WISKUNDIGE GELETTERDHEID
GRAAD 12
NOG OEFENINGE
  
Koers, verhouding en eweredigheid : antwoorde.
  
MATHEMATICAL LITERACY
GRADE 12
MORE EXERCISES
  
Rate, ratio and proportion : answers.
  
  
  
Antwoord 1.

  
     1.1  R(3 + 4) word verdeel in die verhouding 3 : 4
  
            As R7 verdeel word in die verhouding 3 : 4,
  
                   is een deel R3 en die ander deel R4
  
            As R1 verdeel word in die verhouding 3 : 4,
  
                                            3                                         4
                   is een deel R ───   en die ander deel R ───
                                          3+4                                     3+4
  
            As R35 verdeel word in die verhouding 3 : 4,
  
                                  3    35                                     4     35
           is een deel R ─ × ──   en die ander deel R ─ × ──
                                 7      1                                      7      1
  
                      d.i.  R15       en die ander deel      R 20
  
                { Toets : R15 + R20 = R35 }   
  
               OF
  
                                   3
          Een deel is ──── × R35  =  R15    en 
                               3 + 4
  
                                      4
            ander deel is ──── × R35  =  R20
                                   3 + 4
  
  
            { Toets : R15 + R20 = R35 }
  
                                                               Vraag  1.
  
  
   
                   Verdeel
  
        1.2  (2 + 3 + 4) lekkers in die verhouding 2 : 3 : 4
  
                 9 lekkers word verdeel 2 en 3 en 4
  
                                            2           3          4
                1 word verdeel  ──  en  ──  en  ──
                                           9            9          9
  
  
                                            2                 3                 4
                45 word verdeel  ─ × 45  en  ─ × 45  en  ─ × 45
                                            9                 9                 9
  
  
                Dit is                   10   en    15      en   20
  
  
                { Toets : 10 + 15 + 20 = 45 }   
  
               OF
  
                Die dele is
  
  
                       2                      3                   4
                  ──── × 45  ;  ──── × 45 ; ──── × 45
                  2+3+4             2+3+4            2+3+4
  
             Dit is    10                15                   20
  
  
                { Toets : 10 + 15 + 20 = 45 }
  
                                                                Vraag  1.
  
    
   
        1.3  R34 500 in die verhouding 2 : 8 : 15.
  
                                    2                           8
                Die dele is  ── × R34 500  ;  ── × R34 500
                                   25                        25
  
                       15
                en  ── × R34 500
                       25
  
            Dit is      R2 760            R11 040           R20 700
  
  
          { Toets : R2 760 + R11 040 + R20 700 = R34 500 }
  
                                                                  Vraag  1.
  
  
  
  
  
Antwoord 2.

  
     2.1  Die lengte van 4 m word vermeerder na 7 m
  
                                                                              7
           Die lengte van 1 m word vermeerder na   ─ m
                                                                              4
  
                                                           7
            Die lengte 21,6 m word nou  ─ × 21,6  m
                                                          4
  
                                                   d.i.  37,8  m
  
  
            OF
  
                                                    7
            Vermeerderde lengte  = ─ × 21,6 m
                                                   4
  
                                                =  37,8 m
  
                                                                 Vraag  2.
  
   
        2.2  'n Wins van R4 word R5
  
                                                     5
               'n Wins van R1 word R ─
                                                     4
  
                                                            5
               'n Wins van R3 672 word R ─ × 3 672
                                                            4
  
                                                     d.i.  R4 590
  
  
            OF
  
                                                    5
            Vermeerderde wins  = R ─ × 3 672
                                                    4
  
                                             =  R 4 590
  
                                                                  Vraag  2.
  
  
        2.3  'n Massa van 7 kg word 'n massa van 5 kg.
  
                                                                             5
               'n Massa van 1 kg word 'n massa van ─  kg
                                                                             7
  
                                                              5
               'n Massa van 91,2 kg word   ─ × 91,2  kg
                                                             7
  
                                                          d.i.  65,143 kg
  
  
            OF
  
                                                    5
            Vermeerderde massa  = ─ × 91,2  kg
                                                    7
  
                                                 =  65,143 kg
  
                                                                      Vraag  2.
  
  
  
  
Antwoord 3.

  
        3.1  5 km in 25 minute
  
                                25
               1 km in   ───  minute
                                 5
  
               2 km in   5 × 2  minute
  
                           =  10  minute
  
  
        3.2  In 25 minute draf hy 5 km
  
                                                  5
               In 1 minuut draf hy  ───  km
                                                 25
  
                                                    5
               In 65 minute draf hy  ─── ×  65  km
                                                   25
  
                                                   d.i.  13 km
  
            OF
  
        3.1  5 km in 25 minute
  
                                25
               2 km in   ───  ×  2  minute
                                 5
  
                         d.i.  10  minute
  
                                                                  Vraag  3.
  
  
  
  
Antwoord 4.

  
        4.1  Maandag tot Vrydag is 5 dae. 1 Week = 5 dae.
  
               1 Week = 5 dae teen 8 uur per dag.
  
                             = 40 hours.
  
                Weeklikse loon  =  40 uur × R30
  
                                           = R1 200 .
  
  
  
        4.2  Sy verdien R30 in 1 uur
  
                                                  1 170
               Sy verdien R1 170 in ────  uur.
                                                    30
  
                                            d.i. 39 uur.
  
  
  
        4.3  Vir 1 uur verdien sy R30.
  
               Vir 29 uur verdien sy R30 × 29
  
                                           i.e. R870
  
  
                                                                   Vraag  4.
  
  
  
  
Antwoord 5.

  
        5.1  Met 47,32 liter petrol ry die motor 613 km
  
                                                                  613
               Met 1 liter petrol ry die motor ─────  km
                                                                47,32
  
                                                         d.i. 12,95 km
  
               Brandstofvebruik is 12,95 km/ℓ
  
  
  
               Vir 613 km word 47,32 ℓ verbruik
  
                                           47,32
                Vir 1 km word  ─────  ℓ verbruik
                                            613
  
                                               47,32         100
                Vir 100 km word  ─────  ×  ────  ℓ verbruik
                                                613             1
  
                                       d.i. 7,72  ℓ
  
  
                Brandstofvebruik is 7,72 ℓ/100 km
  
                                                                 Vraag  5.
  
  
  
        5.2  Met 47,32 liter petrol ry die motor 613 km
  
                                                             613
               Met 1 ℓ petrol ry die motor ─────  km
                                                            47,32
  
                                                       613          21,4
               Met 21,4 ℓry die motor ─────  ×  ───  km
                                                      47,32           1
  
                                              d.i. 277,22 km
  
  
      OF  gebruik brandstofverbruik : 12,95 km/ℓ
  
            Met 1 liter petrol ry die motor 12,95 km
  
            Met 21,4 liter petrol ry die motor 12,95  ×  21,4 km
  
                                                         d.i. 277,13 km
  
                                                                 Vraag  5.
  
  
        5.3  Vir 613 km word 47,32 ℓ verbruik
  
                                         47,32
               Vir 1 km word ─────  ℓ ℓ verbruik verbruik
                                          613
  
                                             47,32          430
               Vir 430 km word ─────  ×  ──── ℓ verbruik
                                             613              1
  
                                      d.i. 33,19  ℓ
  
  
      OF  gebruik brandstofverbruik : 7,72 ℓ/100 km
  
               Vir 100 km word 7,72 ℓ petrol verbruik.
  
                                        7,72
               Vir 1 km word  ───  ℓ petrol verbruik.
                                        100
  
                                            7,72       430
               Vir 430 km word  ───  ×  ───  ℓ petrol verbruik.
                                            100          1
  
                                      d.i. 33,20 ℓ
  
  
                                                                       Vraag  5.
  
  
  
  
Antwoord 6.

  
        6.1.1  Skaal 1 : 50 1 cm op plan is 50 cm in die gebou
  
                            1 cm op plan is 50 cm in die gebou
  
                            2 cm op plan is 50 × 2  cm in die gebou
  
                                                d.i. 100 cm
  
                                                                 Vraag  6.
  
  
  
        6.1.2      1 mm op plan is 50 mm in die gebou
  
                    35 mm op plan is 50 × 35  mm in die gebou
  
                                           d.i. 1 750 mm
  
                                                                 Vraag  6.
  
  
  
        6.1.3      1 m op plan is 50 m in die gebou
  
                    0,35 m op plan is 50 × 0,35  m in die gebou
  
                                           d.i. 17,50 m
  
                                                                 Vraag  6.
  
  
        6.2.1  Skaal 1 : 50 : 50 m in gebou is 1 m op plan
  
                  50 m in gebou is 1 m op plan
  
                                              1
                  1 m in gebou is  ──  m op plan
                                             50
  
                                                    1
                  35,4 m in gebou is  ──  ×  35,4  m op plan
                                                  50
  
                                        d.i.  0,708 m  of  70,8 cm
  
                                                                 Vraag  6.
  
  
  
        6.2.2  Skaal 1 : 50 : 50 m in gebou is 1 m op plan
  
                     50 m in gebou is 1 m op plan
  
                                                        1
                      1 m in die gebou is  ──  m op plan
                                                       50
  
                                                         1
                    0,4 m in die gebou is  ──  ×  0,4  m op plan
                                                        50
  
                                              d.i.  0,008 m  of  0,8 cm
  
                                                                 Vraag  6.
  
  
        6.2.3  Skaal 1 : 50 : 50 m in gebou is 1 m op plan
  
                  50 m in die gebou is 1 m op plan
  
                                                      1
                    1 m in die gebou is  ──  m op plan
                                                     50
  
                                                           1
                   1,26 m in die gebou is  ──  ×  1,26  m op plan
                                                         50
  
                                              d.i.  0,0252 m  of  2,52 cm
  
  
                                                                       Vraag  6.
  
  
  
  
Antwoord 7.

  
        7.1  Volume is direk eweredig aan massa want
  
                            massa
                         ──────  is 'n konstante.
                           volume
  
                                                                       Vraag  7.
  
  
  
        7.2  1 liter  =  1 000  cm3  en  1 kg  =  1 000 g 
  
                                                        massa
                digtheid van water   =   ─────── 
                                                       volume
  
                                                     1 000 g
                                               =   ───────
                                                     1 000 cm3
  
                                               =  1 g/cm3
  
                                                                       Vraag  7.
  
  
     
                                                        massa
        7.3.1   digtheid van stof   =   ───────
                                                       volume
  
                    massa = digtheid van die stof × volume
  
  
                               massa = 19,3 g/cm3  ×  6 cm3
  
                                          =  115,8 g
  
  
                                                        massa
        7.3.2   digtheid van stof   =   ───────
                                                       volume
  
                                                           massa
                  sodat  volume  =  ──────────────
                                                digtheid van die stof
  
  
                                                  413,02 g
                  sodat  volume  =  ─────────
                                                 19,3 g/cm3
  
                                           =  21,4 cm3
  
  
                                                        massa
        7.3.3   digtheid van stof   =   ───────
                                                       volume
  
                                                        41,2 g
                                                =  ───────
                                                       2,4 cm3
  
                                                =  17,167g/cm3
  
                  Dit is kleiner as die digtheid van suiwer goud en
  
                  dus is die ring nie van suiwer goud gemaak nie.
  
                                                                       Vraag  7.
  
  
  
                                                        massa
        7.4.1   digtheid van lood   =   ───────
                                                       volume
  
  
                                                        24,97 g
                                                =  ───────
                                                       2,2 cm3
  
                                                =  11,35g/cm3
  
                                                        massa
        7.4.2   digtheid van stof   =   ───────
                                                       volume
  
                                                           massa
                  sodat  volume  =  ──────────────
                                                digtheid van die stof
  
  
                                                  181,6 g
                  sodat  volume  =  ─────────
                                                 11,35 g/cm3
  
                                           =  16 cm3
  
  
                                                        massa
        7.4.2   digtheid van stof   =   ───────
                                                       volume
  
                                                        41,2 g
                                                =  ───────
                                                       2,4 cm3
  
                                                =  17,167g/cm3
  
  
        7.4.3   volume van 'n kubus   =  sy3  =  (2 cm)3
  
                                                       = 8 cm3
  
                                                               massa
                         digtheid van stof   =   ───────
                                                              volume
  
                   massa = digtheid van die stof × volume
  
  
                             massa  =  11,35 g/cm3  ×  8 cm3
  
                                          =  90,80 g
  
  
                                                                    Vraag  7.
  
  
  
  
Answer 1.

  
     1.1  R35 is divided in the proportion 3 : 4
  
            If R(3+4) is divided in the proportion 3 : 4,
  
                   one part is R3 and the other R4
  
            If R1 is divided in the proportion 3 : 4,
  
                                           3                                   4
                   one part is R ───   and the other R ───
                                         3+4                               3+4
  
            If R35 is divided in the proportion 3 : 4,
  
                                    3      35                            4     35
              one part is R ─ × ──   and the other R ─ × ──
                                    7      1                              7      1
  
                           i.e.     R15       and the other      R 20
  
                { Test : R15 + R20 = R35 }   
  
               OR
  
                                         3
                One part is  ──── × R35  =  R15    and
                                     3 + 4
  
                                                4
                the other part is  ──── × R35  =  R20
                                            3 + 4
  
  
                { Test : R15 + R20 = R35 }
  
                                                                   Question  1.
  
  
  
                Divide
  
        1.2  (2 + 3 + 4) sweets in the proportion 2 : 3 : 4
  
                 9 sweets are divided 2 and 3 and 4
  
                                      2             3             4
                1 is divided  ──  and  ──  and  ──
                                      9             9             9
  
  
                                         2                   3                   4
                45 are divided  ─ × 45  and  ─ × 45  and  ─ × 45
                                         9                   9                   9
  
  
                That is               10       and    15        and  20
  
  
                { Test : 10 + 15 + 20 = 45 }   
  
               OR
  
                The parts are
  
                         2                      3                   4
                    ──── × 45  ;  ──── × 45 ; ──── × 45
                    2+3+4             2+3+4            2+3+4
  
                 it is      10                    15                20
  
  
                { Test : 10 + 15 + 20 = 45 }
  
  
                                                                   Question  1.
  
  
  
        1.3  R34 500 in the proporion 2 : 8 : 15.
  
                                            2                          8
             The amounts are  ── × R34 500  ;  ── × R34 500
                                           25                        25
  
                         15
                and  ── × R34 500
                         25
  
              That is    R2 760            R11 040           R20 700
  
  
             { Test : R2 760 + R11 040 + R20 700 = R34 500 }
  
                                                                     Question  1.
  
  
  
  
  
Answer 2.

  
     2.1  If the length is 4 m, the increased length is 7 m
  
                                                                                  7
            If the length is 1 m, the increased length is  ─ m
                                                                                  4
  
                                                                 7
            The length of 21,6 m becomes   ─ × 21,6  m
                                                                 4
  
                                                         i.e.  37,8  m
  
  
            OR
  
                                            7
            Increased length  = ─ × 21,6 m
                                            4
  
                                         =  37,8 m
  
                                                                    Question  2.
  
  
        2.2  A profit of R4 becomes R5
  
                                                         5
               A profit of R1 becomes R ─
                                                         4
  
                                                                5
               A profit of R3 672 becomes R ─ × 3 672
                                                                4
  
                                                     i.e.  R4 590
  
  
            OR
  
                                              5
            Increased profit  = R ─ × 3 672
                                              4
  
                                       =  R 4 590
  
                                                              Question  2.
  
  
        2.3  A mass of 7 kg becomes a mass of 5 kg.
  
                                                                          5
               A mass of 1 kg becomes a mass of  ─  kg
                                                                          7
  
                                                               5
               A mass of 91,2 kg becomes   ─ × 91,2  kg
                                                               7
  
                                                       i.e.  65,143 kg
  
  
            OR
  
                                           5
            Increased mass  = ─ × 91,2  kg
                                           7
  
                                         =  65,143 kg
  
                                                                 Question  2.
  
  
  
  
Answer 3.

  
        3.1  5 km in 25 minutes
  
                                25
               1 km in   ───  minutes
                                 5
  
               2 km in   5 × 2  minutes
  
                           =  10  minutes
  
        3.2  In 25 minutes he jogs 5 km
  
                                                  5
               In 1 minute he jogs  ───  km
                                                 25
  
                                                       5
               In 65 minutes he jogs  ─── ×  65  km
                                                     25
  
                                                   d.i.  13 km
  
  
            OR
  
        3.1  5 km in 25 minutes
  
                                25
               2 km in   ───  ×  2  minutes
                                 5
  
                         d.i.  10  minutes
  
                                                                 Question  3.
  
  
  
  
Answer 4.

  
        4.1  Monday to Friday is 5 days. 1 Week = 5 days.
  
               1 Week = 5 days at 8 hours per day.
  
                             = 40 hours.
  
                Weekly wage 40 hours × R30
  
                                   i.e. R1 200 hours.
  
  
  
        4.2  She earns R30 in 1 hour
  
                                                 1 170
               She earns R1 170 in ────  hours.
                                                    30
  
                                                 i.e. 39 hours.
  
  
  
        4.3  For 1 hour she receives R30.
  
               For 29 hours she receives R30 × 29
  
                                                      i.e. R870
  
  
                                                             Question  4.
  
  
  
  
Answer 5.

  
        5.1  Using 47,32 ℓ petrol the car travels 613 km
  
                                                                    613
               Using 1 ℓ petrol the car travels ─────  km
                                                                  47,32
  
                                                            i.e. 12,95 km
  
               Fiel consumption is 12,95 km/ℓ
  
  
  
               Travelling 613 km 47,32   ℓ  is consumed
  
                                           47,32
               Travelling 1 km ─────    ℓ  is consumed
                                            613
  
                                             47,32      100
             Travelling 100 km ───── × ─── ℓ is consumed
                                              613          1
  
                                     i.e. 7,72  ℓ
  
  
                Fuel consumption is 7,72 ℓ/100 km
  
                                                                 Question  5.
  
  
  
        5.2  Using 47,32 ℓ petrol the car travels 613 km
  
                                                                    613
               Using 1 ℓ petrol the car travels ─────  km
                                                                  47,32
  
                                                              613        21,4
               Using 21,4 ℓ the car travels ───── × ──  km
                                                             47,32        1
  
                                                      i.e. 277,22 km
  
  
      OR  use fuel consumption : 12,95 km/ℓ
  
               With 1 ℓ the car travels 12,95 km
  
               With 21,4 ℓ car travels 12,95  ×  21,4 km
  
                                              i.e. 277,13 km
  
                                                                 Question  5.
  
  
        5.3  Driving 613 km the car uses 47,32 ℓ
  
                                                           47,32
               Driving 1 km the car uses ─────   ℓ
                                                             613
  
                                                              47,32          430
               Driving 430 km the car uses ─────  ×  ────  ℓ
                                                                613              1
  
                                                      i.e. 33,19  ℓ
  
  
      OR  use fuel consumption : 7,72 ℓ/100 km
  
             Driving 100 km the car uses 7,72 ℓ.
  
                                                         7,72
             Driving 1 km the car uses  ───  ℓ.
                                                         100
  
                                                             7,72       430
             Driving 430 km the car uses  ───  ×  ─── ℓ
                                                             100          1
  
                                                   i.e. 33,20 ℓ
  
                                                                    Question  5.
  
  
  
  
  
Answer 6.

  
        6.1.1  Scale 1 : 50 : 1 cm on plan is 50 cm in building
  
                     1 cm on the plan is 50 cm in the building
  
                    2 cm on the plan is 50 × 2  cm in the building
  
                                            i.e. 100 cm
  
                                                                 Question  6.
  
  
  
        6.1.2      1 mm on plan is 50 mm in building
  
                    35 mm on plan is 50 × 35  mm in building
  
                                          i.e. 1 750 mm
  
                                                                 Question  6.
  
  
  
        6.1.3      1 m on plan is 50 m in building
  
                    0,35 m on plan is 50 × 0,35  m in building
  
                                           i.e. 17,50 m
  
                                                                 Question  6.
  
  
        6.2.1  Scale 1 : 50 : 50 m in building is 1 m on plan
  
                            50 m in building is 1 m on plan
  
                                                           1
                            1 m in building is  ──  m on plan
                                                          50
  
                                                         1
                     35,4 m in building is  ──  ×  35,4  m on plan
                                                        50
  
                                              i.e.  0,708 m  or  70,8 cm
  
                                                                 Question  6.
  
  
  
        6.2.2  Scale 1 : 50 : 50 m in building is 1 m on plan
  
                    50 m in the building is 1 m on the plan
  
                                                   1
                    1 m in building is  ──  m on plan
                                                  50
  
                                                      1
                    0,4 m in building is  ──  ×  0,4  m on plan
                                                     50
  
                                            i.e.  0,008 m  or  0,8 cm
  
                                                                 Question  6.
  
  
  
        6.2.3  Scale 1 : 50 : 50 m in building is 1 m on plan
  
                    50 m in building is 1 m on plan
  
                                                     1
                      1 m in building is  ──  m on plan
                                                    50
  
                                                        1
                    1,26 m in building is  ──  ×  1,26  m on plan
                                                       50
  
                                            i.e.  0,0252 m  or  2,52 cm
  
  
                                                                    Question  6.
  
  
  
Answer 7.

  
        7.1  Volume is directly proportional to mass because
  
                            mass
                         ──────  is a constant.
                           volume
  
                                                                    Question  7.
  
  
  
        7.2  1 liter  =  1 000  cm3  and  1 kg  =  1 000 g 
  
                                                        mass
                  density of water   =  ───────
                                                       volume
  
                                                     1 000 g
                                              =  ───────
                                                    1 000 cm3
  
                                             =  1 g/cm3
  
                                                                    Question  7.
  
  
  
                                                                 mass
        7.3.1   density of a substance  =   ───────
                                                                volume
  
                    mass  =  density of substance  ×  volume
  
  
                             mass  =  19,3 g/cm3  ×  6 cm3
  
                                        =  115,8 g
  
  
                                                              mass
        7.3.2   density of substance  =   ───────
                                                             volume
  
                                                           mass
                  so that  volume  =  ──────────────
                                                   density of substance
  
  
                                                    413,02 g
                  so that  volume  =  ─────────
                                                   19,3 g/cm3
  
                                             =  21,4 cm3
  
  
                                                                mass
        7.3.3   density of substance  =   ───────
                                                              volume
  
                                                              41,2 g
                                                      =  ───────
                                                             2,4 cm3
  
                                                      =  17,167g/cm3
  
                  It is less then the density of pure gold and
  
                  thus the ring is not made from pure gold.
  
                                                                    Question  7.
  
  
  
                                                      mass
        7.4.1   density of lead  =   ───────
                                                    volume
  
                                                       24,97 g
                                                =  ───────
                                                       2,2 cm3
  
                                                =  11,35g/cm3
  
  
                                                               mass
        7.4.2   density of substance  =   ───────
                                                             volume
  
                                                           mass
                  so that  volume  =  ──────────────
                                                   density of substance
  
  
                                                     181,6 g
                  so that  volume  =  ─────────
                                                   11,35 g/cm3
  
                                             =  16 cm3
  
  
                                                               mass
        7.4.2   density of substance  =   ───────
                                                              volume
  
                                                              41,2 g
                                                      =  ───────
                                                             2,4 cm3
  
                                                      =  17,167g/cm3
  
  
        7.4.3   volume of cube   =  side3  =  (2 cm)3
  
                                                = 8 cm3
  
                                                                  mass
                     density of substance  =   ───────
                                                                volume
  
                    mass  =  density of substance  ×  volume
  
  
                               mass  =  11,35 g/cm3  ×  8 cm3
  
                                          =  90,80 g
  
  
                                                                 Question  7.