WISKUNDIGE GELETTERDHEID
GRAAD 11
NOG OEFENINGE
  
Koers, verhouding en eweredigheid : antwoorde.
  
MATHEMATICAL LITERACY
GRADE 11
MORE EXERCISES
  
Rate, ratio and proportion : answers.
  
 Antwoorde 1 / Answers 1 
  
    
        Hier is die reëls vir deelbaarheid 'n wins.
  
        Die reëls : 'n Getal is deelbaar deur
          2  as die getal op 'n 0, 2, 4, 6 of 8 eindig.
          3  : tel die sysfers waaruit die getal bestaan op.
                As die som uit meer as 1 syfer bestaan, tel
                die syfers weer op, totdat 'n een-syfer getal
                gekry word. Hierdie 1 syfer som word die
                digitale wortel van die oorspronklike getal,
                p, genoem. As die digital wortel 'n
                veelvoud van 3 is, is die oorspronklike
                getal, p, deelbaar deur 3.
          4  as die laaste twee syfers van die getal
              deelbaar is deur 4.
          5  as die getal op 'n 0 of 5 eindig.
          6  as die getal deur beide 2 en 3 deelbaar is.
          7  Daar is geen maklike reël nie.
          8  as die laaste drie syfers waaruit die getal
              bestaan, deelbaar is deur 8.
          9  as die digitale wortel gelyk is aan 9.
        10  as die getal op 'n 0 eindig.
        11 : Tel die syfers in die onewe posisies op.
                Doen dit ook met die syfers in die
                ewe posisies. As hierdie twee somme
                gelyk is, as die getal deelbaar deur 11,
                bv 29189160 is deelbaar deur 11 omdat
                2+1+9+6=18 (som van syfers in die onewe
                posisies) en 9+8+1+0=18 (som van
                syfers in die ewe posisies) en die
                twee somme is gelyk.
        12  as die getal deur beide 3 en 4 deelbaar is.
  
  
       Sommige sakrekenaars, bv. die
       Casio fx-82 ZA PLUS kan die priem faktore van
       'n getal bepaal. Dit is 'n groot bate. Bepaal
       die faktore van die teller en die noemer
       en kanselleer die gemene faktore.
  
  
    
        Here the rules for divisibility are a bonus.
  
        The rules : A number is divisible by
          2  if the number ends on a 0, 2, 4, 6 or 8.
          3  : add the digits that constituate the number.
                If the sum consists of more than 1 digit, add
                the digits again until a one digit number is
                obtained. This 1 digit number is called the
                digital root of the original number, p.
                If the digital root is a multiple of 3, then
                the original number, p, is divisible by 3.
                
          4  if the last two digits in the number is divisible
              by 4.
          5  if the last digit in the number is a 0 or a 5.
          6  if the number is divisible by both 2 and 3.
          7  There is no eady rule.
          8  if the last three digits in the number is
              divisible by 8.
          9  if the digital root is equal to 9.
        10  if the last digit is a 0.
        11 : Add the digits in the odd positions.
                Also add the digits in the even posiions.
                If these two sums are eual, the number is
                divisble by 11
                e.g. 29189160 is divisible by 11 because
                2+1+9+6=18 (sum of the digits in the odd
                positions) and 9+8+1+0=18 (sum of the
                digits in the even positions) and the
                two sums are equal.
        12  if the number is divisible by both 3 and 4.
  
  
       Some pocket calculators, like the
       Casio fx-82 ZA PLUS can calculate the prime
       factors of a number. This is also a huge bonus.
       Get the facors of the numerator and the
       denominator and cancel the common factors.
  
  
    
                 38           2 × 19            2
        1.1  ───   =   ─────   =   ───
                 57           3 × 19            3
                                                                               [ 1.1 ]
  
    
                 52           4 × 13            4
        1.3  ───   =   ─────   =   ───
                 91           7 × 13            7
                                                                               [ 1.3 ]
  
    
                 247            13 × 19           13
        1.5  ────   =   ──────   =   ───
                 437            19 × 23           23
                                                                               [ 1.5 ]
  
  
      
                 42            2 × 3 × 7            2
        1.2  ───   =   ───────   =   ───
                105           3 × 5 × 7            5
                                                                                 [ 1.2 ]
  
    
                136           8 × 17             8
        1.4  ───   =   ──────   =   ───
                221          13 × 17           13
                                                                                 [ 1.4 ]
  
    
                316           4 × 79             4
        1.6  ───   =   ──────   =   ───
                553           7 × 79             7
                                                                               [ 1.6 ]
  
  
  
 Antwoorde 2 / Answers 2 
  
 Antwoorde 2.1 / Answers 2.1 
  
  
           7 albasters / marbles  Piet / Peter kry / gets  3   en / and  Jan / John  kry / gets  4
  
                            3         35                                             4         35
           Piet kry   ──  ×  ───    albasters  en Jan kry   ──  ×  ───    albasters
                            7          1                                              7          1
  
  
           Piet kry   15  albasters  en  Jan kry 20 albasters.
  
  
  
                                                                                                                  3                                                                        4
          Peter gets 3 marbles if there are (3 + 4) marbles, i.e. he gets ──     and John gets 4 marbles, i.e. he gets  ──
                                                                                                                  7                                                                        7
  
  
                                 3          35                                                 4         35
           Peter gets   ──  ×  ───    marbles  and John gets   ──  ×  ───    marbles
                                 7           1                                                  7          1
  
  
           Peter gets   15  marbles  and  John gets 20 marbles.
  
  
  
  
  
                                                                                                                                 OF / OR    
  
  
           7 (3 + 4) albasters word verdeel in die verhouding 3 : 4   /   7 (3 + 4) marbles are divided in the ratio 3 : 4
  
           Verdeel 35 albasters in dieselfde verhouding.                   /   Divide 35 marbles in the same ratio.
  
           Gestel Piet kry x albasters, dan sal Jan 35 ─ x  kry.          / Suppose Peter gets x marbles, then John will get  35 ─ x
  
                                                                                                                   Pieter / Peter          3               x                             
           Verdeel in dieselfde verhouding / Divide in the same ratio  :   ──────────   =  ──   =  ─────
                                                                                                                     Jan / John             4          35 ─ x
  
           Los op deur met KGV te vermenigvuldig /  Solve by multiplying by the LCM    3 × (35 ─ x)  =  4 × x
  
                                                                                                                                                105 ─ 3x  =  4x
  
                                                                                                                                                         105  =  7x
  
                                                                                                                                                           15  =  x
  
          Piet kry / Peter gets  15  albasters / marbles   en / and   Jan kry  /  John gets  35 ─ 15  =  20                                 [ 2.1 ]
  
  
 Antwoorde / Answers 2.2 
  
  
          R 13 (4 + 9) :   A  kry / gets   R 4   en / and  B  kry / gets   R 9
  
                                                             4                                               9
          R 1               :   A kry / gets  R ──   en / and  B  kry  gets  R ──
                                                            13                                            13
  
  
                                                            4        1 378                                               9         1 378
          R 1 378       :  A kry / gets  R ── X ─────   en / and  B  kry  gets  R ─── X ─────
                                                           13           1                                                 13            1
  
  
                                              d.i. / i.e  R 424                                       d.i. / i.e  R 954
  
                                  A kry / gets  R 424     en / and  B  kry  gets  R 954
  
  
  
  
                                                                                                                                 OF / OR    
  
  
                                                                                       4                                                         9
           A kry R4 as daar R(4 + 9) is, d.w.s hy kry   R ──      en B kry R9, d.w.s hy kry   R ──
                                                                                      13                                                      13
  
  
                           3         1 378                             4         1 378
           A kry  R ──  ×  ─────      en B kry  R ──  ×  ─────    
                         13             1                              13            1
  
  
                                  A kry  R 424     en   B  kry  R 954
  
  
                                                                                          4                                                              4
          A gets R 4 if there are R(4 + 9), i.e. he gets R ───     and B gets R 4, i.e. he gets  R ───
                                                                                         13                                                            13
  
  
                               3         1 378                                   4            1 378
           A gets  R ───  ×  ─────      and B gets  R ───  ×  ─────   
                             13             1                                      13              1
  
  
                                  A kry  R 424     en   B  kry  R 954                                                                         [ 2.2 ]
  
  
  
                                                                                                                                 OF / OR    
  
  
           R13 (4 + 9) word verdeel in die verhouding 4 : 9   /   R13 (4 + 9) is divided in the ratio 4 : 9
  
           Verdeel R1 378 in dieselfde verhouding.                   /    Divide R1 378 in the same ratio.
  
           Gestel A kry Rx, dan sal B  R(1 378 ─ x)  kry.          / Suppose A gets Rx, then B will get  R(1 378 ─ x)
  
                                                                                                                     A            R 4                   R x                             
           Verdeel in dieselfde verhouding / Divide in the same ratio   :    ───   =  ────   =  ─────────
                                                                                                                     B            R 9           R(1 378 ─ x)
  
           Los op deur met KGV te vermenigvuldig /  Solve by multiplying by the LCM    4 × (1 378 ─ x)  =  9 × x
  
                                                                                                                                                   5 512 ─ 4x  =  9x
  
                                                                                                                                                            5 512  =  13x
  
                                                                                                                                                               424  =  x
  
          A kry /nbsp;gets  R 424      en / and   B kry  /  John gets  R(1 378 ─ 424)  =  R 954                              [ 2.2 ]

  
  
 Antwoorde / Answers 2.3 
  
  
          R 14 (7 + 5 + 2) :   A  kry / gets   R 7,   B  kry / gets   R 5   en / and  C  kry / gets   R 2
  
                                                               7                                    5                                                2
          R 1               :   A kry / gets  R ─── ,   B kry / gets  R ───   en / and  C  kry  gets  R ───
                                                             14                                  14                                              14
  
  
                                                               7          18 214                                 5       18 214                                                2       18 214
          R 18 214       :  A kry / gets  R ─── X ───── ,    B kry / gets  R ─── X ─────    en / and  C  kry  gets  R ─── X ─────
                                                              14            1                                      14            1                                                   14           1
  
  
                                              d.i. / i.e  R 9 107                                       d.i. / i.e  R 6 505                                        d.i. / i.e  R 2 602
  
  
                                   A kry / gets  R 9 107 ,    B kry / gets  R 6 505     en / and  C  kry  gets  R 2 602
  
  
  
  
                                                                                                                                 OF / OR    
  
  
                                                                                              7                                                        5                                                          2
           A kry R7 as daar R(7 + 5 + 2) is, d.w.s hy kry   R ─── ,     B kry R5, d.w.s hy kry   R ───     en C kry R2, d.w.s hy kry   R ───
                                                                                             14                                                     14                                                        14
  
  
                           7         18 214                          5         18 214                              2         18 214
           A kry  R ───  ×  ─────       B kry  R ───  ×  ─────       en C kry  R ───  ×  ─────    
                           14            1                             14             1                                  14              1
  
  
                        A kry  R 9 107                          B kry  R 6 505                               en   C  kry  R 2 602
  
  
                                                                                                7                                                         5                                                                2
           A gets R7 if there is R(7 + 5 + 2) , i.e. he gets   R ─── ,     B gets R5, i.e. he gets   R ───     and C gets R2, i.e. he gets   R ───
                                                                                                14                                                      14                                                             14
  
  
                               7         18 214                            5         18 214                                  2         18 214
           A gets  R ───  ×  ─────       B gets  R ───  ×  ─────       and C gets  R ───  ×  ─────    
                             14            1                                 14             1                                      14              1
  
  
                        A gets  R 9 107                          B gets  R 6 505                               and   C  gets  R 2 602                                                [ 2.3 ]
  
  
 Antwoorde / Answers 2.4 
  
  
          R 14 (3 + 11) :   C  kry / gets   R 3   en / and  D  kry / gets   R 11
  
                                                              3                                             11
          R 1               :   C kry / gets  R ──   en / and  D  kry  gets  R ──
                                                            14                                             14
  
  
                                                               3          323,64                                              11         323,64
          R 323,64       :  C kry / gets  R ─── X ──────   en / and  D  kry  gets  R ─── X ──────
                                                              14            1                                                    14             1
  
  
                                              d.i. / i.e  R 69,35                                        d.i. / i.e  R 254,29
  
                                  C kry / gets  R 69,35     en / and  D  kry  gets  R 254,29
  
  
  
  
                                                                                                                                 OF / OR    
  
  
                                                                                          3                                                             11
           C kry R3 as daar R(3 + 11) is, d.w.s hy kry   R ───      en D kry R11, d.w.s hy kry   R ───
                                                                                         14                                                            14
  
  
                           3        323,64                            4        323,64
           C kry  R ──  ×  ─────      en D kry  R ──  ×  ─────    
                         14             1                               14            1
  
  
                                  C kry  R 69,35     en   D  kry  R 254,29
  
  
                                                                                             3                                                               11
          C gets R 3 if there are R(3 + 11), i.e. he gets R ───     and D gets R 11, i.e. he gets  R ───
                                                                                            14                                                              14
  
  
                               3          323,64                                11        323,64
           C gets  R ───  ×  ─────      and D gets  R ───  ×  ─────   
                             14             1                                      14              1
  
  
                                  C gets  R 69,35     en   D  kry  R 254,29                                                                       [ 2.4 ]
  
  
  
                                                                                                                                 OF / OR    
  
  
           R14 (3 + 11) word verdeel in die verhouding 3 : 11   /   R14 (3 + 11) is divided in the ratio 3 : 11
  
           Verdeel R323,64 in dieselfde verhouding.                   /    Divide R323,64 in the same ratio.
  
           Gestel C kry Rx, dan sal D  R(323,64 ─ x)  kry.          / Suppose C gets Rx, then D will get  R(323,64 ─ x)
  
                                                                                                                     C            R 3                   R x                             
           Verdeel in dieselfde verhouding / Divide in the same ratio   :    ───   =  ────   =  ─────────
                                                                                                                     D            R 11           R(323,64 ─ x)
  
           Los op deur met KGV te vermenigvuldig /  Solve by multiplying by the LCM    3 × (323,64 ─ x)  =  11 × x
  
                                                                                                                                                   970,92 ─ 3x  =  11x
  
                                                                                                                                                            970,92  =  14x
  
                                                                                                                                                               69,35  =  x
  
          C kry / gets  R 69,35      en / and   D kry  /  gets  R(323,64 ─ 69,35)  =  R 254,29                                 [ 2.4 ]
  
  
 Antwoorde / Answers 2.5 
  
  
           12 (5 + 7) osse / oxen  :   P  kry / gets   5   en / and  Q  kry / gets   7
  
                                                                              5                                               7
           1 os / ox                        :   P kry / gets   ───   en / and  Q  kry  gets   ───
                                                                             12                                            12
  
  
                                                                              5          96                                              7         96
           96 osse / oxen              :  P kry / gets   ─── X ───   en / and  Q  kry  gets  R ─── X ───
                                                                            12           1                                              12         1
  
  
                                              d.i. / i.e   40                                        d.i. / i.e   56
  
                                  P kry / gets  40 osse / oxen   en / and  Q  kry  gets  56  osse / oxen
  
  
  
  
                                                                                                                                 OF / OR    
  
  
                                                                                         5                                                               7
           P kry 5 osse as daar (5 + 7) is, d.w.s hy kry    ───      en Q kry 7 osse, d.w.s hy kry    ───
                                                                                        12                                                             12
  
  
                          5          96                           7           96
           P kry   ───  ×  ───      en Q kry   ───  ×  ───    
                        12           1                           12           1
  
  
                              P kry   40  osse   en   Q  kry   56 osse
  
  
                                                                                             5                                                                   7
           P gets 5 oxen if there are (5 + 7), i.e. he gets    ───      and Q gets 7 oxen, i.e. he gets    ───
                                                                                            12                                                                 12
  
  
                            5          96                                7           96
           P gets   ───  ×  ───      and Q gets   ───  ×  ───    
                           12          1                                12           1
  
  
                              P gets   40  oxen   and   Q  gets   56  oxen
  
  
  
  
  
                                                                                                                                 OF / OR    
  
  
           12 osse (5 + 7) word verdeel in die verhouding 5 : 7   /   12 oxen (3 + 11) are divided in the ratio 5 : 7
  
           Verdeel 96 osse in dieselfde verhouding.                   /    Divide 96 oxen in the same ratio.
  
           Gestel P kry x osse, dan sal Q  (96 ─ x)  osse  kry.          / Suppose Q gets x oxen, then Q will get  (96 ─ x)  oxen
  
                                                                                                                     P             5 osse / oxen                 x  osse / oxen
           Verdeel in dieselfde verhouding / Divide in the same ratio   :    ───   =  ───────────   =  ───────────────
                                                                                                                     Q            7 osse / oxen             (96 ─ x)  osse / oxen
  
           Los op deur met KGV te vermenigvuldig /  Solve by multiplying by the LCM    5 × (96 ─ x)  =  7 × x
  
                                                                                                                                                 480 ─ 5x  =  7x
  
                                                                                                                                                          480  =  12x
  
                                                                                                                                                            40  =  x
  
          P kry / gets   40  osse / oxen    en / and   Q kry  /  gets  (96 ─ 40)  =  56  osse / oxen                             [ 2.5 ]
  
  
 Antwoorde / Answers 2.6 
  
  
           13 skape / sheep  (7 + 4 + 2) :   R  kry / gets    7,   S  kry / gets   4   en / and  T  kry / gets    2
  
                                                                                          7                                 4                                             2
           1  skaap / sheep                      :   R kry / gets   ─── ,   S kry / gets   ───   en / and  T  kry  gets   ───
                                                                                         13                              13                                           13
  
  
                                                                                        7           364                                 4           364                                                2          364
           364  skape / sheep                 :  R kry / gets   ─── X ───── ,    S kry / gets   ─── X ─────    en / and  T  kry  gets   ─── X ─────
                                                                                       13            1                                  13            1                                                 13           1
  
  
                                                                          d.i. / i.e   196                               d.i. / i.e   112                                                d.i. / i.e   56
  
  
                   R kry / gets  196  skape / sheep ,    S kry / gets   112  skape / sheep      en / and  T  kry  gets   56  skape / sheep
  
  
  
  
  
                                                                                                                                 OF / OR    
  
  
                                                                                                          7                                                         5                                                             2
           R kry 7 skape as daar (7 + 4 + 2) skape is, d.w.s hy kry  ─── ,   S kry 4 skape, d.w.s hy kry  ───   en T kry 2 skape, d.w.s hy kry  ───
                                                                                                         13                                                       13                                                          13
  
  
                           7          364                          4          364                              2          364
           R kry   ───  ×  ─────       S kry   ───  ×  ─────       en T kry   ───  ×  ─────    
                        13            1                           13             1                              13           1
  
  
                  R kry   196  skape                 S kry  112  skape              en   T  kry  56  skape
  
  
                                                                                                               7                                                            4                                                               2
           R gets 7 sheep if there are (7 + 4 + 2) sheep, i.e. he gets  ─── ,   S gets 4 sheep, d.w.s hy kry  ───   en T gets 2 sheep, d.w.s hy kry  ───
                                                                                                              13                                                          13                                                            13
  
  
                             7          364                            4            364                                  2            364
           R gets   ───  ×  ─────       S gets   ───  ×  ─────       and T gets   ───  ×  ─────    
                           13            1                             13             1                                   13              1
  
  
                        R gets   196  sheep                         S gets  112  sheep                             and   T  gets  56  sheep
  
  
                                                                                                                                                                           [ 2.6 ]
  
  
  
 Antwoorde / Answers 3 
  
  
            3.1  A : B : C  =  120 000  :  100 000  :  60 000
  
                    A : B : C  =  6  :  5  :  3        vereenvoudig deur te deel met 20 000  /           simplify by dividing by  20 000         [ 3.1 ]
  
  
  
          3.2  R 14 (6 + 5 + 3) :   A  kry / gets   R 6,   B  kry / gets   R 5   en / and  C  kry / gets   R 3
  
                                                                            6                                    5                                                3
                 R 1                      :   A kry / gets  R ─── ,   B kry / gets  R ───   en / and  C  kry  gets  R ───
                                                                           14                                  14                                              14
  
  
                                                                           6          48 601                                   5         48 601                                                3         48 601
                R 48 601             :  A kry / gets  R ─── X ────── ,    B kry / gets  R ─── X ──────    en / and  C  kry  gets  R ─── X ──────
                                                                          14            1                                        14            1                                                     14           1
  
  
                                              d.i. / i.e  R 20 829                                       d.i. / i.e  R 17 357,50                                        d.i. / i.e  R 10 414,50
  
  
                                   A kry / gets  R 20 829 ,    B kry / gets  R 17 357,50     en / and  C  kry  gets  R 10 414,50                                                     [ 3.2 ]
  
  
  
  
                                                                                                                                 OF / OR    
  
  
                                                                                              6                                                        5                                                          3
           A kry R6 as daar R(6 + 5 + 3) is, d.w.s hy kry   R ─── ,     B kry R5, d.w.s hy kry   R ───     en C kry R3, d.w.s hy kry   R ───
                                                                                             14                                                     14                                                        14
  
  
                           6         48 601                          5         48 601                              3         48 601
           A kry  R ───  ×  ─────       B kry  R ───  ×  ─────       en C kry  R ───  ×  ─────    
                           14            1                             14             1                                  14              1
  
  
                        A kry  R 20 829                          B kry  R 17 357,50                               en   C  kry  R 10 414,50
  
  
                                                                                                6                                                         5                                                                3
           A gets R6 if there is R(6 + 5 + 3) , i.e. he gets   R ─── ,     B gets R5, i.e. he gets   R ───     and C gets R3, i.e. he gets   R ───
                                                                                                14                                                      14                                                             14
  
  
                               6         48 601                            5         48 601                                  3         48 601
           A gets  R ───  ×  ─────       B gets  R ───  ×  ─────       and C gets  R ───  ×  ─────    
                             14            1                                 14             1                                      14              1
  
  
                        A gets  R 20 829                          B gets  R 17 357,50                               and   C  gets  R 10 414,50                                  [ 3.2 ]
  
  
  
  
  
          3.3.1   A : B : C  =  120 000  :  100 000  :  60 000  +  5 000
  
                    A : B : C  =  120 000  :  100 000  :  65 000
  
                    A : B : C  =  24  :  20  :  13        vereenvoudig deur te deel met 5 000  /           simplify by dividing by  5 000         [ 3.3.1 ]
  
  
  
          3.3.2    R 57 (24 + 20 + 13) :   A  kry / gets   R 24,   B  kry / gets   R 20   en / and  C  kry / gets   R 13
  
                                                                                        24                                 20                                             13
                       R 1                            :   A kry / gets  R ─── ,   B kry / gets  R ───   en / and  C  kry  gets  R ───
                                                                                        57                                 57                                              57
  
  
                                                                                      24         72 917,25                                  20        72 917,25                                               13        72 917,25
                       R 72 917,25             :  A kry / gets  R ─── X ──────── ,    B kry / gets  R ─── X ────────    en / and  C  kry  gets  R ─── X ────────
                                                                                      57                1                                          57                1                                                       57               1
  
  
                                                                     d.i. / i.e  R 30 702                                       d.i. / i.e  R 25 585                                        d.i. / i.e  R 16 630,25
  
  
                                                             A kry / gets  R 30 702 ,    B kry / gets  R 25 585      en / and  C  kry  gets  R 16 630,25
  
  
                                                                                                                                                                                              [ 3.3.2 ]
  
  
  
  
  
 Antwoorde / Answers 4 
  
  
        4.1  Daar is 24 albasters
  
               As Jan 9 kry, dan kry Martiens (24 - 9) = 15
  
               Verhouding is :  Jan : Martiens  =  9 : 15
  
               Verhouding is :  Jan : Martiens  =  3 : 5
  
                                                            [ 4.1 ]
  
  
  
        4.2  R1 927 word verdeel
  
               As A R1 230 kry en B kry R574, dan
  
               sal C R(1 927 - 1 230 - 574) = R123 kry
  
               Verhouding is :  A : B : C  =  1 230 : 574 : 123
  
                  A : B : C  =  2X3X5X41 : 2X7X41 : 3X41
  
                  A : B : C  =  30 : 14 : 3
  
                                                                                [ 4.2 ]
  
  
  
        4.1  There are 24 marbles
  
                If John gets 9, then Mike will get (24 - 9) = 15
  
               Ratio is :  John : Mike  =  9 : 15
  
               Ratio is :  John : Mike  =  3 : 5
  
                                                                                [ 4.1 ]
  
  
  
        4.2  R1 927 is divided
  
                 If A gets R1 230 and B gets R574, then
  
                 C will get R(1 927 - 1 230 - 574) = R123
  
               Ratio is :  A : B : C  =  1 230 : 574 : 123
  
                  A : B : C  =  2X3X5X41 : 2X7X41 : 3X41
  
                  A : B : C  =  30 : 14 : 3
  
                                                                                [ 4.2 ]
  
  
  
 Antwoorde / Answers 5 
  
  
        5.1  Vermeerdeer 82 kg in die verhouding 4 : 5
  
               4 kg moet nou 5 kg word.
  
                                          5
               1 kg moet nou ── kg word.
                                          4
  
                                            5          82
               82 kg moet nou ──  X  ───    kg word.
                                            4           1
  
                                 d.w.s.  102,5  kg
  
                Die vermeerderde massa is 102,5 kg.
  
  
  
                               OF
  
  
               As die ou massa = 4 kg dan is die nuwe massa = 5 kg
  
               As die ou massa = 82 kg dan is die nuwe massa = x kg
  
                   ou massa             4          82
               ─────────  =  ───  =  ───
                nuwe massa           5           x
  
               Vereenvoudig  :  4 X x  =  5 X 82
  
                                                4x  =  410
  
                                                x  =  102,5
  
                Die vermeerderde massa is 102,5 kg.         [ 5.1 ]
  
  
  
        5.2  Vermeerder / Increase R735
  
               in die verhouding / in the ratio 5 : 9
  
               R5 word / becomes R9.
  
                                                      9
               R1 word / becomes R ──
                                                      5
  
                                                           9         735
               R375 word / becomes R ──  X  ────
                                                           5           1
  
               d.w.s. / i.e.  R 1 323
  
  
  
                               OF / OR
  
  
               R5 word / becomes R9
  
               R735 word / becomes Rx
  
               Die verhouding bly dieselfde /
  
               The ratio remains the same.
  
                                               5           735
                                            ───  =  ────
                                               9             x
  
               Vereenvoudig  :  5 X x  =  9 X 735
  
                                                5x  =  6 615
  
                                                  x  =  1 323
  
                Die vermeerderde bedrag /
  
                The increased amount is R 1 323.
  
                                                                                 [ 5.2 ]
  
  
  
        5.4  Vermeerder / Increase 28,4 kg
  
               in die verhouding / in the ratio 4 : 3
  
               4 kg word / becomes 3 kg.
  
                                                      3
               1 kg word / becomes  ──  kg
                                                      4
  
                                                              3          28,4
               28,4 kg word / becomes    ──  X  ────  kg
                                                              4            1
  
                                         d.w.s. / i.e.  21,3  kg
  
  
  
                               OF / OR
  
  
               4 kg word / becomes 3 kg
  
               28,4 kg word / becomes x kg
  
               Die verhouding bly dieselfde /
  
               The ratio remains the same.
  
                                               4           28,4
                                            ───  =  ────
                                               3             x
  
               Vereenvoudig  :  4 X x  =  3 X 28,4
  
                                                4x  =  85,2 
  
                                                  x  =  21,3
  
                Die verminderde massa /
  
                The decreased mass is 21,3 kg.
  
                                                                                 [ 5.4 ]
  
  
  
        5.6  Vermeerder / Increase 34,2 ℓ
  
               in die verhouding / in the ratio 9 : 5
  
               9 ℓ word / becomes 5 ℓ.
  
                                                    5
               1 ℓ word / becomes  ──  ℓ
                                                    9
  
                                                           5          34,2
               34,2 ℓ word / becomes    ──  X  ────  ℓ
                                                           9            1
  
                                         d.w.s. / i.e.  19  ℓ
  
  
  
                               OF / OR
  
  
               9 ℓ word / becomes 5 ℓ
  
               34,2 ℓ word / becomes x ℓ
  
               Die verhouding bly dieselfde /
  
               The ratio remains the same.
  
                                               9           34,2
                                            ───  =  ────
                                               5             x
  
               Vereenvoudig  :  9 X x  =  5 X 34,2
  
                                                9x  =  171 
  
                                                  x  =  19
  
                Die verminderde volume /
  
                The decreased volume is 19 ℓ.
  
                                                                                 [ 5.6 ]
  
  
  
        5.1  Increase 82 kg in the ratio 4 : 5
  
               4 kg must become 5 kg.
  
                                                 5
               1 kg must become ── kg.
                                                 4
  
                                                   5          82
               82 kg must become ──  X  ───    kg.
                                                   4           1
  
                                 i.e.  102,5  kg
  
                The increased mass is 102,5 kg.
  
  
  
                               OR
  
  
               If the old mass = 4 kg then the new mass = 5 kg
  
               If the old mass = 82 kg then the new mass = x kg
  
                   old mass             4          82
               ─────────  =  ───  =  ───
                  new mass             5           x
  
               Simplify  :  4 X x  =  5 X 82
  
                                      4x  =  410
  
                                        x  =  102,5
  
                The increased mass is 102,5 kg.         [ 5.1 ]
  
  
  
        5.3  Vermeerder / Increase 21,7 m
  
               in die verhouding / in the ratio 7 : 11
  
               7 m word / becomes 11 m.
  
                                                    11
               1 m word / becomes  ──  m
                                                     7
  
                                                            11         21,7
               21,7 m word / becomes    ──  X  ────  m
                                                             7             1
  
                                         d.w.s. / i.e.  34,1  m
  
  
  
                               OF / OR
  
  
               7 m word / becomes 11 m
  
               21,7 word / becomes x m
  
               Die verhouding bly dieselfde /
  
               The ratio remains the same.
  
                                               7          21,7
                                            ───  =  ───  
                                              11           x
  
               Vereenvoudig  :  7 X x  =  11 X 21,7
  
                                                7x  =  238,7 
  
                                                  x  =  34,1 m
  
                Die vermeerderde lengte /
  
                The increased length is 34,1 m.
  
                                                                               [ 5.3 ]
  
  
  
        5.5  Vermeerder / Increase R324
  
               in die verhouding / in the ratio 8 : 5
  
               R8 word / becomes R5.
  
                                                    5
               R1 word / becomes  ──
                                                    8
  
                                                             5         324
               R324 word / becomes   R ──  X  ────
                                                             8           1
  
                                         d.w.s. / i.e.  R 202,5 
  
  
  
                               OF / OR
  
  
                 R8 word / be  comes R5
  
                 R324 word / becomes Rx
  
               Die verhouding bly dieselfde /
  
               The ratio remains the same.
  
                                               8          324
                                            ───  =  ───  
                                               5            x
  
               Vereenvoudig  :  8 X x  =  5 X 324
  
                                                8x  =  1 620
  
                                                  x  =  R202,5 m
  
                Die vermeerderde bedrag /
  
                The increased amount is R 202,5
  
                                                                               [ 5.5 ]
  
  
  
        5.7  Vermeerder / Increase 220 volt
  
               in die verhouding / in the ratio 55 : 3
  
               55 V word / becomes 3 V.
  
                                                      3
               1 V word / becomes  ───
                                                     55
  
                                                              3           220
               220 V word / becomes      ───  X  ────  V
                                                             55            1
  
                                         d.w.s. / i.e.  12 V 
  
  
  
                               OF / OR
  
  
               55 V word / becomes 3 V
  
               220 V word / becomes x V
  
               Die verhouding bly dieselfde /
  
               The ratio remains the same.
  
                                              55        220
                                            ───  =  ───  
                                               3            x
  
               Vereenvoudig  :  55 X x  =  3 X 220
  
                                                55x  =  660
  
                                                    x  =  12
  
                Die vermeerderde volt /
  
                The increased volt is 12  V
  
                                                                               [ 5.7 ]
  
  
  
 Antwoorde / Answers 6 
  
  
              Seuns / boys : dogters / girls = 8 : 7
  
               15 (8 + 7)  leerlinge / pupils  :  8 seuns / boys
  
                                              8
               Seuns / Boys  =  ───  X  45    =  24
                                             15
  
                 Dogters / Girls  =  45 ─ 24  =  21
  
               Daar is 24 seuns en 21 dogters in die klas. /
  
               There are 24 boys and 21 girls in the class.
  
                                                                               [ 6. ]
  
  
  
          Seuns / boys : dogters / girls = 8 : 7
  
               15 (8 + 7)  leerlinge / pupils  :  7 dogters / girls
  
                                              7
               Dogters / Girls  =  ───  X  45    =  21
                                             15
  
                 Seuns / Boys  =  45 ─ 21  =  24
  
               Daar is 24 seuns en 21 dogters in die klas. /
  
               There are 24 boys and 21 girls in the class.
  
                                                                               [ 6. ]
  
  
  
 Antwoorde / Answers 7 
  
  
            7.1  Seuns / boys = 23, dogters / girls = 22
  
                       seuns / boys            23
                     ──────────  =  ────
                      dogters / girls           22
  
  
                   Dit is die eenvoudigste verhouding want
  
                   23 is 'n priemgetal en het nie ander
  
                   faktore nie. / It is the simplest ratio
  
                   because 23 has no other factors.
  
                                                                               [ 7.1 ]
  
  
  
            7.3  Seuns / boys : dogters / girls = 6 : 5
  
               11 (6 + 5)  leerlinge / pupils  :  6 seuns / boys
  
                                                  6
                   Seuns / Boys  =  ───  X  44    =  24
                                                 11
  
                     Dogters / Girls  =  44 ─ 24  =  20
  
                   Daar is 24 seuns en 20 dogters in die klas. /
  
                   There are 24 boys and 20 girls in the class.
  
                   Daar is 24 ─ 23 = 1 seun toegelaat en
  
                   22 ─ 20 = 2 dogters het die klas verlaat. /
  
                   24 ─ 23 = 1 boy was admitted and
  
                   22 ─ 20 = 2 girls left the class.
  
                                                                               [ 7.3 ]
  
  
  
            7.2  Seuns / boys = 23, dogters / girls = 22
  
                     Seuns / Boys = 23 ─ 3  =  20
  
                     Dogters / Girls = 22 + 4  =  26
  
  
                       seuns / boys          20         10
                     ──────────  =  ───  =  ───
                      dogters / girls         26          13
  
  
  
  
  
                                                                               [ 7.2 ]
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
   Antwoorde / Answers 8 
    
  
                     rooi : groen: geel  =  3 : 4 : 5 
  
                     Van 12 (3 + 4 + 5) lekkers,
  
                     is 3 rooi, 4 groen en 5 geel
  
                        3
                     ───  van die lekkers is rooi en dit is 12
                       12
  
  
                        3            x
                     ───  X  ───  =  12
                       12          1
  
  
                                       3x  =  144
  
                                         x  =  48
  
                     Daar is 48 lekkers in die blik.
  
                        4
                     ───  van die lekkers is groen
                       12
  
  
                        4
                     ───  X  48    =  16
                       12
  
  
                     Daar is 16 groen lekkers en
  
                     12 rooi lekkers in die blik.
  
                     Aantal geel lekkers  =  48 ─ (12 + 16)
  
                                                       =  20
  
                     Daar is 12 rooi, 16 groen en
  
                     20 geel lekkers in die blik.
  
                                                                               [ 8. ]
  
  
  
                     red  :  green  :  yellow  =  3 : 4 : 5 
  
                     Of 12 (3 + 4 + 5) sweets,
  
                     3 are red, 4 green and 5 yellow
  
                        3
                     ───  of the sweets are red and that is 12
                       12
  
  
                        3            x
                     ───  X  ───  =  12
                       12          1
  
  
                                       3x  =  144
  
                                         x  =  48
  
                     There are 48 sweets in the tin.
  
                        4
                     ───  of the sweets are green
                       12
  
  
                        4
                     ───  X  48    =  16
                       12
  
  
                     There are 16 green sweets and
  
                     12 red sweets in the tin.
  
                     Number of yellow sweets = 48 ─ (12 + 16)
  
                                                                =  20
  
                     There are 12 red, 16 green and
  
                     20 yellow sweets in the tin.
  
                                                                               [ 8 ]
  
  
  
   Antwoorde / Answers 9 
    
  
          9.1  Die verhouding 3 : 2 : 1 stel die verhouding
  
                  van die plantvoedsels, stikstof (N), fosfor (P)
  
                  en kalium (K), voor.
  
                  Die verhouding word geskryf
  
                         N : P : K = 3 : 2 : 1.
  
                  Die 22 stel die totale persentasie van die
  
                  plantvoedsels, N, P en K,
  
                  in die sak kunsmis voor.
  
                                                                               [ 9.1 ]
  
  
  
          9.2  Massa / mass  = 22% van 10 kg
  
                                                 22
                  Massa / mass  = ────  X  10  kg
                                                100
  
                                           = 2,2  kg
                                                                               [ 9.2 ]
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
          9.1  The ratio 3 : 2 : 1 represents the ratio
  
                  of the plant nutrients, nitrogen (N), phosphorus (P)
  
                  and potassium (K).
  
                  The ratio is written
  
                         N : P : K = 3 : 2 : 1.
  
                  The 22 represents the total percentage of the
  
                  plant nutrients, N, P en K,
  
                  in the bag of fertilizer.
  
                                                                               [ 9.1 ]
  
  
  
          9.3  6 (3+2+1) kg bevatt / contains 3 kg N
  
                                                       3
                  1 kg bevat / contains ──  kg   N
                                                       6
  
                                                          3          2,2
                  2,2 kg bevat / contains ──  X  ───  kg   N
                                                          6           1
  
                                              d.i. / i.e.  1,1  kg N
  
  
                                                          2          2,2
                  2,2 kg bevat / contains ──  X  ───  kg   P
                                                          6           1
  
                                              d.i. / i.e.  0,733  kg P
  
  
                  Massa / mass K  = 2,2 ─ (1,1 + 0,733) kg
  
                                               =  0,367  kg
  
                  Sak bevat / Bag contains  1,1 kg N,
  
                   0,733 kg P en / and 0,367 kg K
                                                                               [ 9.3 ]
  
  
  
   Antwoorde / Answers 10 
    
  
          10.1  oppervlakte / area : volume = 5,2 m2 : 1 ℓ
  
                  5,2 m2 benodig / is covered by 1 ℓ
  
                                                                        1
                  1 m2 benodig / is covered by ────  ℓ
                                                                      5,2
  
                                                                       1        20,8
            20,8 m2 benodig / is covered by ─── X ────  ℓ
                                                                    5,2          1
  
                                                          d.i. / i.e.  4 ℓ
  
  
                    4 ℓ  verf word benodig / paint is needed
  
                                                                               [ 10.1 ]
  
  
  
          10.3  oppervlakte / area : volume = 11,5 m2 : 2,5 ℓ
  
                  11,5 m2 benodig / is covered by 2,5 ℓ  verf / paint
  
                                                                       2,5
                  1 m2 benodig / is covered by ────  ℓ
                                                                      11,5
  
                                                                      2,5        21,4
            21,4 m2 benodig / is covered by ──── X ────  ℓ
                                                                    11,5          1
  
                                                          d.i. / i.e.  4,652 ℓ
  
  
  
  
          10.2  oppervlakte / area : volume = 17,6 m2 : 2,8 ℓ
  
                  2,8 ℓ bedek / covers 17,6 m2
  
                                                     17,6
                  1 ℓ bedek / covers   ────  m2
                                                      2,8
  
                                                    17,6        5,8
              5,8 ℓ bedek / covers   ──── X ────   m2
                                                     2,8           1
  
                                                          d.i. / i.e.  36,457 m2
  
  
                    36,457 m2 word bedek / is covered
  
                                                                               [ 10.2 ]
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
                    Die blik bevat 4,2 ℓ en dit is minder as wat benodig word. Die verf kan dus nie die oppervlakte bedek nie.
  
                    The tin only contains 4,2 ℓ paint and that is less than what is required. The paint can not cover the area.
                                                                                                                                                                                                                                                                                                                                                                                                                                 [ 10.3 ]
  
  
  
   Antwoorde / Answers 11 
    
  
          11.1  afstand / distance : volume = 254 km : 21,082 ℓ
  
                  254 km benodig / needs 21,082 ℓ
  
  
                                                             21,082
                  1 km benodig / needs    ──────  ℓ
                                                                254
  
  
                                                          21,082          613
           613 km benodig / needs   ──────  X  ────  ℓ
                                                             254              1
  
                                              d.i. / i.e.  50,879 ℓ
  
  
                    50,879 ℓ  petrol word verbruik / petrol is needed
  
  
  
  
                                                     OF / OR
  
  
  
                afstand / distance        254               613
               ────────────  =  ──────  =  ────
                   volume petrol            21,082            x
  
  
                                 254 X x  =  613 X 21,082
  
  
                                      254x  =  12 923,2663
  
  
                                            x  =  50,879
  
  
              50,879 ℓ petrol word verbruik/petrol is needed
  
                                                                               [ 11.1 ]
  
  
  
    11.2  afstand/distance : volume = 254 km : 21,082 ℓ
  
             21,082 ℓ verbruik/used  254 km afgelê/travelled
  
  
                                                    254
              1 ℓ verbruik / used  ──────  km afgelê / travelled
                                                  21,082
  
  
                                                   254               14,2
            14,2 ℓ verbruik/used  ──────    X  ────   km afgelê / travelled
                                                 21,082               1
  
                                  d.i. / i.e.  171,084 
  
  
        Afstand afgelê / distance travelled  = 171,084 km
  
  
  
  
                                                     OF / OR
  
  
  
                afstand / distance           254               x
               ────────────  =  ──────  =  ────
                   volume petrol            21,082          14,2
  
  
                            21,082 X x  =  14,2 X 254
  
  
                                 21,082x  =  3 606,8
  
  
                                             x  =  171,084
  
  
              171,084 km word afgelê/is travelled
  
                                                                               [ 11.2 ]
  
  
  
   Antwoorde / Answers 12 
    
  
                                                         36
        12.1  36 minute / minutes  =  ──  uur / hours
                                                         60
  
                                                    = 0,6 h (uur / hours)
  
                 2 uur / hours 36 minute/minutes  = 2 + 0,6 h
  
                                                                       = 2,6 h
   
                                                                                  [ 12.1 ]
  
  
  
                     afstand/distance      253,5            415
        12.3    ─────────── = ──────  =  ────
                       tyd  /  time                2,6                 x
  
  
                               253,5 X x  =  2,6 X 415
  
  
                                    253,5x  =  1 079
  
  
                                             x  =  4,256
  
  
              Tydsduur / time taken  =  4,256 h
  
                                                                               [ 12.3 ]
  
   
  
                                              48
        12.2  3 h 48 min = 3 + ──  h
                                              60
  
                                     = 3,8 h
  
  
                afstand / distance         253,5              x
               ────────────  =  ──────  =  ────
                       tyd  /  time                 2,6              3,8
  
  
                              2,6 X x  =  253,5 X 3,8
  
  
                                 2,6x  =  963,3
  
  
                                             x  =  370,5
  
  
              Afstand afgelê / distance travelled  =  370,5 km
  
                                                                               [ 12.2 ]
  
  
  
  
  
  
  
  
  
   Antwoorde / Answers 13 
    
  
                Hierdie is 'n voorbeeld van omgekeerde
  
                eweredigheid want meer mans sal minder
  
                dae werk en omgekeerd, minder mans sal
  
                langer werk as almal teen dieselfde tempo werk.
  
  
                As a omgekeerd eweredig is aan b, dan
  
                sal a X b = 'n konstante, bv. a X b = 36
  
                Gestel a X b = 36. Let nou op dat
  
                as a groter word, word b in dieselfde
  
                mate kleiner.
  
                As a = 1 is b = 36 want a X b = 36
  
                As a tweemaal groter word, a = 2,
  
                dan is b = 18, b is die helfte kleiner.
  
                
  
                Maar let op dat a X b = 36.
  
                Die produk van die veranderlikes
  
                bly dieselfde.
  
  
  
        13.1  Hier is mans X dae = konstante
  
                  mans X dae = 10 X 3 = 30
  
                  6 mans werk d dae.
  
                  6 X d = 10 X 3
  
                         10 X 3
                  d = ─────    =  5
                              6
  
                  6 mans sal 5 dae werk
  
                                                                             [ 13.1 ]
  
  
  
        13.2  mans/men X dae/days = 30
  
                  12 mans/men werk/word d dae/days
  
                  12 X d = 10 X 3
  
                         10 X 3
                  d = ─────    =  2,5
                              12
  
                  12 mans sal 2,5 dae we+rk /
  
                  12 men will work for 2,5 days.
  
                                                                           [ 13.2 ]
  
  
  
        13.4  mans/men X dae/days = 30
  
                  m mans/men werk/word 2 dae/days
  
                  m X 2 = 10 X 3
  
                         10 X 3
                  m = ─────    =  15
                              2
  
                  15 mans sal 2 dae we+rk /
  
                  15 men will work for 2 days.
  
                                                                           [ 13.4 ]
  
  
  
                This is an example of inverse proportion
  
                because more men will work less days and
  
                inversely, less men will work more days
  
                if everbody works at the same rate.
  
  
                If a is inversely proportional to b, then
  
                a X b = a constant, e.g. a X b = 36
  
                Let a X b = 36. Note that if a
  
                increases, b will decrease at the
  
                same rate.
  
                If a = 1 then b = 36 because a X b = 36
  
                If a is doubled, a = 2, then
  
                b = 18, b is halved.
  
                
  
                But note that a X b = 36.
  
                The product of the variables
  
                remain the same.
  
  
  
        13.1  Here men X days = constant
  
                  mans X dae = 10 X 3 = 30
  
                  6 men werk d days.
  
                  6 X d = 10 X 3
  
                         10 X 3
                  d = ─────    =  5
                              6
  
                  6 men will work 5 days.
  
                                                                             [ 13.1 ]
  
  
  
        13.3  mans/men X dae/days = 30
  
                  m mans/men werk/work 7 dae/days
  
                  m X 7 = 10 X 3
  
                         10 X 3
                  m = ─────    =  4,28571.....
                              7
  
                  Daar bestaan nie 'n breuk van 'n man
  
                  nie en dus moet of 4 of 5 gebruik word.
  
                  4 mans sal meer as 7 dae werk en 5 mans
  
                  sal minder as 7 dae werk. /
  
                  A fraction of a man does not exist and
  
                  therefore either 4 or 5 man must be used.
  
                  4 men will work more than 7 days and
  
                  5 men will word less than 7 days.
  
                                                                           [ 13.3 ]
  
  
  
  
  
  
  
  
  
   Antwoorde / Answers 14 
    
  
      14.1  Hier/Here kinders/children(p) X lekkers/sweets(s) = c
  
                kinders/children(p) X lekkers/sweets(s) = c
  
                p X s = 20 X 8 = 160
  
                40 p kry/ get a s.
  
                40 X a = 20 X 8
  
                       20 X 8
                a = ─────    =  4
                            40
  
                40 kinders kry 4 lekkers elk. /
  
                  40 children get 4 sweets each.
  
                                                                             [ 14.1 ]
  
  
  
      14.3  p X s = 20 X 8 = 160
  
                m p kry/ get 16 s.
  
                m X 16 = 20 X 8
  
                       20 X 8
                m = ─────    =  10
                            16
  
                10 kinders kry 16 lekkers elk. /
  
                10 children get 16 sweets each.
  
                                                                             [ 14.3 ]
  
  
  
        14.2  kinders/children X lekkers / sweets = 20 X 8
  
                  8 kinders kry / children get b lekkers/sweets.
  
                  8 X b = 20 X 8
  
                         20 X 8
                  b = ─────    =  20
                              8
  
                  8 kinders kry 20 lekkers elk. /
  
                  8 children get 20 sweets each.
  
                                                                             [ 14.2 ]
  
  
  
  
  
  
  
        14.4  p X s = 20 X 8
  
                  q kinders kry / children get 4 lekkers/sweets.
  
                  q X 4 = 20 X 8
  
                         20 X 8
                  q = ─────    =  40
                              4
  
                  40 kinders kry 4 lekkers elk. /
  
                  40 children get 4 sweets each.
  
                                                                             [ 14.4 ]