Mathematics Standard • Year 12 • Module 7 • Lesson 3

Comparing Interest Rates — Skill Drill

Build fluency converting nominal rates to effective annual rates, applying flat-rate loan formulas and using the Rule of 72 for quick doubling-time estimates.

Build · Skill Drill

1. Quick recall

Answer each question in the space provided. 1 mark each

Q1.1 Define each term in one short sentence.

Nominal rate: ____________________________________________________________

Effective annual rate: _____________________________________________________

Q1.2 Write the formula for the effective annual rate of a nominal rate r compounded k times per year.

Effective = ____________________________________________

Q1.3 State the Rule of 72.

Doubling time (years) ≈ ____________ ÷ ____________

Stuck? Revisit lesson § Key Ideas — Nominal, Effective rate, Rule of 72.

2. Worked example — compare three savings products

Problem. Three savings products: (A) 5.5% compounded monthly, (B) 5.6% compounded quarterly, (C) 5.7% compounded annually. Which is the best for a saver?

Step 1 — Convert each to an effective annual rate.

A: (1 + 0.055/12)^12 − 1 = (1.004583)^12 − 1 = 1.056408 − 1 = 0.05641 ≈ 5.64%

B: (1 + 0.056/4)^4 − 1 = (1.014)^4 − 1 = 1.057192 − 1 = 0.05719 ≈ 5.72%

C: nominal 5.7% compounded annually = effective 5.70%

Step 2 — Rank.

B (5.72%) > C (5.70%) > A (5.64%).

Conclusion. Product B is the best for a saver — despite having a lower nominal rate than C, its quarterly compounding pushes the effective rate above C's.

3. Faded example — flat-rate car loan

A $25,000 car loan is offered at 7% flat rate over 5 years. Calculate the total interest, total repayment and monthly instalment. Fill in the blanks. 4 marks

Step 1 — Total interest using flat-rate (simple) formula.

I = P × r × n = $25,000 × ________ × ________ = $ ____________

Step 2 — Total repayment.

Total = P + I = $25,000 + $ ________ = $ ____________

Step 3 — Number of monthly instalments and monthly amount.

Months = 5 × 12 = ________

Monthly = $ ________ ÷ ________ = $ ____________

Conclusion. Total interest = $ __________ ; total repayment = $ __________ ; monthly = $ __________.

Stuck? Revisit lesson § Flat Rate — interest charged on the original P, not on the reducing balance.

4. Graduated practice — Effective rates, flat rates and Rule of 72

Show your working. Round effective rates to 2 decimal places of a percent.

Foundation — single conversions (4 questions)

Q	Problem	Answer
4.1 1	Convert 6% p.a. compounded annually to an effective annual rate.
4.2 1	Use the Rule of 72 to estimate the doubling time at 4% p.a.
4.3 1	Use the Rule of 72 to estimate the doubling time at 8% p.a.
4.4 1	Use the Rule of 72 to estimate the doubling time at 12% p.a.

Standard — typical HSC difficulty (6 questions)

4.5 Calculate the effective annual rate of 5.4% p.a. compounded monthly. 2 marks

4.6 Calculate the effective annual rate of 5.5% p.a. compounded quarterly. 2 marks

4.7 Bank A: 5.4% compounded monthly. Bank B: 5.5% compounded quarterly. State which is better for a saver. 2 marks

4.8 A credit card advertises 19.9% p.a. compounded daily (k = 365). Calculate the effective annual rate. 2 marks

4.9 A $20,000 car loan at 6% flat rate over 4 years. Calculate (i) total interest and (ii) the monthly instalment. 2 marks

4.10 A $10,000 loan at 8% flat rate over 4 years. Calculate the monthly instalment. 2 marks

Extension — interpret a deal (2 questions)

4.11 A store offers a $2,000 fridge "0% interest" but charges a $200 establishment fee over the 12-month repayment plan. (i) Calculate the true total amount paid. (ii) Express the $200 as an equivalent simple-interest rate per annum on the $2,000 over the 12 months. 3 marks

4.12 Three savings products: (A) 6.4% compounded monthly, (B) 6.5% compounded quarterly, (C) 6.6% compounded annually. Rank them from best to worst for a saver. 3 marks

Stuck on 4.12? Convert each to its effective rate first — then rank.

5. Self-check the easy 3

Tick the first three once you've checked your method works.

For 4.1 I noted that "compounded annually" means the effective rate equals the nominal rate (k = 1).

For 4.2 I divided 72 by 4 (giving 18 years), not 4 by 72.

For 4.5 I used r/k = 0.054/12 (not 0.054/1) before raising to the 12th power.

How did this worksheet feel?

Got it Partly Lost

What I'll revisit before next class:

Answers — Do not peek before attempting

Q1.1 — Definitions

Nominal rate: the stated (advertised) annual rate before adjusting for the compounding frequency. Effective annual rate: the actual annual rate earned (or paid), accounting for how often interest is compounded.

Q1.2 — Effective annual rate formula

Effective = (1 + r/k)^k − 1.

Q1.3 — Rule of 72

Doubling time (years) ≈ 72 ÷ rate(%).

Q3 — Faded example ($25,000 flat-rate car loan at 7% over 5 years)

Step 1: I = $25,000 × 0.07 × 5 = $8,750.00.
Step 2: Total = $25,000 + $8,750 = $33,750.00.
Step 3: Months = 60. Monthly = $33,750 ÷ 60 = $562.50.
Conclusion: Interest = $8,750; total = $33,750; monthly = $562.50.

Q4.1 — Effective annual rate at 6% p.a. (k=1)

Effective = (1.06)^1 − 1 = 0.06 = 6.00% (nominal = effective when k = 1).

Q4.2 — Rule of 72 at 4%

72 ÷ 4 = 18 years.

Q4.3 — Rule of 72 at 8%

72 ÷ 8 = 9 years.

Q4.4 — Rule of 72 at 12%

72 ÷ 12 = 6 years.

Q4.5 — Effective rate of 5.4% monthly

Effective = (1 + 0.054/12)^12 − 1 = (1.0045)^12 − 1 = 1.05536 − 1 = 5.54% p.a.

Q4.6 — Effective rate of 5.5% quarterly

Effective = (1 + 0.055/4)^4 − 1 = (1.01375)^4 − 1 = 1.05614 − 1 = 5.61% p.a.

Q4.7 — Bank A vs Bank B

Bank A effective = 5.54% (from Q4.5). Bank B effective = 5.61% (from Q4.6). Bank B is better for a saver — both the nominal rate and the effective rate are higher.

Q4.8 — Effective rate of 19.9% daily

r/k = 0.199/365 = 0.000545. Effective = (1.000545)^365 − 1 = 1.22010 − 1 = 22.01% p.a.

Q4.9 — Flat-rate $20,000 at 6% for 4 years

(i) I = 20,000 × 0.06 × 4 = $4,800.00.
(ii) Total = $24,800. Monthly = $24,800 / 48 = $516.67.

Q4.10 — Flat-rate $10,000 at 8% for 4 years

I = 10,000 × 0.08 × 4 = $3,200. Total = $13,200. Monthly = $13,200 / 48 = $275.00.

Q4.11 — "0% interest" with $200 fee

(i) Total paid = $2,000 + $200 = $2,200.00.
(ii) Equivalent simple-interest rate: r = I / (Pn) = 200 / (2,000 × 1) = 0.10 = 10% p.a. The "0%" advertising is misleading — the $200 fee is functionally identical to a 10% simple-interest loan.

Q4.12 — Rank A, B, C at 6.4%, 6.5%, 6.6%

A: (1 + 0.064/12)^12 − 1 = (1.005333)^12 − 1 = 1.06592 − 1 = 6.59%.
B: (1 + 0.065/4)^4 − 1 = (1.01625)^4 − 1 = 1.06660 − 1 = 6.66%.
C: (1.066)^1 − 1 = 6.60%.
Ranking (best → worst): B (6.66%) > C (6.60%) > A (6.59%). Quarterly compounding on B narrowly beats the higher nominal rate of C.