Mathematics Standard • Year 12 • Module 7 • Lesson 3
Comparing Interest Rates — Problem Set
Apply effective-rate, flat-rate and doubling-time reasoning to choose between real Australian banking, credit and lending offers.
Problem 1 — Two high-interest savings accounts
Daniel is choosing where to park $40,000 of savings for 3 years.
Bank A: 6.2% p.a. compounded monthly.
Bank B: 6.25% p.a. compounded semi-annually.
Set up: What are we solving for?
(i) Calculate the effective annual rate for Bank A, to 2 decimal places of a percent. 2 marks
(ii) Calculate the effective annual rate for Bank B, to 2 decimal places of a percent. 2 marks
(iii) Recommend a bank for Daniel and justify in one sentence using the effective rates. 1 mark
Stuck? Revisit lesson § Comparing Rates — the higher nominal rate does NOT always win.Problem 2 — Flat-rate vs reducing-balance car loan
Hailey is buying a $15,000 used car and has two financing options over 3 years.
Loan A (flat rate): 8% p.a. flat-rate simple interest.
Loan B (reducing balance): 10% p.a. reducing balance, monthly repayment of $484.01.
Set up: What are we solving for?
(i) Calculate the total interest payable on Loan A. 1 mark
(ii) Calculate the total amount paid on Loan B (over 36 months). 1 mark
(iii) Which loan is cheaper for Hailey and by how much? 2 marks
(iv) Explain in one sentence why a 10% reducing-balance loan can actually be cheaper than an 8% flat-rate loan. 1 mark
Stuck? Revisit lesson § Flat Rate — flat-rate interest is charged on the original principal even after you've repaid most of it.Problem 3 — "0% interest" store finance
A furniture store advertises a $3,600 lounge suite on "0% interest for 18 months", but adds a $360 establishment fee plus a $5 per month account-keeping fee. The total ($3,600 + $360 + 18 × $5 = $4,050) is then spread over 18 equal monthly payments.
Set up: What are we solving for?
(i) Calculate the total amount actually paid by the customer. 1 mark
(ii) Calculate the monthly instalment. 1 mark
(iii) Express the $450 of extra charges as an equivalent simple-interest rate per annum on $3,600 over 1.5 years. 2 marks
(iv) Explain in one sentence why "0% interest" is misleading marketing. 1 mark
Stuck? Revisit lesson § Activity 2 Q1 — convert any add-on fee to an equivalent interest rate.Problem 4 — Estimating doubling times
A financial adviser claims that any investment doubling in 12 years or less is "good enough", and advises clients accordingly.
Set up: What are we solving for?
(i) Using the Rule of 72, estimate the minimum annual rate of return required to double money in 12 years. 1 mark
(ii) Using the Rule of 72, estimate the doubling times for 3%, 6% and 9% p.a. 2 marks
(iii) An investment at 6% p.a. compounded annually actually doubles in 11.9 years (exact). Comment in one sentence on how close the Rule of 72 estimate is, and why it is useful as a quick mental check. 2 marks
Stuck? Revisit lesson § Rule of 72 — handy for sanity-checking calculator answers.Problem 5 — Three-way savings comparison
Mei has $25,000 to invest for 4 years and receives three offers.
Offer 1: 5.2% p.a. compounded monthly.
Offer 2: 5.3% p.a. compounded quarterly.
Offer 3: 5.35% p.a. compounded annually.
Set up: What are we solving for?
(i) Calculate the effective annual rate of each offer, to 2 decimal places of a percent. 3 marks
(ii) Rank the three offers from best to worst for a saver. 1 mark
(iii) Calculate the final value of Mei's investment under her best option, after 4 years. 2 marks
Stuck? Revisit lesson § Worked Example — compare three products by converting to effective rates first.How did this worksheet feel?
What I'll revisit before next class:
Problem 1 — Bank A vs Bank B
Set up. Convert both nominal rates to effective annual rates, then compare.
(i) Bank A: r/k = 0.062/12 = 0.005167. Effective = (1.005167)^12 − 1 = 1.06383 − 1 = 6.38% p.a.
(ii) Bank B: r/k = 0.0625/2 = 0.03125. Effective = (1.03125)^2 − 1 = 1.063477 − 1 = 6.35% p.a.
(iii) Bank A is better for Daniel — its effective rate (6.38%) is higher than Bank B's (6.35%), despite Bank A having the lower nominal rate. Monthly compounding outweighs the small nominal-rate gap.
Problem 2 — Flat-rate vs reducing-balance car loan
Set up. Compute Loan A's total interest and total paid, then Loan B's total paid (over 36 months), then compare and explain.
(i) Loan A interest = 15,000 × 0.08 × 3 = $3,600.00. Total = $18,600.
(ii) Loan B total = $484.01 × 36 = $17,424.36.
(iii) Loan B is cheaper by $18,600 − $17,424.36 = $1,175.64 over the 3 years.
(iv) A flat-rate loan charges interest on the full $15,000 every year, even after you have repaid most of it; a reducing-balance loan charges interest only on the outstanding balance, so even at a higher nominal rate the total interest paid can be lower.
Problem 3 — "0% interest" lounge suite
Set up. Compute the all-in total, monthly amount, equivalent rate, then explain.
(i) Total = $3,600 + $360 + 18 × $5 = $3,600 + $360 + $90 = $4,050.00.
(ii) Monthly = $4,050 / 18 = $225.00.
(iii) Extra paid = $4,050 − $3,600 = $450. r = I / (P n) = 450 / (3,600 × 1.5) = 450 / 5,400 = 0.0833... = 8.33% p.a. simple interest.
(iv) The label "0% interest" technically refers only to the interest rate on the principal — but the fees add up to the equivalent of an 8.33% p.a. loan, so the customer is paying more than the sticker price even though no "interest" is named.
Problem 4 — Rule of 72 estimates
Set up. Use 72 / rate(%) ≈ doubling time, then compare the Rule of 72 estimate against the exact value.
(i) 72 / 12 = 6, so a rate of 6% p.a. doubles money in approximately 12 years.
(ii) 3%: 72/3 = 24 years. 6%: 72/6 = 12 years. 9%: 72/9 = 8 years.
(iii) The Rule of 72 estimate (12 years) is within 0.1 years of the exact value (11.9 years) — a percentage error well under 1%. It is useful because a calculator-free check makes it easy to spot grossly wrong answers in exam conditions.
Problem 5 — Three-way comparison on $25,000 for 4 years
Set up. Convert each to an effective annual rate, rank, then apply the winner to $25,000 for 4 years.
(i) Offer 1: (1 + 0.052/12)^12 − 1 = (1.004333)^12 − 1 = 0.05326 = 5.33%.
Offer 2: (1 + 0.053/4)^4 − 1 = (1.01325)^4 − 1 = 0.05408 = 5.41%.
Offer 3: (1.0535)^1 − 1 = 5.35%.
(ii) Best → worst: Offer 2 (5.41%) > Offer 3 (5.35%) > Offer 1 (5.33%).
(iii) Under Offer 2 (5.3% compounded quarterly, kn = 16): A = 25,000 × (1.01325)^16 = 25,000 × 1.23396 = $30,848.93.