Simple vs Compound Interest
Compare simple and compound interest for NSW Year 12 Mathematics Standard 1 (NESA outcome MST-12-S1-02). Simple interest is earned on the principal only, so it grows in a straight line; compound interest also earns interest on past interest, so it grows faster, in an upward-bending curve.
You will set the two kinds of growth side by side using numbers or a graph you are given, work out how much extra compound interest earns, and choose the better account — a key skill for making smart decisions about where to save your money.
Theory
Simple interest is earned on the principal only, so it grows in a straight line; compound interest is also earned on past interest, so it grows faster, in an upward-bending curve. This NSW Year 12 Mathematics Standard 1 guide shows how to compare simple vs compound interest by working out final values or interest for both, and how to choose the better account.
Simple interest earns the same interest, \(Pr\), every year because it is worked out on the principal only. The interest is \(I=Prn\) and the total value is \(A=P+Prn\). Its graph is a straight line.
Compound interest adds each year's interest to the balance, where it earns interest too. The total value is \(FV=PV(1+r)^n\) and the interest is \(FV-PV\). Its graph is an upward-bending curve.
For the same principal and rate the two are equal after \(1\) year; after that, compound interest is always worth more, and the gap keeps growing. To compare them, work out the same thing for both — both final values, or both amounts of interest — then subtract.
Simple interest — interest and total value:
Compound interest — total value and interest:
How to compare simple and compound interest
- Decide what to compare — the final values or the amounts of interest.
- Calculate the simple amount with \(I=Prn\) (and \(A=P+I\)) and the compound amount with \(FV=PV(1+r)^n\) (and \(I=FV-PV\)).
- Subtract to find the difference and state which option is better.
Find both totals, then subtract.
| \(\text{Simple: } A\) | \(=\) | \(6000+6000\times0.07\times3\) |
| \(\) | \(=\) | \(6000+1260 = 7260.00\) |
| \(\text{Compound: } FV\) | \(=\) | \(6000(1.07)^3\) |
| \(\) | \(=\) | \(6000\times1.225043\) |
| \(\) | \(=\) | \(7350.26\) |
| \(\text{Difference}\) | \(=\) | \(7350.26-7260.00\) |
| \(\) | \(=\) | \(90.26\) |
The compound value is \(\$90.26\) greater.
Work out each amount of interest.
| \(\text{Simple: } I\) | \(=\) | \(2500\times0.04\times4 = 400.00\) |
| \(\text{Compound: } I\) | \(=\) | \(2500(1.04)^4-2500\) |
| \(\) | \(=\) | \(2924.65-2500 = 424.65\) |
| \(\text{Extra}\) | \(=\) | \(424.65-400.00 = 24.65\) |
Compound interest earns \(\$24.65\) more.
Simple adds \(\$60\) each year; compound multiplies by \(1.06\) each year.
| Year | Simple | Compound |
|---|---|---|
| 0 | \(\$1000.00\) | \(\$1000.00\) |
| 1 | \(\$1060.00\) | \(\$1060.00\) |
| 2 | \(\$1120.00\) | \(\$1123.60\) |
| 3 | \(\$1180.00\) | \(\$1191.02\) |
They are equal after year \(1\); from year \(2\) the compound value is ahead, and the gap grows (\(\$3.60\), then \(\$11.02\)).
Compound interest is worth \(\$11.02\) more after \(3\) years.
Compare the final values.
| \(\text{A: } A\) | \(=\) | \(4000+4000\times0.055\times10\) |
| \(\) | \(=\) | \(4000+2200 = 6200.00\) |
| \(\text{B: } FV\) | \(=\) | \(4000(1.05)^{10}\) |
| \(\) | \(=\) | \(4000\times1.628895\) |
| \(\) | \(=\) | \(6515.58\) |
\(6515.58 > 6200.00\), so Bank B, even with the lower rate, gives \(\$315.58\) more.
Common pitfalls
Frequently asked questions
What is the difference between simple and compound interest?
Simple interest is worked out on the original principal only, so the same interest is added every year and the value grows in a straight line. Compound interest is worked out on the balance, including past interest, so it grows faster in an upward-bending curve.
Which is better, simple or compound interest?
For an investment over more than one year at the same rate, compound interest gives more. Over a long time compound interest can even beat a higher simple rate: $4000 for 10 years at 5% compound gives $6515.58, more than $6200 at 5.5% simple.
When are simple and compound interest the same?
For the same principal and rate, simple and compound interest are equal after 1 year, because the first year's interest is worked out on the principal either way. After that compound interest is always ahead.
How do you compare simple and compound interest?
Work out the same thing for both, either both final values or both amounts of interest, then subtract. For example, $6000 at 7% for 3 years gives $7260.00 simple and $7350.26 compound, a difference of $90.26.
Why does compound interest grow faster than simple interest?
With compound interest, each year's interest is added to the balance and then earns interest itself, so the interest gets bigger every year. Simple interest stays the same each year.