WISKUNDE
GRAAD 12
NOG OEFENINGE
Enkelvoudige en saamgestelde rente, huurkoop : antwoorde.
MATHEMATICS
GRADE 12
MORE EXERCISES
Simple and compound interest and hire purchase : answers.
Antwoord / Answer 1
Gebruik die volgende formules : / Use the following formulae :
I = Pin en / and A = P + I
waar / where I = rente / interest; P = kapitaal / capital; i = r/100;
A = die bedrag uitbetaal / amount paid out en / and
n = tydperk in jaar / duration (time) in years.
ook / also
Prn
820 × 6 × 2
Prn
110 200 × 8,5 × 3
I = ──── = R ──────────
1.2
I = ──── = R ───────────────
100
100
100
100
= R98,40
= R28 101
A = P + I= R820 + 98,40
A = P + I= R110 200 + 28 101
= R918,40
= R138 301
a = 98,40; b =918,40
c = 28 101; d =138 301
Prn
14 600 × 6,2 × 4
Prn
118 150 × 7,8 × 5
I = ──── = R ──────────
I = ──── = R ───────────────
100
100
100
100
= R3 620,80
= R46 117,50
A = P + I= R14 600 + 3 620,80
A = P + I= R118 150 + 46 117,50
= R18 220,80
= R164 367,50
e = 3 620,80; f =18 220,80
g = 46 117,50; h =164 367,50
Ant. / Ans. 1.5
Ant. / Ans. 1.6
Prn
120 218 × 13,3 × 2,5
Prn
820 020 × 9,8 × 1,25
I = ──── = R ────────────────
I = ──── = R ───────────────
100
100
100
100
= R39 972,49
= R100 452,50
A = P + I= R120 218 + 39 972,49
A = P + I= R820 020 + 100 452,50
= R160 190,49
= R920 472,50
k = 39 972,49; m =160 190,49
n = 100 452,50; p =920 472,50
Ant. / Ans. 1.7
Ant. / Ans. 1.8
I = A − P = R 495 − 450
I = A − P = R 28 320 − 23 600
= R 45
= R 4 720
100I
100 × 45
100I
100 × 4 720
n = ──── = ────────
n = ──── = ─────────
Pr
Pr
450 × 5
23 600 × 8
= 2
= 2,5
q = 2 t = 45
s = 2,5 t = 4 720
Ant. / Ans. 1.9
Ant. / Ans. 1.10
I = A − P = R 919 921 − 720 800
I = A − P = R 36 957,04 − 31 620
= R 199 121
= R 5 337,04
100I
100 × 199 121
100I
100 × 5 337,04
n = ──── = ───────────
n = ──── = ───────────
Pr
Pr
720 800 × 8,5
31 620 × 11,25
= 3,25
= 1,5
u = 3,25; v = 199 121
w = 1,5; x = 5 337,04
Ant. / Ans. 1.11
Ant. / Ans. 1.12
I = A − P = R 918,40 − 820
I = A − P = R 63 569,70 − 51 060
= R 98,40
= R 12 509,70
100I
100 × 98,40
100I
100 × 12 509,70
r = ──── = ───────────
r = ──── = ───────────
Pn
Pn
820 × 2
51 060 × 3,5
= 6
= 7
y = 6; z = 98,40
aa = 7; ab = 12 509,70
Ant. / Ans. 1.13
Ant. / Ans. 1.14
I = A − P = R 745 188,92 − 616 814,42
I = A − P = R 18 538,01 − 15 861,4
= R 128 374,50
= R 2 676,61
100I
100 × 128 374,50
100I
100 × 2 676,61
r = ──── = ─────────────
r = ──── = ─────────────
Pn
Pn
616 814,42 × 2,25
15 861,40 × 1,25
= 9,25
= 13,50
ac = 9,25; ad = 128 374,50
ae = 13,50; af = 2 676,61
Ant. / Ans. 1.15
Ant. / Ans. 1.16
I = A − P = R1 051 708,20 − 725 316
P = A − I = R 572,40 − 42,4
= R 326 392,20
= R 530
100I
100 × 326 392,20
100I
100 × 42,40
r = ──── = ─────────────
n = ──── = ─────────
Pn
Pr
725 316 × 2,5
530 × 4
= 18
= 2
ag = 18; ah = 326 392,20
ai = 530; aj = 2 2
Ant. / Ans. 1.17
Ant. / Ans. 1.18
P = A − I = R 25 249,80 − 3 631,82
P = A − I = R 36 044,15 − 3 528
= R 21 617,98
= R 32 516,15
100I
100 × 3 631,82
100I
100 × 3 528
r = ──── = ─────────────
n = ──── = ─────────
Pr
Pr
21 617,98 × 5,6
32 516,15 × 1,75
= 3
= 6,2
ak = 21 617,98; am = 3
an = 32 516,15; ap = 6,2
Ant. / Ans. 1.19
Ant. / Ans. 1.20
P = A − I = R 24 649,21 − 5 779
P = A − I = R 91 601,56 − 32 976,56
= R 18 870,21
= R 58 625
100I
100 × 5 779
100I
100 × 32 976,56
r = ──── = ─────────────
n = ──── = ─────────
Pr
Pr
18 870,21 × 8,75
58 625 × 12,5
= 3,5
= 4,5
aq = 18 870,21; ar = 3,5
as = 58 625; at = 4,5
Antwoorde / Answers 2.
Gestel die kapitaal vir eerste periode is P1
Let the principal for the first period be P1
die rentekoers is r% per jaar, die rente
the interest rate is r% per annum (p.a.), the
word m keer per jaar saamgestel en die
interest is compounded m times per annum
bedrag word vir t jaar belê.
and the amount is invested for t years.
Bereken nou die rentekoers per periode,
Calculate the interest rate per period, i, and
i en die aantal periodes.
the number of periods.
Die aantal periodes = n = m x t
The number of periods = n = m x t
r
r
i (rentekoers per periode) = ──────
i (interest rate per period) = ──────
100 × m
100 × m
Bereken die rente vir die eerste periode, I1
Calculate the interest for the first period, I1
I1 = P1 × i × 1
I1 = P1 × i × 1
Tel dit by P1 om die kapitaal vir die
Add it to P1 to form the principal for
2de periode te vorm, P2
the 2nd period, P2
Herhaal die proses vir die totale
Repeat the process for the total number
aantal periodes.
of periods.
Antwoord / Answer 2.1
P = P1 = R18 320; r = 3,2% p.a.,
t = 3 jaar / years en rente word jaarliks saamgestel
/
and interest is compounded yearly.
m (aantal periodes per jaar / number of periods per year) = 1 {jaarliks saamgestel / compounded yearly}
n (totale aantal periodes / total number of periods) = 1 x 3 = 3
r
3,2
i (rentekoers per periode / interest rate per period) = ────────
= ───────
100 × m
100 × 1
Rente 1ste periode / Interest 1st period =
I1 = Pi1
= R18 320 × 0,032 × 1 = R586,24
Kapitaal 2de periode / Principal 2nd period =
P2 = P1 + I1
= R18 320 + R586,24 = R18 906,24
Rente 2de periode / Interest 2nd period =
I2 = P2i1
= R18 906,24 × 0,032 × 1 = R605,00
Kapitaal 3de periode / Principal 3rd period =
P3 = P2 + I2
= R18 605,24 + R605,00 = R19 511,24
Rente 3de periode / Interest 3rd period =
I3 = P3i1
= R19 511,24 × 0,032 × 1 = R624,36
Bedrag uitbetaalbaar / Amount paid out = A =
P3 + I3
= R19 511,24 + R624,36 = R20 135,00
Rente / Interest = A − P = R20 135,60 − R18 320,00 = R1 815,60
a = R1 815,60; b = R20 135,60
Antwoord / Answer 2.2
P = P1 = R58 610; r = 4,8% p.a.,
t = 1,5 jaar / years en rente word half-jaarliks saamgestel
/
and interest is compounded semi-annually.
m (aantal periodes per jaar / number of periods per year) = 2
n (totale aantal periodes / total number of periods) = 1,5 x 2 = 3
r
4,8
i (rentekoers per periode / interest rate per period) = ────────
= ───────
100 × m
100 × 2
Rente 1ste periode / Interest 1st period =
I1 = Pin
= R58 610 × 0,024 × 1 = R1 406,64
Kapitaal 2de periode / Principal 2nd period =
P2 = P1 + I1
= R58 610 + R1 406,64 = R60 016,64
Rente 2de periode / Interest 2nd period =
I2 = P2in
= R60 016,64 × 0,024 × 1 = R1 440,40
Kapitaal 3de periode / Principal 3rd period =
P3 = P2 + I2
= R60 016,64 + R1 440,40 = R61 457,04
Rente 3de periode / Interest 3rd period =
I3 = P3i1
= R19 511,24 × 0,032 × 1 = R1 474,97
Bedrag uitbetaalbaar / Amount paid out = A =
P3 + I3
= R61 457,04 + R1 440,40 = R62 932,01
Rente / Interest = A − P = R62 932,01 − R58 610,00 = R4 322,01
c = R4 322,01; d = R62 932,01
Antwoord / Answer 2.3
Antwoord / Answer 2.4
2.3 e = R14 907,48 en / and f = R128 757,48
2.4 g = R18 410,28 en / and h = R429 220,28
Antwoord / Answer 2.5
Antwoord / Answer 2.6
2.5 k = R41 669,31 en / and
2.6 n = R2 496 839,19 en / and
m = R1 262 529,31
p = R10 927 629,19
Antwoord / Answer 2.7
Antwoord / Answer 2.8
2.7 q = R353 900,11 en / and r = R785 200,11
2.8 s = R5 255,78 en / and t = R90 905,78
Antwoord / Answer 2.9
Antwoord / Answer 2.10
2.9 u = R43 817,88 en / and v = R475 317,88
2.10 w = R1 095 289,38 en / and
x = R49 411 539,38
Antwoord / Answer 3.
3.1 Rentekoers / Interest rate = 10,2% p.j. / p.a.
30
3.2 Aantal jaar / Number of years = ─── = 2,5 jaar / years
12
r
10,2
3.3 i = ──── = ──── = 0,102
100
100
Rente / Interest = I = Pin
= R18 352,00 × 0,102 × 2,5
= R4 679,76
3.4 Totale bedrag / Total amount = Uitstaande bedrag / Outstanding amount + Rente / Interest
= R18 352,00 + R4 679,76
= R23 031,76
Totale bedrag / Total amount
3.5 Maandelikse paaiement / Monthly installmen = ───────────────────────
Aantal maande / Number of months
R23 031,76
= ─────────
30
= R767,73
3.6 Bedrag betaal / Amount paid = R30 × 767,73 = R23 031,90
Dit is R0,14 te veel. / It is R0,14 too much.
Finale paaiement / Final instalment = Totale bedrag / Total amount − 29 paaiemente / instalments
= R(23 031,76 − 29 × 767,73)
= R767,59
Antwoord / Answer 4.
Opsie / Option 1 :
P = R28 650,00; r = 8,5%; t = 1,5 jaar / years
en die rente word maandeliks saamgestel. / and the interest is compounded monthly.
Aantal periodes per jaar / Number of periods per year = 12
Totale aantal periodes / Total number of periods = 12 × 1,5 = 18
8,5
i = ────── = 0,00708333
100 × 12
Rente 1ste periode / Interest 1st period = I1 = P1 × i
= R28 650,00 × 0,00708333 = R202,94
Bedrag / Amount = A1 = P2 = P1 + I1
P2 = R28 650,00 + R202,94 = R28 852,94
P3 = P2 + I2 = P2 + P2 × i
= R28 852,94 × 0,00708333 = R28 852,94 + R204,37 = R29 057,31
Herhaal tot / Repeat until A = P19
OF / OR P3 = P2 + + P2 × i
= P2 × (1 + i) = P2 × (1 + 0,00708333) = P2 × 1,00708333
P4 = R29 057,31 × 1,00708333 = R29 263,13
P5 = R29 263,13 × 1,00708333 = R29 470,41
P6 = R29 470,41 × 1,00708333 = R29 679,16
P7 = R29 679,16 × 1,00708333 = R29 889,39
P8 = R29 889,39 × 1,00708333 = R30 101,11
P9 = R30 101,11 × 1,00708333 = R30 314,33
P10 = R30 314,33 × 1,00708333 = R30 529,06
P11 = R30 529,06 × 1,00708333 = R30 745,31
P12 = R30 745,31 × 1,00708333 = R30 963,09
P13 = R30 963,09 × 1,00708333 = R31 182,41
P14 = R31 182,41 × 1,00708333 = R31 403,29
P15 = R31 403,29 × 1,00708333 = R31 625,73
P16 = R31 625,73 × 1,00708333 = R31 849,75
P17 = R31 849,35 × 1,00708333 = R32 075,35
P18 = R32 075,35 × 1,00708333 = R32 302,55
P19 = R32 302,55 × 1,00708333 = R32 531,36
Die bedrag / The amount = R32 531,36
Opsie 2 / Option 2 :
P = R28 650,00; r = 10%; t = 2,5 jaar / years
en die rente word kwartaalliks saamgestel / and the interest is compounded quarterly
Aantal periodes per jaar / Number of periods per year = 4
Total aantal periodes / Total number of periods = 2,5 × 4 = 10
10
i = ─────── = 0,025
100 × 4
Net soos in Opsie 1 / Like in Option 1 Pn = Pn − 1 + + Pn − 1 × i
= Pn − 1 × (1 + i) = Pn − 1 × (1 + 0,025) = Pn − 1 × 1,025
P2 = R28 650,00 × 1,025 = R29 366,25
P3 = R29 366,25 × 1,025 = R30 100,41
P4 = R30 100,41 × 1,025 = R30 852,92
P5 = R30 852,92 × 1,025 = R31 624,24
P6 = R31 624,24 × 1,025 = R32 414,85
P7 = R32 414,85 × 1,025 = R33 225,22
P8 = R33 225,22 × 1,025 = R34 055,85
P9 = R34 055,85 × 1,025 = R34 907,25
P10 = R34 907,25 × 1,025 = R35 779,93
P11 = R35 779,93 × 1,025 = R36 674,43
Total bedrag / Total amount = R36 674,43
Opsie 1 is die goedkoopste. / Option 1 is the cheaper one.
Antwoord / Answer 5.
5.1 3 jaar. / 3 years.
5.2 Inflasie word m.b.v. enkelvoudige renteformules bereken. /
Inflation is calculated by using simple interest formulae.
P = R18 550;
r = 5% en / and t = 3 jaar / years
5
i = ──── = 0,05
100
A = P(1 + in)
= R18 550(1 + 0,05 × 3)
= R21 332,50
Na 3 jaar kos die meubels R21 332,50 / After 3 years the price of the furniture is R21 332,50
5.3 A = P(1 + in)
R21 332,50 = R15 700 (1 + i × 3)
21 332,50
1 + 3i = ──────── = 1,3588579....
15 700
3i = 0,3588579....
0,3588579....
i = ──────── = 0,11958....
3
r = 0,11958.... × 100% = 12%
Die rentekoers is 12% per jaar. / The interest rate is 12% p.a.
Antwoord / Answer 6.
Deposito / Deposit
100
6.1 Persentasie deposito / Percentage of deposit = ──────────────── × ──── %
Koopprys / Cost Price
1
250
100
= ───── × ──── %
2500
1
= 10%
6.2 Bedrag verskuldig / Outstanding amount = Koopprys / Cost price − deposito / deposit
= R2 500 − R250
= R2 250
10
24
Rente op uitstaande bedrag / Interest on outstanding amount = Pin = R2 250 × ──── × ────
100
12
= R450
Totale bedrag verskuldig / Total outstanding amount = R2 250 + R450
= R2 700
2 700
Maandelikse paaiement / Monthly instalment = R ─────
24
= R112,50
6.3 Total bedrag betaal / Total amount paid = Deposito / Deposit + 24 paaiemente / instalments
= R250 + R2 700
= R2 950
Antwoord / Answer 7.
7.1 Deposito betaal / Deposit paid = 10% van koopprys / of the cost price
10
= R8 500 × ────
100
= R850
7.2 Bedrag geleen / Amount borrowed = R8 500 − R850
= R7 650
8
24
Rente verskuldig / Interest due = R7 650 × ──── × ────
100
12
= R1 224
Totale lening / Total amount due = R7 650 + R1 224
= R8 874
Bedrag verskuldig / Amount due
R8 874
Paaiement betaalbaar / Instalment due = ──────────────────────── = ────────
Aantal maande / Number of months
24
= R369,75
Antwoord / Answer 8.
8.1 Deposito betaal / Deposit paid = 10% van koopprys / of the cost price
10
= R95 000 × ────
100
= R9 500
8.2 Bedrag geleen / Amount of loan = Koopprys / Cost price − Deposito / Deposit
= R95 000 − R9 500
= R85 500
13
54
8.3 Rente verskuldig / Interest due = R85 500 × ──── × ───
100
12
= R50 017,50
Totale lening / Total amount due = R85 500 + R50 017,50
= R135 517,50
Bedrag verskuldig / Amount due
R135 517,50
Paaiement betaalbaar / Instalment due = ──────────────────────── = ──────────
Aantal maande / Number of months
54
= R2 509,58
8.4 Finale paaiement = Totale uitstaande bedrag − Totaal van 53 paaiemente. /
Final instalment = Total outstanding amount − Total of 53 instalments.
Finale paaiement / Final instalment = R135 517,50 − 53 × R2 250,00
= R1 957,50
Antwoord / Answer 9.
9.1 Deposito betaal / Deposit paid = 15% van koopprys / of the cost price
10
= R25 000 × ────
100
= R3 750
9.2 Bedrag geleen / Amount of loan = Koopprys / Cost price − Deposito / Deposit
= R25 000 − R3 750
= R21 250
12
24
Rente verskuldig / Interest due = R21 250 × ──── × ───
100
12
= R5 100,00
Totale bedrag verskuldig / Total amount due = R21 250 + R5 100,00
= R26 350,00
Bedrag verskuldig / Amount due
R26 350,00
9.3 Paaiement betaalbaar / Instalment due = ──────────────────────── = ──────────
Aantal maande / Number of months
24
= R1 097,9166666... = R1 097,92
9.4 Koste / Cost = Deposito / Deposit + 24 paaiemente / instalments
= R3 750 + 24 × R1 097,92
= R30 100,08
9.5 Finale paaiement = Totale uitstaande bedrag − Totaal van 23 paaiemente. /
Final instalment = Total outstanding amount − Total of 23 instalments.
Finale paaiement / Final instalment = R26 350,00 − 23 × R1 100,00
= R1 050,00
Antwoord / Answer 10.
12
10.1 Deposito / Deposit = R38 500 × ────
100
= R4 620
Lening / Loan = Koopprys / Cost price − deposito / deposit
= R38 500 − 4 620
= R33 880
14
Rente / Interest = R33 880 × ────
100
= R11 858
Totale lening / Total Loan = (Koopprys / Cost price − deposito / deposit) + rente / interest
= (R38 500 − 4 600) + R11 858
= R45 738
R45 738
Paaiement / Instalment = ───────
30
= R1 524,60
10.2 Hy betaal / He pays = deposito / deposit + paaiemente / instalments
= R4 620 + 30 × R1 524,60
= R50 358
R45 738
10.3.1 Aantal paaiemente / Number of instalments = ────────
R1 630
= 28,06
Hy betaal 28 paaiemente. / He pays 28 instalments.
10.3.2 Laaste paaiement / Final instalment = R45 738 − 27 × R1 630
= R1 728
10.3.3 Hy betaal nou / He now pays = deposisto / deposit + 28 paaiemente / instalments
= R4 620 + 27 × R1 630 + R1 728
= R50 358
10.3.4 Hy spaar niks nie - die koste van die meubels is dieslfde. /
He does not save anything - the cost of the furniture remains the same.
Antwoord / Answer 11.
10
11.1 Deposito / Deposit = R35 500 × ────
100
= R3 500
Lening / Loan = Koopprys / Cost price − deposito / deposit
= R35 000 − 3 500
= R31 500
15
24
Rente / Interest = R31 500 × ──── × ────
100
12
= R9 450
Lening / Loan = (Koopprys / Cost price − deposito / deposit) + rente / interest
= (R35 000 − 3 500) + R9 450
= R40 950
11.2 Versekerings bedrag = 18% van die totale uitstaande bedrag /
Amount of insurance = 18% of the total amount owed
18
Versekering / Insurance = R40 950 × ────
100
= R7 371
11.3 Totale bedrag verskuldig / Total amount due = R40 950 + R7 391
= R40 950 + R7 391
= R48 321
R48 321
Paaiememt / Instalment = R40 950 × ───────
24
= R2 013,38
48 321
11.4.1 Aantal paaiemente / Number of instalments = ───────
2 100
= 23,01
Hy betaal 23 paaiemente - 22 gelykes en 'n verskillende laaste paaiement. /
He pays 23 instalments - 22 equal instalments and a different last instalment.
11.4.2 Laaste paaiement = Uitstaande bedrag − 22 paaiemente /
Final instalment = Amount due − 22 instalments
Laaste paaiement / Final instalment = R48 321 − 22 × R2 100
= R21
Antwoord / Answer 12.
12.1 Totale bedrag verskuldig / Total amount owed = bedrag "geleen" / amount of loan + rente / interest
Gestel hy leen / Suppose his loan is Rx
9
20 × 12 × 1 560 = x + x × ──── × 20
100
374 400 = x + 1,8x × x
2,8x = 374 400
x = 133 714,29
Sy lening is / His loan is R133 714 (korrek tot naaste Rand / Correct to nearest Rand)
12.2 Totale bedrag verskuldig / Total amount owed = bedrag "geleen" / amount of loan + rente / interest
8
= R133 714 + R133 714 × ──── × 15
100
= R294 170,80
R294 170,80
Paaiement / Instalment = ────────────
15 × 12
= R1 634,28
12.3 Rente betaal = Total bedrag betaal − bedrag geleen
Interest paid = Total amount paid − amount of loan
Rente betaal Bank A / Total amount Bank A = R374 400 − R133 714
= R240 685,20
Rente betaal Bank B / Total amount Bank B = R294 170 − R133 714
= R160 457,15
Bank B verhaal minder rente en dus is dit die beter opsie. /
Bank B charges less interest and therfore it is the better option.
Antwoord / Answer 13.
8
i = ─── en / and n = 2 jaar / years
100
Waarde na 2 jaar / Value after 2 years : A = P(1 + in)
= R15 000 (1 + 0,08 × 2)
= R17 400
Kapitaal belë na 2 jaar / Capital invested after 2 years = A(2 jaar / years) + Nog / Another
= R17 400 + 3 000 = R20 400
7
i = ─── en / and n = 3 jaar / years
100
Waarde na 5 jaar / Value after 5 years : A = P(1 + in)
= R20 400 (1 + 0,07 × 3)
= R24 684