Worked examples · reveal each step
PROBLEM 1 · COMPARING THREE PRODUCTS
Compare: (A) 5.5% compounded monthly, (B) 5.6% compounded quarterly, (C) 5.7% compounded annually. Find the best effective rate for a saver.
1
Product A: $r = 0.055$, $k = 12$
Identify nominal rate and compounding frequency for each product
2
$r_A = (1 + 0.055/12)^{12} - 1 = 5.641\%$
Apply the effective rate formula to Product A
3
$r_B = (1 + 0.056/4)^{4} - 1 = 5.718\%$ $r_C = 5.700\%$
Calculate effective rates for B (quarterly) and C (annual, so effective = nominal)
4
Ranking: B (5.718%) > C (5.700%) > A (5.641%)
Product B is best for a saver despite having the middle nominal rate, quarterly compounding on 5.6% beats annual on 5.7%
PROBLEM 2 · FLAT RATE LOAN
A car loan advertises 7% flat rate over 5 years on $25,000. Find the total repayment and monthly repayment.
1
Total interest = $25{,}000 \times 0.07 \times 5 = \$8{,}750$
Flat rate: interest on full principal for all 5 years
2
Total repayment = $25{,}000 + \$8{,}750 = \$33{,}750$
Add principal and total interest
3
Monthly repayment = $\$33{,}750 \div 60 = \$562.50$
Divide total by 60 months (5 years × 12)
4
True rate ≈ 13–14% effective
Because you're paying interest on $25,000 the whole time, even as you repay the principal, the true rate is nearly double 7%
Beyond the syllabus. Comparing products across compounding frequencies is core, and effective annual rate is a supporting method. The Rule of 72, its logarithmic derivation and doubling-time proofs are extension — useful mental arithmetic, but not something the exam assesses.
08
Rule of 72, quick mental estimation
core concept
Flat-rate loans charge simple interest on the original principal throughout the term, giving a stated rate that looks low but a true effective rate roughly twice as high. Switching from comparing advertised rates to comparing effective rates: the Rule of 72 provides a fast mental check, divide 72 by the annual interest rate (%) to estimate how many years it takes for money to double. At 6% p.a., doubling time ≈ 72 ÷ 6 = 12 years.
The Rule of 72 estimates how long an investment takes to double in value:
$$\text{Doubling time} \approx \frac{72}{\text{interest rate (\%)}}$$
Examples:
- At 6% p.a.: doubles in approx. $72 \div 6 = 12$ years. Exact: $\ln(2)/\ln(1.06) = 11.9$ years.
- At 4% p.a.: doubles in approx. $72 \div 4 = 18$ years.
- At 8% p.a.: doubles in approx. $72 \div 8 = 9$ years.
- At 12% p.a.: doubles in approx. $72 \div 12 = 6$ years.
Why it works: The Rule of 72 is a clever approximation of the exact formula $t = \ln(2)/\ln(1+r)$. Because $\ln(2) \approx 0.693$ and for small $r$, $\ln(1+r) \approx r$, we get $t \approx 0.693/r \approx 70/r$. Using 72 instead of 69.3 gives slightly better accuracy for typical interest rates of 4–12%.
HSC application: Use the Rule of 72 for quick sanity checks and estimates. For exact answers, use the compound interest formula or logarithms.
Rule of 72: years to double ≈ 72 ÷ annual interest rate (%). For example, at 6% p.a. money doubles in about 72/6 = 12 years. Use for quick mental estimation only, it is most accurate for rates between 6% and 10%.
Pause, copy the Rule of 72: years to double ≈ 72 ÷ r%, with one example (e.g., 8% → doubles in ~9 years), and note the accuracy range: most reliable for rates between 6% and 10% p.a. into your book.
