Comparing Simple and Compound Interest
Every question with a fully worked solution.
- Comparing Simple and Compound Interest - Video - Difference between Simple Interest and Compound Interest Watch
Theory
Simple interest is worked out on the original amount only, using \(I = Prn\). Compound interest is simple interest worked out again each period on the growing balance. This Year 10 maths page shows how to find simple interest and the total amount, solve for the principal, rate or time, read simple interest graphs, and compare simple interest with compound interest, built up as repeated simple interest.
The principal (\(P\)) is the amount invested or borrowed. Interest (\(I\)) is the extra money earned on an investment or paid on a loan.
With simple interest, the interest is worked out on the principal only, so the same amount is added every period. The rate is usually per annum (p.a.), which means per year.
The amount or future value at the end is the principal plus the interest: \(A = P + I\). A flat-rate loan charges simple interest on the amount borrowed.
\(r\) is the rate per period as a decimal (\(4.5\% = 0.045\)) and \(n\) is the number of periods.
How to solve a simple interest problem
- List \(P\), \(r\) and \(n\), with the rate as a decimal and the time in the same period as the rate.
- Substitute into \(I = Prn\). If \(I\) is known, solve the equation for the unknown.
- Add the interest to the principal if the total amount, value or amount owed is asked for.
Other questions you will meet
- Value known, principal unknown: find the total percentage added, rate \(\times\) time, then divide the value by \(1 +\) that decimal (Example 3).
- Value known, time unknown: interest \(=\) value \(-\) principal, then divide by one year's interest.
- Days: write the time as a fraction of a year, \(n = \dfrac{\text{days}}{365}\).
- Simple against compound: work out each final value (or each total interest) and subtract. If the rates differ, tabulate both values year by year to see when compound first gets ahead.
- Comparing offers: change rates to the same period. \(1.2\%\) per quarter is \(4 \times 1.2\% = 4.8\%\) p.a.
- Short loans: to give a rate for a few months as a yearly rate, multiply by the number of those periods in a year.
- Doubling: the money has doubled when the interest equals the principal, so \(n = 1 \div r\).
- Money split between two accounts: let \(x\) be the amount in one account and the rest be (total \(- x\)), then write total interest as an equation.
- Graphs: the interest or value grows by the same amount each year. Read values from the graph; the rate is one year's interest \(\div\) principal. Where two value lines cross, the investments are worth the same.
(i) the interest earned
(ii) the value of the investment at the end.
The rate is per year, so write the time in years: \(30\) months \(= \dfrac{30}{12} = 2.5\) years.
| \(I\) | \(=\) | \(Prn\) |
| \(\) | \(=\) | \(8400 \times 0.045 \times 2.5\) |
| \(\) | \(=\) | \(\mathbf{\$945.00}\) |
(ii) The value at the end is the principal plus the interest.
| \(A\) | \(=\) | \(P + I\) |
| \(\) | \(=\) | \(8400 + 945\) |
| \(\) | \(=\) | \(\mathbf{\$9345.00}\) |
(ii) How long will it take \(\$6200\) at \(3.5\%\) p.a. to earn \(\$542.50\) simple interest? Answer in months.
Substitute what you know into \(I = Prn\) and solve for the unknown.
(i)
| \(337.50\) | \(=\) | \(2500 \times r \times 3\) |
| \(337.50\) | \(=\) | \(7500r\) |
| \(r\) | \(=\) | \(337.50 \div 7500\) |
| \(\) | \(=\) | \(0.045\) |
| \(\) | \(=\) | \(\mathbf{4.5\%} \text{ p.a.}\) |
(ii) One year's interest is \(6200 \times 0.035 = \$217.00\).
| \(542.50\) | \(=\) | \(217 \times n\) |
| \(n\) | \(=\) | \(542.50 \div 217\) |
| \(\) | \(=\) | \(2.5 \text{ years}\) |
| \(\) | \(=\) | \(2.5 \times 12\) |
| \(\) | \(=\) | \(\mathbf{30} \text{ months}\) |
Find the total percentage added over the whole loan, then undo it.
| \(n\) | \(=\) | \(\dfrac{15}{12}\) |
| \(\) | \(=\) | \(1.25 \text{ years}\) |
| \(\text{total rate}\) | \(=\) | \(8.4\% \times 1.25\) |
| \(\) | \(=\) | \(10.5\%\) |
So \(\$7735\) is \(110.5\%\) of the amount borrowed.
| \(1.105 \times P\) | \(=\) | \(7735\) |
| \(P\) | \(=\) | \(7735 \div 1.105\) |
| \(\) | \(=\) | \(\mathbf{\$7000.00}\) |
Check: \(7000 \times 0.084 \times 1.25 = 735\), and \(7000 + 735 = 7735\).
(i) Find the value each year if the interest is compounded annually, by working out one year of simple interest at a time. Check with \(P(1 + r)^n\).
(ii) How much more interest is earned than with simple interest?
(i) Each year's interest is \(4\%\) of the balance at the start of that year.
| Year | Start | Interest | End |
|---|---|---|---|
| \(1\) | \(\$3500.00\) | \(\$140.00\) | \(\$3640.00\) |
| \(2\) | \(\$3640.00\) | \(\$145.60\) | \(\$3785.60\) |
| \(3\) | \(\$3785.60\) | \(\$151.42\) | \(\$3937.02\) |
| \(3500 \times 1.04^3\) | \(=\) | \(3937.024\) |
| \(\) | \(\approx\) | \(\mathbf{\$3937.02}\) |
(ii) Compare the total interest:
| \(\text{compound } I\) | \(=\) | \(3937.02 - 3500\) |
| \(\) | \(=\) | \(\$437.02\) |
| \(\text{simple } I\) | \(=\) | \(3500 \times 0.04 \times 3\) |
| \(\) | \(=\) | \(\$420.00\) |
| \(\text{difference}\) | \(=\) | \(\mathbf{\$17.02}\) |
Simple interest adds \(\$140\) every year; compound interest adds a little more each year because it earns interest on interest.
Common pitfalls
Frequently asked questions
What is the formula for simple interest?
The formula is \(I = Prn\), where \(I\) is the interest, \(P\) is the principal, \(r\) is the rate per period as a decimal and \(n\) is the number of periods.
How do you calculate simple interest for months?
Write the months as a fraction of a year by dividing by \(12\), then use that value for \(n\) with the yearly rate. For example, \(21\) months is \(1.75\) years.
How do you find the interest rate in simple interest?
Rearrange \(I = Prn\) to get \(r = \dfrac{I}{Pn}\). Multiply the decimal by \(100\) to write it as a percentage per annum.
What does p.a. mean?
p.a. stands for per annum, which means per year. A rate of \(5\%\) p.a. adds \(5\%\) of the principal each year with simple interest.
Why is compound interest more than simple interest?
Simple interest is always worked out on the original principal. Compound interest is worked out on the balance, which includes earlier interest, so from the second year on it earns interest on interest.