Practice Maths

Compound Interest

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Key Terms

Compound interest
earns interest on both the principal and previously earned interest.
Each compounding period, interest is added to the balance, which grows the next period’s interest.
More frequent compounding means slightly more interest earned.
Effective annual rate (EAR)
= (1 + r/n)n − 1 — the true yearly return when compounding more than once per year.
Compound interest grows exponentially; simple interest grows linearly.
📚 QCAA Formula Sheet
Compound interest: A = P(1 + r/n)nt
Where: P = principal, r = annual rate (decimal), n = compoundings per year, t = time in years
Simple interest for comparison: A = P(1 + rt)
Compounding Period n (per year) Formula term
Annual1A = P(1 + r)t
Semi-annual2A = P(1 + r/2)2t
Quarterly4A = P(1 + r/4)4t
Monthly12A = P(1 + r/12)12t
Daily365A = P(1 + r/365)365t
Time (years) Amount ($) 0 2 4 6 8 10 P 1.5P 2P 2.6P Compound interest Simple interest A = P(1+r/n)^(nt) A = P(1+rt)
Hot Tip Always convert the annual rate to a decimal before substituting: 6% → r = 0.06. Then divide by n to get the rate per period, and multiply t by n to get total number of periods.

Worked Example 1 — Calculate compound interest

Question: $5 000 is invested at 6% p.a. compounding quarterly for 3 years. Find the final amount and total interest.

Identify values: P = 5000, r = 0.06, n = 4, t = 3

Substitute: A = 5000 × (1 + 0.06/4)4×3 = 5000 × (1.015)12

Calculate: (1.015)12 ≈ 1.19562 → A ≈ $5 978.09

Interest earned: $5 978.09 − $5 000 = $978.09

Worked Example 2 — Compare compounding periods

Question: $10 000 invested at 5% p.a. for 4 years. Compare annual vs monthly compounding.

Annual: A = 10000 × (1.05)4 = 10000 × 1.21551 ≈ $12 155.06

Monthly: A = 10000 × (1 + 0.05/12)48 = 10000 × (1.004167)48 ≈ 10000 × 1.22079 ≈ $12 207.94

Difference: Monthly compounding earns $52.88 more over 4 years.

Introduction

When you invest money in a bank account or take out a loan, interest is usually calculated as compound interest — interest is earned on interest. This is fundamentally different from simple interest, where interest is only ever earned on the original principal. Understanding compound interest is essential for making smart financial decisions.

How Compound Interest Works

At the end of each compounding period, the interest earned is added to the balance. The next period’s interest is then calculated on this larger balance. This “snowball effect” means the balance grows faster and faster over time — exponential growth rather than linear growth.

Example: Seeing compound interest in action

$1 000 at 10% p.a. compounding annually for 3 years:

Year 1: A = 1000 × 1.10 = $1 100 (interest = $100)

Year 2: A = 1100 × 1.10 = $1 210 (interest = $110 — more than Year 1!)

Year 3: A = 1210 × 1.10 = $1 331 (interest = $121)

Compare to simple interest: A = 1000(1 + 0.10×3) = $1 300. Compound earns $31 more.

Using the Formula

The compound interest formula A = P(1 + r/n)nt condenses this period-by-period calculation into one step. Each factor (1 + r/n) represents one compounding period’s growth multiplier, and we apply it nt times in total.

Worked Example — Finding the principal needed

How much must be invested now at 4% p.a. compounding monthly to have $20 000 in 5 years?

Step 1: A = 20000, r = 0.04, n = 12, t = 5

Step 2: 20000 = P × (1 + 0.04/12)60 = P × (1.003333)60

Step 3: (1.003333)60 ≈ 1.22099

Step 4: P = 20000 / 1.22099 ≈ $16 380.98

Effective Annual Rate

When comparing accounts with different compounding periods, the effective annual rate (EAR) puts them on a fair basis:

EAR = (1 + r/n)n − 1

For example, 6% p.a. compounding monthly: EAR = (1 + 0.06/12)12 − 1 = (1.005)12 − 1 ≈ 0.06168 = 6.168%. This is the true annual return.

💡 Key Reminder: When the question asks for “total interest earned”, calculate A first, then subtract the principal: Interest = A − P. Never confuse the final amount A with the interest earned.

Summary

Compound interest: A = P(1 + r/n)nt. More compounding periods per year means slightly more interest. The effective annual rate allows fair comparison between different compounding frequencies. Compound interest grows exponentially; simple interest grows linearly.

Mastery Practice

See Answers ➔
  1. Fluency

    Calculate the final amount for each compound interest investment.

    1. (a) P = $2 000, r = 5% p.a., compounding annually, t = 3 years
    2. (b) P = $8 000, r = 4% p.a., compounding semi-annually, t = 2 years
    3. (c) P = $15 000, r = 6% p.a., compounding quarterly, t = 5 years
    4. (d) P = $500, r = 3% p.a., compounding monthly, t = 1 year
  2. Fluency

    Find the total interest earned for each investment in Question 1.

  3. Fluency

    For each investment, calculate the effective annual rate (EAR) to 3 decimal places.

    1. (a) 6% p.a. compounding semi-annually
    2. (b) 8% p.a. compounding quarterly
    3. (c) 12% p.a. compounding monthly
  4. Understanding

    $6 000 is invested at 5.4% p.a. compounding monthly.

    1. (a) How much is in the account after 3 years?
    2. (b) How much interest has been earned?
    3. (c) Compare this to simple interest at the same rate over the same period. How much more does compound interest earn?
  5. Understanding

    $10 000 is invested at 6% p.a. for 5 years. Compare the final amounts for each compounding period.

    1. (a) Annual compounding
    2. (b) Quarterly compounding
    3. (c) Monthly compounding
    4. (d) What pattern do you notice as compounding becomes more frequent?
  6. Understanding

    How much must be invested now to reach each target?

    1. (a) $15 000 in 4 years at 5% p.a. compounding annually
    2. (b) $50 000 in 10 years at 4% p.a. compounding monthly
  7. Understanding

    Asha invests $3 000 at 7% p.a. compounding annually.

    1. (a) Complete a table showing the balance at the end of each year for 5 years.
    2. (b) In which year does the balance first exceed $4 000?
    3. (c) How does this compare to a simple interest account at 7% p.a.?
  8. Understanding

    A bank offers two accounts: Account A pays 5.8% p.a. compounding annually; Account B pays 5.6% p.a. compounding monthly.

    1. (a) Calculate the EAR for each account.
    2. (b) Which account gives a better return?
    3. (c) How much more would $20 000 earn in the better account over 3 years?
  9. Problem Solving

    How long does it take for an investment to double?

    Challenge. $5 000 is invested at 6% p.a. compounding annually.
    1. (a) By substituting values of t = 1, 2, 3, ..., find when the investment first exceeds $10 000.
    2. (b) The “Rule of 72” estimates doubling time as 72 ÷ interest rate %. Use this to estimate when $5 000 doubles at 6% p.a. How close is this to your answer in (a)?
  10. Problem Solving

    Marcus takes out a $25 000 car loan at 7.2% p.a. compounding monthly for 5 years. (Treat this as a compound interest calculation on the full amount for the full term.)

    1. (a) What is the total amount owed after 5 years?
    2. (b) How much interest does Marcus pay in total?
    3. (c) Compare this to a simple interest loan at 7.2% p.a. for 5 years. Which costs more, and by how much?