Mathimatika A

512

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= = .