Mathematics Standard • Year 12 • Module 7 • Lesson 2

Compound Interest — Skill Drill

Build fluency with A = P(1 + r/k)^(kn): convert nominal rates by compounding period, compute final amounts and effective annual rates.

Build · Skill Drill

1. Quick recall

Answer each question in the space provided. 1 mark each

Q1.1 Write down the value of k for each compounding frequency.

Annually: k = ______ Semi-annually: k = ______ Quarterly: k = ______ Monthly: k = ______ Daily: k = ______

Q1.2 Complete the compound-interest formula for k compounding periods per year over n years.

A = P (1 + ______ )^( ______ )

Q1.3 Write the effective annual rate formula and convert each nominal rate to its rate-per-period.

Effective rate = (1 + ______ )^( ______ ) − 1

6% p.a. compounded monthly: rate per period = ____________

8% p.a. compounded quarterly: rate per period = ____________

Stuck? Revisit lesson § Key Ideas — Compounding Frequency table and Effective Annual Rate.

2. Worked example — A = P(1 + r/k)^(kn)

Follow each line carefully.

Problem. $10,000 is invested at 8% p.a. compounded quarterly for 5 years. Find the final amount A and the effective annual rate.

Step 1 — Identify P, r, k, n.

P = $10,000 r = 0.08 (nominal, per year) k = 4 (quarterly) n = 5 years

Reason: quarterly means 4 compounding periods per year.

Step 2 — Rate per period and total periods.

r / k = 0.08 / 4 = 0.02 per quarter; k × n = 4 × 5 = 20 quarters.

Step 3 — Substitute into A = P(1 + r/k)^(kn).

A = 10,000 × (1.02)^20 = 10,000 × 1.485947 = $14,859.47

Step 4 — Effective annual rate.

Effective = (1.02)^4 − 1 = 1.082432 − 1 = 0.08243 ≈ 8.24% p.a.

Conclusion. A = $14,859.47, effective annual rate ≈ 8.24% p.a.

3. Faded example — fill in the missing steps

$5,000 is invested at 6% p.a. compounded monthly for 3 years. Find A and the effective annual rate. Fill in each blank. 4 marks

Step 1 — Identify P, r, k, n:

P = $ ________ r = ________ k = ________ n = ________ years

Step 2 — Rate per period and total periods:

r / k = ________ / ________ = ________ kn = ________ × ________ = ________ periods

Step 3 — Final amount:

A = 5,000 × (1 + ________ )^( ________ ) = 5,000 × ________ = $ ____________

Step 4 — Effective annual rate:

Effective = (1 + ________ )^( ________ ) − 1 = ________ − 1 = ________ ≈ ________ % p.a.

Conclusion. A = $ __________ Effective rate = __________ % p.a.

Stuck? Revisit lesson § Worked Example — $10,000 at 8% compounded quarterly.

4. Graduated practice — Compound-interest calculations

Show your working below each part. Round dollar amounts to 2 decimal places and rates to 2 decimal places of a percent.

Foundation — annual compounding (4 questions)

Q	Problem	Answer
4.1 1	$2,000 at 5% p.a. compounded annually for 3 years. Find A.
4.2 1	$4,000 at 5% p.a. compounded annually for 6 years. Find A.
4.3 1	$1,500 at 4% p.a. compounded annually for 10 years. Find A.
4.4 1	$3,000 at 6% p.a. compounded annually for 5 years. Find total interest I.

Standard — different compounding frequencies (6 questions)

Show one line of substitution and label your final answer with units.

4.5 $8,000 at 4.8% p.a. compounded semi-annually for 4 years. Find A. 2 marks

4.6 $6,000 at 7.2% p.a. compounded quarterly for 5 years. Find A and total interest. 2 marks

4.7 $7,500 at 5.4% p.a. compounded quarterly for 5 years. Find A. 2 marks

4.8 $5,000 at 6% p.a. compounded monthly for 3 years. Find A. 2 marks

4.9 Calculate the effective annual rate of 4.8% p.a. compounded semi-annually. 2 marks

4.10 Calculate the effective annual rate of 7.2% p.a. compounded quarterly. 2 marks

Extension — compare and interpret (2 questions)

4.11 $2,000 is invested at 5% p.a. compounded annually for 3 years. Calculate (i) the total interest earned under compound interest, and (ii) the total interest under simple interest at the same nominal rate. State the dollar gap. 3 marks

4.12 A credit card charges 18% p.a. compounded daily (k = 365). Calculate the effective annual rate to 2 decimal places of a percent. 3 marks

Stuck on 4.12? Compute r/k = 0.18/365 carefully, then raise (1 + r/k) to the 365th power.

5. Self-check the easy 3

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

For 4.1 I raised (1.05) to the power 3, not multiplied by 3.

For 4.5 I used r/k = 0.048/2 = 0.024 per half-year, and kn = 8 periods.

For 4.9 I wrote the answer as a rate (e.g. 4.86%), not as an amount in dollars.

How did this worksheet feel?

Got it Partly Lost

What I'll revisit before next class:

Answers — Do not peek before attempting

Q1.1 — Compounding-frequency values of k

Annually: 1. Semi-annually: 2. Quarterly: 4. Monthly: 12. Daily: 365.

Q1.2 — Formula

A = P(1 + r/k)^(kn).

Q1.3 — Rate-per-period and effective rate

Effective rate = (1 + r/k)^k − 1.

6% p.a. monthly: r/k = 0.06/12 = 0.005 per month.
8% p.a. quarterly: r/k = 0.08/4 = 0.02 per quarter.

Q3 — Faded example ($5,000 at 6% monthly for 3 years)

Step 1: P = $5,000, r = 0.06, k = 12, n = 3.
Step 2: r/k = 0.06/12 = 0.005, kn = 12 × 3 = 36.
Step 3: A = 5,000 × (1.005)^36 = 5,000 × 1.196680 = $5,983.40.
Step 4: Effective = (1.005)^12 − 1 = 1.061678 − 1 = 0.06168 ≈ 6.17% p.a.

Q4.1 — $2,000 at 5% annually for 3 years

A = 2,000 × (1.05)^3 = 2,000 × 1.157625 = $2,315.25.

Q4.2 — $4,000 at 5% annually for 6 years

A = 4,000 × (1.05)^6 = 4,000 × 1.340096 = $5,360.38.

Q4.3 — $1,500 at 4% annually for 10 years

A = 1,500 × (1.04)^10 = 1,500 × 1.480244 = $2,220.37.

Q4.4 — $3,000 at 6% annually for 5 years

A = 3,000 × (1.06)^5 = 3,000 × 1.338226 = $4,014.68. I = A − P = $1,014.68.

Q4.5 — $8,000 at 4.8% semi-annually for 4 years

r/k = 0.024, kn = 8. A = 8,000 × (1.024)^8 = 8,000 × 1.20902 = $9,672.16.

Q4.6 — $6,000 at 7.2% quarterly for 5 years

r/k = 0.018, kn = 20. A = 6,000 × (1.018)^20 = 6,000 × 1.42825 = $8,569.50. I = A − P = $2,569.50.

Q4.7 — $7,500 at 5.4% quarterly for 5 years

r/k = 0.0135, kn = 20. A = 7,500 × (1.0135)^20 = 7,500 × 1.30694 = $9,802.05.

Q4.8 — $5,000 at 6% monthly for 3 years

r/k = 0.005, kn = 36. A = 5,000 × (1.005)^36 = 5,000 × 1.19668 = $5,983.40.

Q4.9 — Effective annual rate of 4.8% semi-annually

Effective = (1.024)^2 − 1 = 1.048576 − 1 = 0.04858 = 4.86% p.a.

Q4.10 — Effective annual rate of 7.2% quarterly

Effective = (1.018)^4 − 1 = 1.073929 − 1 = 0.07393 = 7.39% p.a.

Q4.11 — Compound vs simple on $2,000 at 5% for 3 years

(i) Compound: A = 2,000 × (1.05)^3 = $2,315.25. Interest = $315.25.
(ii) Simple: I = 2,000 × 0.05 × 3 = $300.00.
Compound earns $15.25 more. (Over decades the gap becomes huge.)

Q4.12 — Effective annual rate of 18% daily

r/k = 0.18/365 = 0.000493. Effective = (1.000493)^365 − 1 = 1.19716 − 1 = 0.19716 ≈ 19.72% p.a. (Daily compounding on a 18% nominal card lifts the true cost by almost 1.72 percentage points per year.)