Mathematics Standard • Year 12 • Module 7 • Lesson 3
Comparing Interest Rates — Past-Paper Style
Practise HSC-style short answers and one extended response on comparing rates — effective rates, flat-rate loans and Rule of 72.
1. Short-answer questions
1.1 Calculate the effective annual interest rate for a savings product advertised at 6.4% per annum, compounded monthly. Give your answer to 2 decimal places of a percent. 2 marks Band 3
1.2 A $20,000 car loan is offered at 6% per annum flat rate over 4 years, with equal monthly repayments.
(a) Calculate the total interest payable.
(b) Calculate the equal monthly repayment.
(c) Explain in one sentence why the true interest rate on this loan is higher than 6% p.a. 3 marks Band 3-4
1.3 A bank advertises "Your savings double in under 8 years!"
(a) Use the Rule of 72 to determine the minimum annual rate of return that would make this claim true.
(b) Compare your answer in (a) with an actual effective rate of 9% p.a. compounded annually: how many years does an investment at this rate take to double, to 1 decimal place? 4 marks Band 4
2. Extended response
2.1 Sarah is comparing three offers for a $30,000 personal loan that she needs to repay over 5 years.
Offer P (flat-rate loan): 6.5% per annum flat-rate simple interest, equal monthly repayments.
Offer Q (reducing-balance loan): total of $35,460 to be repaid in 60 equal monthly instalments.
Offer R (credit-card transfer): 9.9% p.a. compounded monthly on the outstanding balance, equal monthly repayments of $635.85 for 5 years.
(a) Calculate the total amount Sarah would repay on Offer P (interest + principal).
(b) Calculate the monthly repayment on Offer Q.
(c) Calculate the total amount repaid on Offer R.
(d) Determine the effective annual rate of Offer R, to 2 decimal places of a percent.
(e) Rank the three offers from cheapest to most expensive total cost and write a one-sentence recommendation. 7 marks Band 5-6
Explicit marking criteria
Part (a) — 1 mark
• 1 mark — correct total = P + flat-rate interest, using I = 30,000 × 0.065 × 5.
Part (b) — 1 mark
• 1 mark — correct monthly = $35,460 ÷ 60.
Part (c) — 1 mark
• 1 mark — correct total = $635.85 × 60.
Part (d) — 2 marks
• 1 mark — correct r/k = 0.0099/12 (and k = 12) identified or used.
• 1 mark — correct effective rate to 2 d.p. of a percent.
Part (e) — 2 marks
• 1 mark — correct ranking of the three total-cost figures.
• 1 mark — explicit recommendation sentence naming the cheapest offer and the dollar gap to the next-cheapest.
Your response:
Stuck on (e)? After listing the three total-cost figures, the cheapest is the smallest — and the recommendation sentence is essential for the final mark.How did this worksheet feel?
What I'll revisit before next class:
1.1 — Effective rate of 6.4% monthly (2 marks)
Sample response. Effective = (1 + 0.064/12)^12 − 1 = (1.005333)^12 − 1 = 1.06592 − 1 = 0.06592 ≈ 6.59% p.a.
Marking notes. 1 mark — correct r/k = 0.005333. 1 mark — correct effective rate to 2 d.p. (must include % sign and "p.a." or per annum).
1.2 — Flat-rate $20,000 at 6% over 4 years (3 marks)
(a) Sample response. I = 20,000 × 0.06 × 4 = $4,800.00.
(b) Sample response. Total = $20,000 + $4,800 = $24,800. Monthly = $24,800 / 48 = $516.67.
(c) Sample response. Flat-rate interest is calculated on the original $20,000 even after the borrower has repaid part of the principal, so the borrower is paying interest on money they no longer owe — the true (reducing-balance equivalent) rate is approximately 11–12% p.a.
Marking notes. (a) 1 mark — correct I. (b) 1 mark — correct monthly (rounded). (c) 1 mark — explanation must specifically mention interest charged on the original balance, not just say "it's higher".
1.3 — Rule of 72 vs exact 9% doubling (4 marks)
(a) Sample response. 72 / r = 8 ⇒ r = 9. 9% p.a.
(b) Sample response. Test (1.09)^n = 2 by trial:
n = 8: (1.09)^8 = 1.9926 (under 2)
n = 9: (1.09)^9 = 2.1719 (over 2)
Between 8 and 9: try n = 8.04: (1.09)^8.04 ≈ 2.000. So exact doubling time ≈ 8.0 years (to 1 d.p.).
Marking notes. (a) 1 mark — correct rule-of-72 estimate of 9%. (b) 1 mark — n = 8 evaluation under 2. 1 mark — n = 9 evaluation over 2. 1 mark — exact ≈ 8.0 years (accept 8.0 or 8.04).
2.1 — Three personal-loan offers (7 marks): sample Band-6 response with annotations
Sample Band-6 response.
(a) Offer P total repayment.
Interest = 30,000 × 0.065 × 5 = $9,750. Total = $30,000 + $9,750 = $39,750.00. [1 mark — correct total for Offer P.]
(b) Offer Q monthly repayment.
Monthly = $35,460 ÷ 60 = $591.00. [1 mark — correct monthly for Offer Q.]
(c) Offer R total repaid.
Total = $635.85 × 60 = $38,151.00. [1 mark — correct total for Offer R.]
(d) Offer R effective rate.
r/k = 0.099 / 12 = 0.00825. [1 mark — correct r/k identified.]
Effective = (1.00825)^12 − 1 = 1.10363 − 1 = 0.10363 ≈ 10.36% p.a. [1 mark — correct effective rate to 2 d.p.]
(e) Ranking and recommendation.
Total costs (cheapest → most expensive): Offer Q ($35,460) < Offer R ($38,151) < Offer P ($39,750). [1 mark — correct ranking.]
Recommendation: Sarah should take Offer Q — a reducing-balance loan with total cost $35,460 — saving $2,691 over Offer R and $4,290 over Offer P. [1 mark — recommendation names the cheapest offer and the dollar gap to next-cheapest.]
Total: 7/7.
Band descriptors for marker.
Band 3: Offer P total correct, but uses flat-rate working on Offers Q or R; effective rate omitted. ≈ 2-3 marks.
Band 4: Totals for all three offers correct, effective rate for R attempted but with wrong r/k; no ranking. ≈ 4-5 marks.
Band 5: Full numerical solution including correct effective rate and ranking, but recommendation is bare numbers ("Q is best") without naming the dollar gap. ≈ 6 marks.
Band 6: Complete: three correct totals, correct effective rate, correct ranking, AND recommendation sentence that names Offer Q and the dollar gap. 7/7.