Simple and Compound Interest

Interest is money a bank pays you for keeping your savings with it. The money you put in is the amount invested. The rate of interest is a percentage, usually per year. Per annum also means per year.

There are two kinds. Simple interest is the same amount every year. Compound interest is added on at the end of each year, and the next year's interest is worked out on the new, bigger amount.

Some values work the other way. The value of something, such as a car, goes down by a percentage of its value each year.

In the exam you are usually asked to work out simple interest, compound interest or a value that goes down by a percentage each year, on a calculator paper, for 2 or 3 marks. Harder questions, worth 4 or 5 marks, ask you to compare two accounts or two values and decide which is bigger. Always check whether the question wants the total interest or the amount in the account at the end, because they are different answers.

With simple interest, the interest is always a percentage of the amount first invested, so you get the same amount every year. For example, £500 at 4\% simple interest earns 4\% of £500, which is £20, every year.

On a calculator, change the percentage to a decimal by dividing by 100. So 4\% = 0.04, and one year's interest is 500 \times 0.04 = 20

Then multiply by the number of years:

\text{total interest} = \text{one year's interest} \times \text{number of years}

In 3 years the £500 earns 20 \times 3 = 60, which is £60. The amount in the account at the end is 500 + 60 = 560, which is £560.

Some questions work the other way. If £500 earns £60 in 3 years, one year's interest is 60 \div 3 = 20, and £20 as a percentage of £500 is \frac{20}{500} \times 100 = 4\%. If you know one year's interest, the number of years is the total interest divided by one year's interest.

With compound interest, each year's interest is added to the account. The next year's interest is a percentage of this new, bigger amount, so you also earn interest on your interest.

For example, £2000 at 10\% compound interest earns £200 in year 1, giving £2200. In year 2 it earns 10\% of £2200, which is £220, giving £2420. In year 3 it earns 10\% of £2420, which is £242, giving £2662.

So work one year at a time: find the interest, add it on, then use the new amount for the next year. Multiplying the first year's interest by the number of years gives simple interest, not compound interest.

On a calculator, a multiplier is quicker. An increase of 3\% leaves you with 100\% + 3\% = 103\% of the amount, so you multiply by 1.03 each year. Take care: 3\% gives 1.03, not 1.3

For 3 years, multiply by 1.03 three times. You can write 1.03 \times 1.03 \times 1.03 as the power 1.03^3, and use the power key on your calculator:

\text{amount after } n \text{ years} = \text{amount invested} \times \text{multiplier}^n

So £2500 at 3\% compound interest for 3 years becomes 2500 \times 1.03^3 = 2731.8175, which is £2731.82 to the nearest penny.

If the rate changes from one year to the next, use each year's multiplier in turn. £2000 at 5\% in year 1 and then 3\% in year 2 becomes 2000 \times 1.05 \times 1.03 = 2163, which is £2163.

Since 2022 a formulae sheet has been given in the exam, and the compound interest formula is on it, but learn it anyway. It is written P\left(1 + \frac{r}{100}\right)^n, where P is the amount invested, r is the rate and n is the number of years. It is the same calculation as using a multiplier. Use it only for compound interest, never for simple interest.

When a value goes down by a percentage each year, the multiplier is less than 1. A loss of 15\% leaves 100\% - 15\% = 85\%, so multiply by 0.85 each year. After 3 years the value is the starting value \times\, 0.85^3

Don't add the percentages together. A 15\% loss for 3 years is not a 45\% loss, because each year's 15\% is taken from a smaller value. In fact 0.85^3 = 0.614125, so the value falls by about 39\%

Some questions ask after how many years a value first goes above or below an amount. Work out the value at the end of each year in turn, and stop at the first year that passes the amount. This works for savings growing with compound interest too.

Some questions give you two accounts, or two values, and ask which is bigger. Work out each one in full, compare them, then write a sentence that gives your decision and the amounts.

Read each account carefully. One may pay simple interest and the other compound interest. A rate may change after the first year, or money may be paid in or taken out part of the way through. Do the years in order, and add or take away any money at the point where it happens.

A higher rate does not always win. Over a few years, simple interest at a higher rate can beat compound interest at a lower rate, and a bigger starting amount can make up for a lower rate.

• Using simple interest when the question says compound. 4\% for 3 years is not 12\%: add each year's interest on before working out the next, or multiply by the multiplier once for each year.
• Using compound interest when the question says simple. Simple interest is the same amount every year, so multiply one year's interest by the number of years.
• Using the wrong multiplier. A 4\% increase is \times 1.04, not \times 1.4, a 1.5\% increase is \times 1.015, not \times 1.15, and a 20\% loss is \times 0.8, not \times 0.2.
• Forgetting to divide the rate by 100. 2.5\% is 0.025 as a decimal, so multiply by 0.025, not by 2.5.
• Giving the final amount when the question asks for the interest. For the interest, subtract the amount invested; for the amount in the account, include it.
• Rounding too early. Keep every figure on your calculator display until the end, then round money to the nearest penny.