WISKUNDIGE GELETTERDHEID
GRAAD 12
NOG OEFENINGE
  
Koers, verhouding en eweredigheid : antwoorde.
  
MATHEMATICAL LITERACY
GRADE 12
MORE EXERCISES
  
Rate, ratio and proportion : answers.
  
  
Antwoorde  / Answers  1  
  
  
    1.1  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 R ─ × ──
                             7      1                           7      1
  
                      d.i.  R15       en 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
  
  
             45 word verdeel
  
                    2               3                4
                   ─ × 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  ── × R34500 ; ── × R34500
                               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.
  
  
  
  
     1.1  If R7 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 ─ × ──  ; 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
  
  
              45 are divided
  
                    2                 3                  4
                   ─ × 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 parts are ── × R34500 ; ── × R34500
                                 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.
  
  
  
  
Antwoorde  / Answers  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 5 kg.
  
                                               5
          'n Massa van 1 kg word ─  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.
  
  
  
    2.1  The length of 4 m, is increased to 7 m
  
                                                             7
          The length of 1 m, is increased to  ─ 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.
  
  
  
Antwoorde  / Answers  3  
  
  
    3.1  5 km in 25 minute
  
                         25
          1 km in   ───  minute
                          5
  
          2 km in   5 × 2  minute
  
                     =  10  minute
  
                                                Vraag  3.
  
  
  
    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
  
                                                Vraag  3.
  
  
  
            OF
  
  
    3.1  5 km in 25 minute
  
                         25
          2 km in   ───  ×  2  minute
                          5
  
                  d.i.  10  minute
  
                                                Vraag  3.
  
  
  
    3.1  5 km in 25 minute
  
                            25
           1 km in   ───  minutes
                             5
  
           2 km in   5 × 2  minutes
  
                      =  10  minutes
  
                                                   Question  3.
  
  
  
    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
  
                                             i.e.  13 km
  
                                                    Question  3.
  
  
  
            OR
  
  
    3.1  5 km in 25 minutes
  
                                25
               2 km in   ───  ×  2  minutes
                                 5
  
                         d.i.  10  minutes
  
                                                    Question  3.
  
  
  
Antwoorde  / Answers  4  
  
  
    4.1  Maandag tot Vrydag is 5 dae.
  
          1 Week = 5 dae.
  
          1 Week = 5 dae teen 8 uur per dag.
  
                      = 40 uur.
  
          Weeklikse loon  =  40 uur × R30
  
                                  = R1 200 .
  
                                                Vraag  4.
  
  
  
    4.2  Sy verdien R30 in 1 uur
  
                                          1 170
          Sy verdien R1 170 in ────  uur.
                                            30
  
                                    d.i. 39 uur.
  
                                                Vraag  4.
  
  
    4.3  Vir 1 uur verdien sy R30.
  
          Vir 29 uur verdien sy R30 × 29
  
                                   i.e. R870
  
  
                                              Vraag  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
  
                                 = R1 200 hours.
  
                                                Question 4.
  
  
  
        4.2  She earns R30 in 1 hour
  
                                              1 170
               She earns R1 170 in ────  hours.
                                              30
  
                                         i.e. 39 hours.
  
                                                Question 4.
  
  
        4.3  For 1 hour she receives R30.
  
               For 29 hours she receives R30 × 29
  
                                                 i.e. R870
  
  
                                                   Question  4.
  
  
  
Antwoorde  / Answers  5  
  
  
    5.1  Met 47,32 liter petrol ry hy 613 km
  
                                               613
          Met 1 liter petrol ry hy ─────  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 hy 613 km
  
                                                    613
          Met 1 ℓ petrol ry die motor ─────  km
                                                  47,32
  
                                    613          21,4
          Met 21,4 ℓry hy ─────  ×  ───  km
                                  47,32           1
  
                           d.i. 277,22 km
  
  
      OF  gebruik brandstofverbruik : 12,95 km/ℓ
  
       Met 1 liter petrol ry hy 12,95 km
  
       Met 21,4 liter petrol ry hy 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 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 verbruik.
                                 100         1
  
                          d.i. 33,20 ℓ
  
  
                                              Vraag  5.
  
  
  
  
    5.1  Using 47,32 ℓ petrol it travels 613 km
  
                                                   613
           Using 1 ℓ petrol it 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 it travels 613 km
  
                                                  613
          Using 1 ℓ petrol it 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 it travels 12,95 km
  
            With 21,4 ℓ it 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 it uses ─────  ×  ────  ℓ
                                              613            1
  
                                       i.e. 33,19  ℓ
  
  
      OR
  
         use fuel consumption : 7,72 ℓ/100 km
  
         Driving 100 km it uses 7,72 ℓ.
  
                                        7,72
         Driving 1 km it uses  ───  ℓ.
                                        100
  
                                          7,72       430
        Driving 430 km it uses  ───  ×  ─── ℓ
                                           100         1
  
                                   i.e. 33,20 ℓ
  
  
                                                  Question  5.
  
  
  
  
Antwoorde  / Answers  6  
  
  
    6.1.1  Skaal 1 : 50
  
             1 cm op plan is 50 cm in gebou
  
             2 cm op plan is 50 × 2  cm in gebou
  
                              d.i. 100 cm
  
                                                  Vraag  6.
  
  
    6.1.2  1 mm op plan is 50 mm in gebou
  
           35 mm op plan is 50 × 35  mm in 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 gebou
  
                                d.i. 17,50 m
                                                    Vraag  6.
  
  
    6.2.1     Skaal 1 : 50
  
              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
  
                                                         [ V 6.]
  
  
    6.2.2  Skaal 1 : 50
  
             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 gebou is  ── × 0,4  m op plan
                                       50
  
                               d.i.  0,008 m  of  0,8 cm
  
                                                      [ V 6]
  
  
    6.2.3  Skaal 1 : 50
  
             50 m in gebou is 1 m op plan
  
                                        1
              1 m in gebou is  ──  m op plan
                                       50
  
                                       1
         1,26 m in gebou is  ── × 1,26 m op plan
                                      50
  
                            d.i.  0,0252 m  of  2,52 cm
  
                                                        [ V 6]
  
  
  
    6.1.1  Scale 1 : 50 :
  
             1 cm on the plan is 50 cm in the building
  
             2 cm on plan is 50 × 2  cm in 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
  
                                            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
  
                                                          [ Q 6 ]
  
  
    6.2.2  Scale 1 : 50 :
  
             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
  
                                                            [ Q 6 }
  
  
    6.2.3  Scale 1 : 50 :
  
              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
  
                                                            [ Q 6 ]
  
  
  
Antwoorde  / Answers  7  
  
  
    7.1  Volume is direk eweredig aan massa
  
                      massa
           want  ──────  is 'n konstante.
                      volume
                                                     [ V 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
  
                                                         [ V 7 ]
  
  
  
                                                 massa
    7.3.1   digtheid van stof   =   ───────
                                                 volume
  
              massa = digtheid van stof × volume
  
  
                massa = 19,3 g/cm3  ×  6 cm3
  
                          =  115,8 g
                                                       [ V 7 ]
  
  
  
                                                 massa
    7.3.2   digtheid van stof   =   ───────
                                                 volume
  
                                               massa
              sodat volume  = ───────────
                                       digtheid van stof
  
  
                                           413,02 g
                       volume  =  ─────────
                                         19,3 g/cm3
  
                                   =  21,4 cm3
  
                                                   [ V 7 ]
  
  
  
                                                 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.
                                                    [ V 7 ]
  
  
  
  
                                                 massa
    7.4.1   digtheid van lood   =   ───────
                                                 volume
  
                                                24,97 g
                                          =  ───────
                                                2,2 cm3
  
                                          =  11,35g/cm3
  
                                                      [ V 7 ]
  
  
  
                                                 massa
    7.4.2   digtheid van stof   =   ───────
                                                 volume
  
                                               massa
              sodat  volume = ───────────
                                       digtheid van stof
  
  
                                 181,6 g
             volume  =  ─────────
                              11,35 g/cm3
  
                         =  16 cm3
  
  
                                                  [ V 7 ]
  
  
  
    7.4.3   Volume van kubus   =  sy3
  
                                             = (2 cm)3
  
                                             = 8 cm3
  
                                                massa
               digtheid van stof   =   ──────
                                                volume
  
            massa = digtheid van stof × volume
  
              massa  =  11,35 g/cm3  ×  8 cm3
  
                         =  90,80 g
  
  
                                                      [ V 7 ]
  
  
  
    7.1  Volume is directly proportional to mass
  
                             mass
           because  ──────  is a constant.
                            volume
                                                     [ Q 7 ]
  
  
  
  
    7.2  1 liter = 1 000 cm3 en 1 kg = 1 000 g
  
                                           mass
           density of water = ─────── 
                                          volume
  
                                          1 000 g
                                   =  ───────
                                        1 000 cm3
  
                                   =  1 g/cm3
  
                                                         [ Q 7 ]
  
  
  
                                                     mass
    7.3.1   density of substance = ───────
                                                    volume
  
             mass = density of substance × volume
  
  
               mass  =  19,3 g/cm3  ×  6 cm3
  
                         =  115,8 g
                                                         [ Q 7 ]
  
  
  
                                                     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
  
                                                         [ Q 7 ]
  
  
  
                                                     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.
  
                                                                 [ Q 7 ]
  
  
  
  
                                               mass
    7.4.1   density of lead  =   ───────
                                              volume
  
                                            24,97 g
                                      =  ───────
                                            2,2 cm3
  
                                     =  11,35 g/cm3
  
                                                          [ Q 7.4 ]
  
  
  
                                                        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
  
  
                                                          [ Q 7.4 ]
  
  
  
  
    7.4.3   Volume of a 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
  
  
                                                          [ Q 7.3 ]