... , , , ... . o . .

; ; ; ; . .. (3,5 ) , ( 2,8 ) Ox, Oy, . (, ) .

86.

. (, ) . (0, 0 ) .

; ( ). . .

.

; ; ; ; (m). : ( dam ) :1dam = 10m ( hm ) :1hm = 100m ( km ) :1km = 1000m : ( dm ) :1dm = 0,1m (cm ) :1cm = 0, 01m ( mm ) :1mm = 0, 001m : (yrd), (ft), (in), . :

- - - - - - .

87.

1yrd = 3ft = 36in1ft = 12in

1yrd = 0,9144m

1ft = 0,3048m 1in = 0, 0254m1 = 1609m 1 = 1852m.

; . , , . .

; ; . .

1. . 2. Ox, y O (0, 0 ) . 3. (1m). 4. .

- - - - - - .

88.

: . 3 . 5 . 2,5 . 3,8

A (3, 0 )

B (5, 0 )

( 2,5, 0 )

(3,8, 0 )

2, 3, 8. : ( 2,3) , (3, 2 ) , ( 2,8 ) , (8, 2 ) , (3,8) , (8,3) : ( 2, 3 ) , (1, 0 )

( 3, 5 ) , ( 2.5,1.5 )

3. 3 xx

- - - - - - .

89.

yy 3.

, 2,6 . , .

12 9. . 6,5Km m . 12500m Km. . 6,5m = (6,51000 ) m = 6500m . 12500m = (12500 :1000 ) Km = 12,5Km dm cm : . 960mm . 6,4m . 0,5Km . 0,9m . 960mm = 960 :100 = 9, 6dm 960mm = 960 :10 = 96cm . 6, 4m = 6, 410 = 64dm

6, 4m = 6, 4100 = 640cm. 0,9m = 0,910 = 9dm

. 0,5Km = 0,51000 = 500m = 50010 = 5000dm

0,5Km = 500m = 500100 = 50000cm m : . 3dm 5cm . 2m, 8cm 9mm . 3dm 5cm =3 5 m = 0,35m 10 100 8 9 = 2, 089m 100 1000

0,9m = 0,9100 = 90cm

. 12m 3dm 4mm

. 2m 8cm 9mm = 2m

- - - - - - .

90.

. 12m 3dm 4mm = 12m

3 4 m = 12,304m 10 1000

= 305m, = 100m, = 500m, = 180m. m Km. : = + + + = 305 + 100 + 500 + 180 = 1085m = 1, 085Km . 7,5dm = 7,5100 = 750mm

10cm = 1010 = 100mm : AB + B + + = 750 + 400 + 100 + 200 = 1450mm = 5,5cm, = 3in = 0,5ft. : = + + = 5, 5cm + 32,54cm + 0,530, 48cm= 5, 5 + 7, 62 + 15, 24 = 28, 36cm

. . 120dm . 9m . 15000mm . 0,01m : 120dm =

120 = 12m 10 15000 15000mm = = 15m 1000 0, 01Km = 0, 011000 = 10cm

: 15000mm > 120dm > 0, 01Km > 9m 2,34 2,93 yrd. ;

- - - - - - .

91.

: .

2,93 = 3, 20 . 0,9144

1. () 50 (). (5)

2. , , , 1, 3, 5, 12 . , , .

3. 6 4. 1,25 2,38. 5. dm cm :. 360mm . 2,5m . 0,4m . 0,1m

6. N :. 165cm m . 2550mm cm . 350mm m

7. N cm :. 2dm 5cm . 2m 3cm 5mm . 38m 3dm 6mm

8. = 105,8m, = 30m, =15,8m = 20m. m, Km, dm, cm, mm.

- - - - - - .

92.

9.

O AB = 3, 5m , B = 45dm , = 260cm ,A = 56dm . m, dm, cm.

10. N 2850mm 2,85m. 11. .. 1,2m . 102dm . 10300mm . 1035cm . 0,015Km

12. N :A

mm

cm

2,5d

A

1,5dm

B

m

45

14

20

0m

B

m

2,5dm

5dm

13. :. 3,28m . 0,09m . 0,014m . 15,308mm

14.

40yrd 200. .

15. . 130yrd, 100yrd 6in 80yrd 3ft. 8yrd .

- - - - - - .

93.

1 , , .

2 ; ;

3 . (0, 0 ) ;

1 100m, 150dm. 4 . ;

2 .

- - - - - - .

8

; ; (1), (1) (2) . (1). .

. (m2) . (m2) 1 m. :2 2 - (dm ) :1dm =

.

1 m2 100 1 m2 10000 1 m2 1000000

- (cm 2 ) :1cm 2 =

- ( mm 2 ) :1mm 2 = :

- (dam 2 ) :1dam 2 = 100m 22 2 - ( hm ) :1hm = 10000m

96.

2 2 2 - ( Km ) = 1Km = 1000000m

: 1 = 1000m2.

; :

.

; ; , . .

1. 1m2. 2. To m2 :

- - - .

97.

3. E = , . 4. = 2 .

m2 :

= 1, 5dm 2

= 12255mm 2

= 0, 01dm 2

= 135000cm 2

: = 1,5 :100 = 0, 015m 2

= 12255 :1000000 = 0, 012255m 2 = 0, 01:100 = 0, 0001m 2 = 135000 :10000 = 13,5m 2 dm2 :

- - - .

98.

= 3, 5m 2

= 45500cm 2

= 0,12Km 2

= 1550000mm 2

: = 3,5m 2 = 3,51000000 = 3500000m 2 = 3500000100 = 350000000dm 2

= 45500 :100 = 455dm 2 = 0,12Km 2 = 0,121000000 = 120000m 2 = 12000000dm 2 = 1550000mm 2 = 1550000 :10000 = 155dm 2 . . 40000m2 . 156500m2 . 1240m2 = 40000 :1000 = 40 .

= 156500 :1000 = 156,5 .

= 1240 :1000 = 1, 24 . .

8m 2dm. 2dm = 2 :10 = 0, 2m . E = = 80, 2 = 1, 6m 2 . 400m2.

- - - .

99.

E = 2

= 400cm2 2 = 400cm2 = 20cm . 27m2 30cm. . : E = 2 = (30cm ) = 900cm 2 = 0, 09m 22

: 27 : 0, 09 = 300 . 0,25m 15cm. 15cm = 0,15m , :20, 25 + 20,15 = 0, 5 + 70,3 = 0,8m

: E = = 0, 250,15 = 0,37m 2 . 35dm 240cm. 1m2 5 . 35dm = 3,5m 240cm = 2, 4m E = = 3,524 = 8, 4m 2 . 8, 45 = 42 14,4m 300 60000. . 600 : 300 = 200m2 . x : 14, 4x = 200 x = 200 :14, 4 13,8m . .

- - - .

100.

: = 4221 = 882m 2 . = 1212 = 144m 2 . = 882 144 = 738m 2 .

1. 18,5dm2. 3,4m2 . 2,5dm2

m 2 , cm 2 , mm 2 .

2. cm2 :. 15625mm2 . 0,6m2 . 0,02dm2

3. m2 :. 2Km2 . 570dm2 . 85000cm2 . 2800000mm2

4. .

5. 6dm 16dm. 6. H 45m2 30cm. .

- - - .

101.

7. 3600m2. A 7200cm . 7,2m 1,5m. .

8. 60m 1000 12000. .

9. 96m. .

10. 30m 12m. ;

- - - .

102.

1 ;

2 m2; ;

1 180m 10m. .

2 .

- - - .

9

; ; (1) (1) (2) . (1). .

; m3 ; (m3), 1m. : (dm3 ) :1dm3 = 0, 001m3 (cm 3 ) :1cm 3 = 0, 000001m 3 ( mm 3 ) :1mm 3 = 0, 00000001m 3

( ) :1

= 1dm3 ( m )1m = 1cm3 .

;

104.

:

; ; , .

V = , , .

. : V = 3

1. . 2. 1m3. 3. (1

)

.

1 = 1000cm3 = 1000ml = 1dm3 4. m 3 , :

- -

105.

5. V = 3 6. , , V =

dm 3 : . 3m 3 . 25cm 3 . 5,256m3

. 3m3 = 31000 = 3000dm3 . 25cm3 = 25 :1000 = 0, 025dm3 . 5, 256m3 = 5, 2561000 = 5256dm 3 cm 3 : . 0, 08dm 3 . 30mm3 . 0, 645m 3

- -

106.

. 0, 08dm3 = 0, 081000 = 80cm3 . 30mm3 = 30 :1000 = 0, 03cm3 . 0, 645m3 = 0, 6451000000 = 645000cm3 : . 25dm3 . 25dm3 = 25 . 6m3 = 61000 = 6000 . 525m = 525 :1000 = 0, 525 mm 3 : . 0,12m 3 . 5, 75dm 3 . 3, 465cm 33 . 6m

. 525ml

. 0,12m3 = 0,125109 = 125000000mm3 . 5, 75dm3 = 5, 751000000 = 5750000mm3 . 3, 46cm3 = 3, 4651000 = 3465mm3 m 3 : . 5200000cm 3 .5200000 = 5, 2m 3 1000000

. 55850124mm 355850124 = 0,55850124m 3 1000000000

.

2,5m, 15dm, 1150mm. 15dm =15 1150 = 1,5m 1150mm = = 1,15m . 10 1000

V = = 2, 51,51,15 = 4,312m 3 3,2m : . m 3 . dm 3 . cm 3

- -

107.

V = 2 = 3, 23 = 32,8m3 = 32,81000 = 32800dm3 = 328001000 = 32800000cm3 0, 216m 3 . .

V = 3

3 3 3 = 0, 216 = 0, 6 = 0, 6m V = 0, 216m 3

60cm, 15cm, 10cm, 3 13, 5m .

V = 601510 = 9000cm3 9000 :1000000m3 = 0, 009m3 13,5: 0,009 = 1500 . 4, 5m, 20dm, 150cm . 20,15 . 4,5m = 4,510 = 45dm 150cm = 150 :10 = 15dm V = 451520 = 13500dm3 . 1 20,15 , 60 : 6020,15 = 1209 : 13500 :1209 = 11 .

- -

108.

1. . 0,3m3

dm3 :. 10500cm3 . 7000000mm3

2. cm3 :. 0,32dm3 . 128mm3 . 0,18m3

3. mm3 :. 2m3 . 0,12dm3 . 25cm3

4. m3 :. 35450000mm3. 5400000cm3

5. :. 10m3 . 530m3

6. :

7. 81cm2 . . 8. 1331cm3 . .

- -

109.

9. 45m cm3 10.

.

N , 3m, 25dm, 200cm.

11. 30dm 6m. 1980m3 .

12. 4m, 30dm 150cm.. . 1800 ;

13. cm2 2160

.

14. 10ft. .

- -

110.

1 , ;

2 ;

3 ;

1 5,5mm. 30m 40m ;

2 8 3cm. .

- -