Use the compound interest formula to calculate future value, understand the effect of compounding frequency, and compare compound with simple interest.
Use the PDF for classwork, homework or revision. It includes key ideas, activities, questions, an extend task and success-criteria proof.
Albert Einstein allegedly called compound interest "the eighth wonder of the world." Whether he said it or not, the idea is worth sitting with: what if your bank didn't just pay you interest on your original deposit — what if it also paid you interest on last year's interest, and then interest on that, and so on? After 10 years, the difference between simple and compound interest on the same deposit can be thousands of dollars. And in reverse — for a loan — compound interest is what makes debt grow frighteningly fast if you don't repay it. Before any formulas: why do you think compound interest grows so much faster than simple interest over time?
Apply $A = P(1+r)^n$ to find the future value of a compound interest investment or loan
Adjust $r$ and $n$ correctly for different compounding frequencies (monthly, quarterly, semi-annual)
Compare simple and compound interest and quantify the difference in interest earned
Find interest earned by subtracting the principal from the final amount: $I = A - P$
For a nominal annual rate of 8% over 3 years — how $r$ and $n$ change with compounding frequency:
| Frequency | Periods/year | $r$ per period | $n$ for 3 years |
|---|---|---|---|
| Annually | 1 | 0.08 | 3 |
| Semi-annually | 2 | 0.04 | 6 |
| Quarterly | 4 | 0.02 | 12 |
| Monthly | 12 | 0.08 ÷ 12 | 36 |
Rule: divide the annual rate by the number of periods per year to get $r$; multiply years by periods per year to get $n$.
Wrong: Compound interest is calculated the same way as simple interest.
Right: Compound interest uses A = P(1 + r)^n, earning interest on interest. Simple interest uses I = Prn. Over time, compound interest grows much faster.
Compound interest adds interest to the principal at the end of each period, so the next period's interest is calculated on a larger balance — creating exponential rather than linear growth.
Under simple interest, a $10,000 investment at 5% per annum earns $500 every year — always on the original $10,000. Under compound interest at the same rate:
The formula $A = P(1 + r)^n$ captures this: each period the balance is multiplied by $(1 + r)$. After $n$ periods, the original principal has been multiplied $n$ times. The difference between compound and simple interest becomes more pronounced as $n$ increases.
The more frequently interest compounds, the more interest is earned — but the rate per period must be adjusted to match the compounding frequency.
When applying $A = P(1 + r)^n$ with a non-annual compounding frequency, always adjust both $r$ and $n$ before substituting:
For monthly compounding at 9% per annum: $r = 0.09 \div 12 = 0.0075$ per month. Keep this as a fraction rather than rounding to 0.0075 — rounding introduces cumulative error over many periods.
Simple and compound interest produce the same result after one period — the difference emerges from the second period onwards and grows with time.
For short time periods (1–2 years), the difference between simple and compound interest is small. Over longer periods, compound interest significantly outperforms simple interest for investors. When comparing:
Key distinction: simple interest is linear (A = P(1 + rn)); compound interest is exponential (A = P(1 + r)^n). These are different formulas — do not mix them in the same calculation.
Compound interest questions become much easier when you follow the same setup every time instead of trying to do adjustments mentally.
| Step | What to do |
|---|---|
| 1 | Write the principal $P$ |
| 2 | Identify the annual nominal rate |
| 3 | Convert to rate per period: $r = \text{annual rate} \div \text{periods per year}$ |
| 4 | Convert time to total periods: $n = \text{years} \times \text{periods per year}$ |
| 5 | Substitute into $A = P(1+r)^n$ with brackets |
| 6 | If needed, subtract $P$ to find the interest earned |
$14,000 is invested at 6% per annum compounding quarterly for 4 years. Calculate: (a) the final amount, and (b) the interest earned.
Compounding frequency: quarterly = 4 periods per year.
$$r = 0.06 \div 4 = 0.015 \text{ per quarter}$$
$$n = 4 \times 4 = 16 \text{ quarters}$$
Adjust both $r$ and $n$ before substituting. The annual rate is divided by 4; the years are multiplied by 4.
$20,000 is invested for 5 years at a nominal rate of 4.5% per annum. Compare the total interest earned under: (a) simple interest, and (b) compound interest compounding annually. Which earns more, and by how much?
$$I = Prn = \$20{,}000 \times 0.045 \times 5 = \$4{,}500.00$$
Linear calculation — the same $900 interest is added each year.
Mia takes out a loan of $7,500 at 9% per annum compounding monthly. She makes no repayments. What does she owe after 2 years?
$$r = 0.09 \div 12 = 0.0075 \text{ per month}$$
$$n = 2 \times 12 = 24 \text{ months}$$
Monthly compounding: divide annual rate by 12 for $r$; multiply years by 12 for $n$. Keep $r$ as 0.0075 — do not round further.
$10,000 is invested for 3 years at 7.2% per annum. Compare the final amount if interest is compounded annually versus monthly.
Which option gives more, and by how much?
$$A_{\text{annual}} = \$10{,}000 \times (1.072)^3$$
$$A_{\text{annual}} = \$10{,}000 \times 1.231925248 = \$12{,}319.25$$
With annual compounding, the rate stays as 0.072 and the number of periods is 3.
Look back at what you wrote in the Think First section. What has changed? What did you get right? What surprised you?
5 random questions from a replayable lesson bank — feedback shown immediately
What is the interest rate per compounding period if a 7.2% per annum rate compounds monthly?
An investment of $15,000 compounds annually at 5% for 4 years. How much interest is earned?
$12,000 is invested at 6% per annum compounding quarterly for 2 years. What is the number of compounding periods?
These questions focus on adjusting the compounding setup correctly and separating total amount from interest earned.
Calculate the final amount for $9,500 invested at 4.8% per annum compounding annually for 6 years.
A loan of $6,800 compounds monthly at 8.4% per annum for 18 months. What is the amount owed at the end?
Compare the interest earned on $18,000 over 4 years at 5.5% per annum under simple interest and compound interest compounding annually.
The ultimate Module 3 challenge — defeat the boss using all your financial maths knowledge. Pool: lessons 1–12.
Use this as a fast recognition drill: spot the compounding setup before reaching for the calculator.
If interest compounds quarterly for 5 years, what should happen to the number of periods?
What does the compound interest formula give directly?
Which setup is correct for 6% per annum compounding monthly over 2 years?
Why does monthly compounding usually give a slightly larger final amount than annual compounding at the same nominal annual rate?