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.