Practice Maths

Solutions — Compound Interest

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  1. Q1 — Calculate the final amount (compound interest)

    Use A = P(1 + r/n)nt for each part.

    (a) P = 2000, r = 0.05, n = 1, t = 3
    A = 2000 × (1 + 0.05/1)1×3 = 2000 × (1.05)3
    (1.05)3 = 1.157625
    A = 2000 × 1.157625 = $2 315.25

    (b) P = 8000, r = 0.04, n = 2, t = 2
    A = 8000 × (1 + 0.04/2)2×2 = 8000 × (1.02)4
    (1.02)4 = 1.08243216
    A = 8000 × 1.08243216 = $8 659.46

    (c) P = 15000, r = 0.06, n = 4, t = 5
    A = 15000 × (1 + 0.06/4)4×5 = 15000 × (1.015)20
    (1.015)20 = 1.34685501
    A = 15000 × 1.34685501 = $20 202.83

    (d) P = 500, r = 0.03, n = 12, t = 1
    A = 500 × (1 + 0.03/12)12×1 = 500 × (1.0025)12
    (1.0025)12 = 1.03041596
    A = 500 × 1.03041596 = $515.21

  2. Q2 — Total interest earned from Q1

    Interest earned = A − P for each part.

    (a) I = $2 315.25 − $2 000 = $315.25

    (b) I = $8 659.46 − $8 000 = $659.46

    (c) I = $20 202.83 − $15 000 = $5 202.83

    (d) I = $515.21 − $500 = $15.21

  3. Q3 — Effective Annual Rate (EAR)

    Use EAR = (1 + r/n)n − 1. Express as a percentage to 3 decimal places.

    (a) r = 0.06, n = 2 (semi-annual)
    EAR = (1 + 0.06/2)2 − 1 = (1.03)2 − 1 = 1.0609 − 1 = 0.0609
    EAR = 6.090%

    (b) r = 0.08, n = 4 (quarterly)
    EAR = (1 + 0.08/4)4 − 1 = (1.02)4 − 1 = 1.08243216 − 1 = 0.08243216
    EAR = 8.243%

    (c) r = 0.12, n = 12 (monthly)
    EAR = (1 + 0.12/12)12 − 1 = (1.01)12 − 1 = 1.12682503 − 1 = 0.12682503
    EAR = 12.683%

  4. Q4 — $6 000 at 5.4% p.a. compounding monthly

    (a) P = 6000, r = 0.054, n = 12, t = 3
    A = 6000 × (1 + 0.054/12)12×3 = 6000 × (1.0045)36
    (1.0045)36 = 1.17398854
    A = 6000 × 1.17398854 = $7 043.93

    (b) Interest earned = $7 043.93 − $6 000 = $1 043.93

    (c) Simple interest: A = P(1 + rt) = 6000 × (1 + 0.054 × 3) = 6000 × 1.162 = $6 972.00
    Simple interest earned = $6 972.00 − $6 000 = $972.00
    Extra earned with compound interest = $1 043.93 − $972.00 = $71.93 more with compound interest

  5. Q5 — $10 000 at 6% p.a. comparing compounding periods

    P = 10 000, r = 0.06, t = 5 years.

    (a) Annual (n = 1):
    A = 10000 × (1.06)5 = 10000 × 1.33822558 = $13 382.26

    (b) Quarterly (n = 4):
    A = 10000 × (1 + 0.06/4)20 = 10000 × (1.015)20
    (1.015)20 = 1.34685501
    A = 10000 × 1.34685501 = $13 468.55

    (c) Monthly (n = 12):
    A = 10000 × (1 + 0.06/12)60 = 10000 × (1.005)60
    (1.005)60 = 1.34885015
    A = 10000 × 1.34885015 = $13 488.50

    (d) Pattern: As compounding becomes more frequent, the final amount increases, but by smaller and smaller amounts. Annual gives $13 382.26, quarterly gives $13 468.55 (an increase of $86.29), and monthly gives $13 488.50 (an additional increase of only $19.95). Increasing compounding frequency always increases the final amount, but with diminishing returns as frequency increases.

  6. Q6 — Find the principal needed to reach a target

    Rearrange A = P(1 + r/n)nt to find P = A ÷ (1 + r/n)nt.

    (a) A = 15000, r = 0.05, n = 1, t = 4
    Growth factor = (1.05)4 = 1.21550625
    P = 15000 ÷ 1.21550625 = $12 340.00 (to nearest cent)

    (b) A = 50000, r = 0.04, n = 12, t = 10
    Growth factor = (1 + 0.04/12)120 = (1.003333)120
    (1.003333)120 = 1.49083241
    P = 50000 ÷ 1.49083241 = $33 540.46

  7. Q7 — Asha’s investment: $3 000 at 7% p.a. annual compounding

    (a) A = 3000 × (1.07)t for t = 1, 2, 3, 4, 5:

    Year (t) Balance (A) Interest that year
    1$3 210.00$210.00
    2$3 434.70$224.70
    3$3 675.13$240.43
    4$3 932.39$257.26
    5$4 207.66$275.27

    (b) At the end of Year 4, the balance is $3 932.39, which is still under $4 000.
    At the end of Year 5, the balance is $4 207.66, which exceeds $4 000.
    The balance first exceeds $4 000 at the end of Year 5.

    (c) Simple interest at 7% p.a.: A = 3000(1 + 0.07t)
    Year 1: $3 210 — same as compound (both earn $210 first year)
    Year 2: $3 420 (compound = $3 434.70, difference = $14.70)
    Year 3: $3 630 (compound = $3 675.13, difference = $45.13)
    Year 4: $3 840 (compound = $3 932.39, difference = $92.39)
    Year 5: $4 050 (compound = $4 207.66, difference = $157.66)
    With simple interest, the balance reaches $4 000 during Year 5 (at t = 4.76 years), but the compound account gets there earlier, at the end of Year 5. The compound account pulls further ahead each year as interest on interest accumulates.

  8. Q8 — Account A (5.8% annual) vs Account B (5.6% monthly)

    (a) EAR = (1 + r/n)n − 1
    Account A (annual, n = 1): EAR = (1.058)1 − 1 = 5.800%
    Account B (monthly, n = 12): EAR = (1 + 0.056/12)12 − 1 = (1.004667)12 − 1
    (1.004667)12 = 1.05750786
    EAR = 1.05750786 − 1 = 0.05750786 = 5.751%

    (b) Account A EAR = 5.800%, Account B EAR = 5.751%.
    Account A gives a better return despite Account B compounding more frequently, because Account A’s nominal rate (5.8%) is high enough to overcome the benefit of more frequent compounding.

    (c) Calculate final amount for $20 000 over 3 years in each account:
    Account A: A = 20000 × (1.058)3 = 20000 × 1.18416378 = $23 683.28
    Account B: A = 20000 × (1.004667)36 = 20000 × (1.004667)36
    (1.004667)36 = 1.18367613
    Account B: A = 20000 × 1.18367613 = $23 673.52
    Difference = $23 683.28 − $23 673.52 = Account A earns $9.76 more over 3 years.

  9. Q9 — Doubling time for $5 000 at 6% p.a. annual

    (a) A = 5000 × (1.06)t. We need A > $10 000, i.e., (1.06)t > 2.

    t (years) A ($)
    1$5 300.00
    2$5 618.00
    4$6 312.38
    6$7 092.60
    8$7 969.24
    10$8 954.24
    11$9 491.50
    12$10 061.99

    At t = 11, A = $9 491.50 (below $10 000). At t = 12, A = $10 061.99 (above $10 000).
    The investment first exceeds $10 000 after 12 years.

    (b) Rule of 72: estimated doubling time = 72 ÷ 6 = 12 years.
    The Rule of 72 gives exactly 12 years, which matches the trial-and-error result perfectly. The Rule of 72 is a remarkably accurate shortcut for interest rates in the typical range of 4–12%.

  10. Q10 — Marcus’s car loan: $25 000 at 7.2% monthly for 5 years

    (a) P = 25000, r = 0.072, n = 12, t = 5
    A = 25000 × (1 + 0.072/12)12×5 = 25000 × (1.006)60
    (1.006)60 = 1.43204457
    A = 25000 × 1.43204457 = $35 801.11

    (b) Total interest = $35 801.11 − $25 000 = $10 801.11

    (c) Simple interest at 7.2% for 5 years:
    A = 25000 × (1 + 0.072 × 5) = 25000 × 1.36 = $34 000.00
    Simple interest cost = $34 000 − $25 000 = $9 000.00
    Compound interest cost = $10 801.11
    Difference = $10 801.11 − $9 000.00 = $1 801.11 more under compound interest.
    The compound interest loan is significantly more expensive because interest is calculated on the growing balance each month, not just on the original $25 000.