Simple interest is principal times rate times time
Definition
Principal: the original sum of money saved or borrowed, before any interest is added.
Simple interest: interest calculated only on the principal, not on any interest already earned or owed.
- Multiply the principal by the rate as a decimal, then by the number of years, and the answer is the interest in pounds.
- Add the interest to the principal when a question asks for the total saved or the total owed, since the two are different figures.
Work out the interest earned on savings
- Convert the percentage to a decimal first, so 3% becomes 0.03, because multiplying by 3 gives an answer a hundred times too big.
- Use one year unless the question names a longer period, and multiply by the number of years if it does.
Example
- A saver puts £5,000 into an account paying 3% simple interest.
Step 1: multiply the principal by the rate to find one year's interest:
£5,000×0.03=£150 \pounds5{,}000 \times 0.03 = \pounds150 £5,000×0.03=£150Step 2: add it to the principal to find the total after one year:
£5,000+£150=£5,150 \pounds5{,}000 + \pounds150 = \pounds5{,}150 £5,000+£150=£5,150Step 3: if the money is left for three years, multiply the yearly interest by the time:
£5,000×0.03×3=£450 \pounds5{,}000 \times 0.03 \times 3 = \pounds450 £5,000×0.03×3=£450- The account earns £150 a year and £450 over three years, so the total after three years is £5,450.
Compare two rates to find the effect
- Work out the interest at the old rate, then at the new one, and subtract to find what the change is worth.
- Say who gains and who loses, because the same change that rewards a saver costs a borrower.
Example
- The same £5,000 saved for one year, at three different rates.
| Interest rate | Interest for one year | Total after one year |
|---|---|---|
| 3% | £150 | £5,150 |
| 4% | £200 | £5,200 |
| 5% | £250 | £5,250 |
Step 1: subtract the interest at 3% from the interest at 5%:
£250−£150=£100 \pounds250 - \pounds150 = \pounds100 £250−£150=£100- A rise from 3% to 5% is worth an extra £100 a year to this saver, which is why a rate change moves the level of saving in 2.8.4.
Borrowing works the same way, often monthly
- The calculation is identical, but the answer is a cost rather than a reward, so it is money the borrower pays out.
- Loans are usually repaid monthly, so divide the yearly interest by 12 when a question asks for a monthly figure.
- Watch for a period mentioned only as background, because a rate agreed some years ago does not mean the interest must be multiplied by those years.
Example
- A household has a £200,000 mortgage at 4.45%, and the rate then rises to 5.45%.
Step 1: find the interest for one year at the old rate:
£200,000×0.0445=£8,900 \pounds200{,}000 \times 0.0445 = \pounds8{,}900 £200,000×0.0445=£8,900Step 2: divide by 12 to get the monthly interest:
£8,90012=£741.67 \frac{\pounds8{,}900}{12} = \pounds741.67 12£8,900=£741.67Step 3: repeat at the new rate of 5.45%:
£200,000×0.0545=£10,900£10,90012=£908.33 \pounds200{,}000 \times 0.0545 = \pounds10{,}900 \qquad \frac{\pounds10{,}900}{12} = \pounds908.33 £200,000×0.0545=£10,90012£10,900=£908.33Step 4: subtract the yearly figures, then divide by 12 to find what the rise costs each month:
£10,900−£8,900=£2,000£2,00012=£166.67 \pounds10{,}900 - \pounds8{,}900 = \pounds2{,}000 \qquad \frac{\pounds2{,}000}{12} = \pounds166.67 £10,900−£8,900=£2,00012£2,000=£166.67- A one percentage point rise costs this household about £167 a month, which is money no longer available to spend on anything else.
Checking an interest calculation looks right
- Check the rate was converted to a decimal, since an answer a hundred times too large almost always means it was not.
- Check whether the question wants the interest or the total, because the total includes the principal and the interest does not.
- Check the period, so a monthly answer has been divided by 12 and a multi-year answer multiplied by the years.
Exam technique
- Show the working line by line, because a question asking you to calculate usually gives marks for the method as well as the answer.
- Convert an annual figure to a month by dividing by 12 whenever the question asks per month, since that step often carries its own mark.
- Use simple interest unless a question clearly asks otherwise, and do not compound a figure over several years just because a date is mentioned.
Self review
- State the formula for simple interest.
- How much interest does £4,000 earn at 5% for one year?
- What is the total in the account after that year?
- How much interest does £4,000 earn at 5% over three years?
- A £150,000 mortgage is charged 4%. What is the interest per month?
