Practice Maths

Compound Interest

Key Ideas

Key Terms

compound interest
Interest calculated on both the original principal and any interest already earned, causing the balance to grow exponentially.
principal (P)
The initial amount of money invested or borrowed before any interest is applied.
rate (r)
The annual interest rate expressed as a decimal in the compound interest formula — e.g. 5% per year means r = 0.05.
periods (n)
The total number of compounding periods — e.g. 3 years compounded annually means n = 3.
future value (A)
The total amount after interest is applied — calculated using A = P(1 + r)n.
depreciation
A decrease in value over time modelled by A = P(1 − r)n, where r is the annual depreciation rate.
Formula Meaning Use when…
A = P(1 + r)nFuture value of investmentCompound growth (r as decimal)
CI = A − PInterest earnedFinding just the interest
A = P(1 − r)nDepreciated valueValue decreases each period
SI = P × r × nSimple interestComparing with CI (r as decimal)
Hot Tip Always convert the percentage rate to a decimal first (e.g. 6% → r = 0.06). Check: (1 + r) should be greater than 1 for growth, and (1 − r) should be between 0 and 1 for depreciation.

Worked Example — Compound Interest

Question: $5000 is invested at 6% p.a. compound interest for 3 years. Find the total amount and interest earned.

Step 1 — Identify P, r, n.
P = $5000,   r = 0.06,   n = 3

Step 2 — Apply the formula.
A = 5000 × (1.06)3 = 5000 × 1.191016 = $5955.08

Step 3 — Find interest earned.
CI = $5955.08 − $5000 = $955.08

Worked Example — Depreciation

Question: A car costs $32 000 and depreciates at 12% p.a. Find its value after 4 years.

A = 32 000 × (1 − 0.12)4 = 32 000 × (0.88)4 = 32 000 × 0.59970 ≈ $19 190.25

What Is Compound Interest?

With simple interest, you only ever earn interest on the original principal. With compound interest, you earn interest on the principal and on all the interest that has already been added. In other words, your interest earns interest — this is why investments (and debts) grow much faster under compound interest over time.

Real-world example: most savings accounts, home loans, and credit cards use compound interest. This is why credit card debt can spiral quickly if you only make minimum repayments.

The Compound Interest Formula

The formula for the final amount under compound interest is:

A = P(1 + r/n)nt, where:

A = final amount, P = principal, r = annual interest rate as a decimal, n = number of compounding periods per year, t = time in years.

If interest compounds annually (n = 1): A = P(1 + r)t.

If interest compounds quarterly (n = 4): each period the rate is r/4 and there are 4t periods total.

Example: $3000 invested at 6% per year compounded monthly for 5 years. A = 3000(1 + 0.06/12)12×5 = 3000(1.005)60 ≈ $4046.55.

Finding the Interest Earned

The interest earned is simply: I = A − P.

From the example above: I = $4046.55 − $3000 = $1046.55. Compare this to simple interest: I = 3000 × 0.06 × 5 = $900. Compound interest earns $146.55 more over 5 years — and the gap widens the longer the money is invested.

Simple vs Compound Interest Over Time

Under simple interest, the balance grows in a straight line (linear). Under compound interest, the balance grows exponentially — the graph curves upward more and more steeply over time.

For short time periods, the difference is small. Over many years (like a superannuation account over 40 years of work), the difference is enormous. This is why starting to save early is so powerful — compound interest has more time to work.

Exam tip: The most common error with compound interest is using the wrong value of n. If the problem says "compounded monthly" then n = 12, r/n = annual rate / 12, and nt = 12 × years. If it says "compounded quarterly" then n = 4. Do not confuse the rate per period (r/n) with the annual rate. Always write out each variable before substituting into the formula.

Mastery Practice

  1. Find the total amount A after compound interest. Round answers to the nearest cent. Fluency

     Principal (P)Rate (r % p.a.)Years (n)Total Amount A
    (a)$2 0005%2 
    (b)$3 0004%3 
    (c)$10 0003%5 
    (d)$1 5008%4 
    (e)$7 5006%3 
    (f)$5002%10 
    (g)$4 0005.5%6 
    (h)$20 0003.5%4 

  2. Find the compound interest earned (CI = A − P) for each investment below. Fluency

     Pr % p.a.n yearsCI Earned
    (a)$6 0004%3 
    (b)$8 0005%2 
    (c)$12 0003%4 
    (d)$9 5006%5 

  3. Calculate the depreciated value using A = P(1 − r)n. Fluency

     Item / Starting ValueRateYearsDepreciated Value
    (a)Laptop, $2 40020%3 
    (b)Machinery, $50 00015%5 
    (c)Phone, $90030%2 
    (d)Boat, $40 00010%6 

  4. True or False? Write T or F and briefly explain. Fluency

     StatementT / F
    (a)Compound interest always earns more than simple interest for the same P, r, and n (when n > 1). 
    (b)For n = 1, simple interest and compound interest give the same result. 
    (c)Doubling the number of years doubles the compound interest earned. 
    (d)In the depreciation formula, (1 − r) must be between 0 and 1 for the formula to make sense. 

  5. Savings account growth. Understanding

    Starting out. Lena opens a savings account with $4 000 at 3.5% p.a. compound interest.
    1. How much does she have after 2 years?
    2. How much interest has she earned after 2 years?
    3. If she had used a simple interest account at the same rate for the same period, how much interest would she have earned?
    4. How much extra does compound interest earn compared to simple interest over these 2 years?

  6. Comparing simple and compound interest over time. Understanding

    Seeing the difference grow. P = $5000, r = 6% p.a. Complete the table showing the total amount under each scheme at the end of each year.
    YearSimple Interest TotalCompound Interest TotalCI − SI Difference
    1   
    2   
    3   
    4   

    What pattern do you notice in the CI − SI difference column?

  7. Choosing an investment. Understanding

    Two options. An investor has $10 000 to invest for 5 years. Option A offers 7% p.a. simple interest. Option B offers 6% p.a. compound interest.
    1. Calculate the total amount under Option A after 5 years.
    2. Calculate the total amount under Option B after 5 years.
    3. Which option gives more money and by how much?
    4. If the investment period were 10 years instead of 5, which option would be better? (Calculate and compare.)

  8. Finding the principal. Understanding

    Working backwards. James invests some money at 5% p.a. compound interest. After 2 years he has $11 025 in the account.
    1. Write an equation using A = P(1 + r)n to represent this situation.
    2. Solve the equation to find the original principal P.
    3. How much compound interest did James earn over the 2 years?

  9. Reaching a savings goal. Problem Solving

    How long will it take? Mia invests $15 000 at 4.5% p.a. compound interest. She wants her investment to grow to at least $18 000.
    1. By testing n = 4 and n = 5, determine the minimum whole number of years needed.
    2. If Mia instead invested at 6% p.a. compound interest, would she reach $18 000 in fewer years? Test n = 3 and n = 4.
    3. Explain in your own words why a higher interest rate means a shorter time to reach the same goal.

  10. Car depreciation investigation. Problem Solving

    Losing value. A car is purchased new for $45 000 and depreciates at 18% p.a. using the reducing-balance method.
    1. Find the car’s value after 3 years (round to the nearest cent).
    2. By testing n = 4 and n = 5, find after how many complete years the car’s value first falls below $20 000.
    3. A second car originally cost $60 000 and depreciates at 15% p.a. After 3 years, which car is worth more?
    4. After how many years would the $45 000 car be worth less than half its original value? Test values of n until you find the answer.
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