Mathematics • Year 10 • Unit 1 • Lesson 6
Compound Interest — Mixed Challenge
Pull together everything from Lesson 6 (and the simple interest rule from earlier in the unit): different compounding periods, comparisons, working backwards, and a classic "spot the mistake" question. Choose the right tool for each problem.
1. Mixed problems — choose the right setup
Each problem mixes compounding frequency, simple vs compound, or "find the missing number". Decide which setup applies before you start. Show working. 3 marks each
1.1 $9,500 is invested at 5.8% p.a. compounded annually for 6 years. Find the total amount and the compound interest earned.
1.2 $5,000 is invested at 6% p.a. for 4 years. Compare: (a) simple interest total, (b) annually compounded total, (c) monthly compounded total. State which is largest.
1.3 A super fund balance grows from $40,000 to $48,620 in 4 years with annual compounding. Use trial values (4%, 4.5%, 5%, 5.5%, 6%) to find the interest rate p.a. correct to the nearest 0.5%.
1.4 $2,000 is invested at 7% p.a. compounded annually. After how many whole years will the balance first exceed $5,000?
1.5 An ubank Save account earns 5.1% p.a. compounded monthly. Aisha deposits $4,500 and leaves it for 30 months. Find the final balance and the interest earned.
1.6 An account compounded quarterly grows from $8,000 to $9,212.96 in 3 years. Show that the annual interest rate is approximately 4.7% p.a. Use trial values 4.4%, 4.7%, 5.0%.
2. Find the mistake
A student is asked to find the total amount when $10,000 is invested at 6% p.a. compounded quarterly for 2 years. Their working is below. Exactly one line contains a mistake. Spot it, explain why, then re-do the working correctly. 3 marks
Student's working — $10,000 at 6% p.a. quarterly for 2 years:
Line 1: P = 10,000, annual rate = 6% = 0.06, T = 2 years
Line 2: Quarterly rate R = 0.06 ÷ 4 = 0.015
Line 3: Number of periods n = 2 (because 2 years)
Line 4: A = 10,000 × (1.015)² = 10,000 × 1.030225 = $10,302.25
Line 5: So the total is $10,302.25.
(a) Which line contains the mistake?
(b) Explain in one or two sentences why that line is wrong.
(c) Write out the corrected working in full, including the corrected final answer.
Stuck? Lesson § "Misconceptions" warns about this exact slip — when you divide the rate by 4, you must also multiply the number of years by 4.3. Open-ended challenge — the "Rule of 72"
This question has more than one valid answer — there are several setups that work. 4 marks
3.1 The "Rule of 72" is a banking shortcut: it claims that an investment at r% p.a. compounded annually will roughly double in 72 ÷ r years. For example, at 8% p.a. it should take about 9 years to double.
Find two different (rate, time) pairs that the Rule of 72 predicts will double an investment. For each pair:
(i) State the rate and the predicted doubling time.
(ii) Use the compound interest formula to compute the actual factor (1 + R)ⁿ.
(iii) Comment on whether the rule's prediction is close (within about 5% of doubling, i.e. final factor between 1.95 and 2.05).
Bonus: For one of your pairs, find the smallest whole-year time that gives a factor of at least 2.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — $9,500 at 5.8% annually for 6 years
A = 9,500 × (1.058)⁶ = 9,500 × 1.402552 = $13,324.24.
Compound interest = 13,324.24 − 9,500 = $3,824.24.
1.2 — $5,000 at 6% for 4 years (three methods)
(a) Simple: I = 5,000 × 0.06 × 4 = $1,200 → Total = $6,200.00.
(b) Annual compound: 5,000 × (1.06)⁴ = 5,000 × 1.262477 = $6,312.38.
(c) Monthly compound: R = 0.005, n = 48. A = 5,000 × (1.005)⁴⁸ = 5,000 × 1.270489 = $6,352.45.
Largest: monthly compound, as the lesson predicts (more frequent compounding wins).
1.3 — Find the rate (super: $40,000 → $48,620 in 4 years)
We need (1 + r)⁴ = 48,620 / 40,000 = 1.21550.
Trials: (1.04)⁴ = 1.1699; (1.045)⁴ = 1.1925; (1.05)⁴ = 1.2155 ✓; (1.055)⁴ = 1.2388.
Rate ≈ 5% p.a.
1.4 — $2,000 at 7% to exceed $5,000
Try n = 13: 2,000 × (1.07)¹³ = 2,000 × 2.4098 = $4,819.65 — under.
Try n = 14: 2,000 × (1.07)¹⁴ = 2,000 × 2.5785 = $5,157.06 — over!
Answer: 14 years.
1.5 — ubank Save: $4,500 at 5.1% monthly for 30 months
R = 0.051 ÷ 12 = 0.00425; n = 30.
A = 4,500 × (1.00425)³⁰ = 4,500 × 1.136058 = $5,112.26.
Interest earned = 5,112.26 − 4,500 = $612.26.
1.6 — Find the rate (quarterly: $8,000 → $9,212.96 in 3 years)
Need (1 + R)¹² = 9,212.96 / 8,000 = 1.15162.
Trial 4.4%: R = 0.011, (1.011)¹² = 1.1404 — under.
Trial 4.7%: R = 0.01175, (1.01175)¹² = 1.1505 — close ✓.
Trial 5.0%: R = 0.0125, (1.0125)¹² = 1.1608 — over.
Annual rate ≈ 4.7% p.a.
2 — Find the mistake
(a) The mistake is on Line 3.
(b) The student forgot that when the rate is divided by 4 (quarterly), the number of periods must also be multiplied by 4. n is the number of compounding periods, not the number of years. So n = 2 × 4 = 8, not 2.
(c) Corrected working:
P = 10,000, R = 0.015 per quarter, n = 2 × 4 = 8 quarters.
A = 10,000 × (1.015)⁸ = 10,000 × 1.126493 = $11,264.93.
This is the exact pitfall flagged in the lesson's "Misconceptions" card — quarterly = ÷ 4 for the rate AND × 4 for the periods.
3 — Rule of 72 (sample solution)
Pair 1: r = 8% p.a., predicted doubling time = 72 ÷ 8 = 9 years.
Actual factor: (1.08)⁹ = 1.9990. Within 5% of 2 → rule is excellent here.
Pair 2: r = 6% p.a., predicted doubling time = 72 ÷ 6 = 12 years.
Actual factor: (1.06)¹² = 2.0122. Within 5% of 2 → rule is excellent.
Other valid pairs: r = 12% with n = 6 gives (1.12)⁶ = 1.9738 (close, just under); r = 4% with n = 18 gives (1.04)¹⁸ = 2.0258 (close, just over). The rule works best for rates between roughly 4% and 12%.
Bonus. Using Pair 1 (r = 8%): try n = 8 → 1.851; n = 9 → 1.999; n = 10 → 2.159. The smallest whole-year time where the factor reaches 2 is n = 10 years (the rule's "9 years" is just shy).
Marking: 2 marks per valid pair (one for the calculation, one for the "close to 2" judgement). 4 in total. Bonus is recognition — award full marks for any complete pair.