Simple Interest - the Formula
Learn the simple interest formula for NSW Year 12 Mathematics Standard 1 (NESA outcome MST-12-S1-02). Simple interest is worked out on the original amount only, so the same interest is earned every year. The formula \(I = Prn\) uses the principal \(P\), the interest rate \(r\) per period and the number of periods \(n\).
You will work out simple interest at different rates and over different lengths of time, find the total value of an investment or the amount to repay on a loan, and compare two investments — skills for everyday decisions about savings, term deposits and borrowing money.
Theory
Simple interest is interest worked out on the original amount only, so the same interest is earned every year and \(I=Prn\) gives the total interest. This Year 12 Standard 1 (NSW) guide shows how to substitute into the simple interest formula, find the total value of an investment or the amount to repay on a loan, and compare two investments.
Simple interest is interest worked out on the original amount only. The amount invested or borrowed is called the principal, \(P\). Because the principal never changes, every year earns the same interest, \(P\times r\).
The interest after \(n\) years is \(I=Prn\), where \(r\) is the interest rate per time period written as a decimal (usually per year, per annum: \(4\%=0.04\)) and \(n\) is the number of time periods — it must match the rate (a rate per annum means \(n\) is in years).
The total value of an investment, or the total to repay on a loan, is the principal plus the interest: \(A=P+I\). Because the same amount is added every year, the total value grows in a straight line when it is graphed against time (it starts at \(P\)). Compound interest gives a curve instead.
The simple interest earned on a principal \(P\) at rate \(r\) per year for \(n\) years is:
The total value (or amount to repay) adds the interest to the principal:
How to solve a simple interest problem
- List \(P\), \(r\) (as a decimal) and \(n\) (in years).
- Substitute into \(I=Prn\) and multiply to find the interest.
- Add the principal if the question asks for the total value or the amount to repay: \(A=P+I\).
List \(P=3500\), \(r=0.04\), \(n=5\), then substitute into \(I=Prn\).
| \(I\) | \(=\) | \(Prn\) |
| \(\) | \(=\) | \(3500\times0.04\times5\) |
| \(\) | \(=\) | \(140\times5\) |
| \(\) | \(=\) | \(700\) |
The simple interest is \(\$700\).
Find the interest first, then add it to the principal.
| \(I\) | \(=\) | \(7500\times0.03\times4\) |
| \(\) | \(=\) | \(225\times4 = 900\) |
| \(A\) | \(=\) | \(P+I\) |
| \(\) | \(=\) | \(7500+900\) |
| \(\) | \(=\) | \(8400\) |
The total value is \(\$8400\).
The interest is charged on the amount borrowed.
| \(\text{(i) } I\) | \(=\) | \(4800\times0.09\times3\) |
| \(\) | \(=\) | \(432\times3 = 1296\) |
| \(\text{(ii) } A\) | \(=\) | \(P+I\) |
| \(\) | \(=\) | \(4800+1296 = 6096\) |
(i) \(\$1296\) interest. (ii) \(\$6096\) to repay.
Use \(I=Prn\) for each option, then compare.
| \(\text{A: } I\) | \(=\) | \(6000\times0.045\times3\) |
| \(\) | \(=\) | \(270\times3 = 810\) |
| \(\text{B: } I\) | \(=\) | \(5000\times0.05\times4\) |
| \(\) | \(=\) | \(250\times4 = 1000\) |
\(1000 > 810\), so Option B earns more interest (\(\$1000\) compared with \(\$810\)).
Common pitfalls
Frequently asked questions
What is simple interest?
Simple interest is interest worked out on the original amount invested or borrowed (the principal) only. The principal never changes, so the same amount of interest is earned every year.
What is the simple interest formula?
The simple interest formula is I equals P times r times n, where P is the principal, r is the interest rate per year written as a decimal and n is the number of years.
How do you find the total value of a simple interest investment?
Work out the interest with I equals Prn, then add it to the principal: A equals P plus I. For example, $7500 at 3% per annum for 4 years earns $900 interest, so the total value is $8400.
How do you write the interest rate as a decimal?
Divide the percentage by 100. For example, 4% is 0.04, 9% is 0.09 and 4.5% is 0.045. Using 4 or 0.4 instead of 0.04 gives an answer that is far too big.
Is simple interest worked out on the new balance each year?
No. Simple interest is always worked out on the original principal, so the interest is the same every year. Interest worked out on the growing balance is compound interest.