# Understanding Interest

Making something of a generalisation, most currencies are now decimal. They work in tens and hundreds. There are 100 pence in a UK pound, 100 cents in a dollar, and 100 cents in a Euro.

This means that it’s quite easy to do basic sums involving money; you just do the sum as you would any other decimal numbers, and then add the currency sign.

The difficult bit comes in managing your money: living within your budget, understanding interest rates for savings and for loans and credit cards, and comparing prices. This page focuses on understanding interest. Together with our page on Savings and Loans, it should help you make informed decisions about borrowing and saving.

Our page on Real World Maths explains more about comparing prices, and our page on Budgeting explains how to plan your spending to live within your means.

## What is Interest?

Interest is the premium paid for the use of money.

If you have allowed the bank to use your money, which is what you do when you put it in a savings account, then the bank pays you. If the bank (or another organisation) has lent you money, then you pay them for the privilege.

Interest is calculated per period, usually but not always monthly or annually. It is almost invariably quoted as a percentage of the value of the money.

### Simple and Compound Interest

There are two ways in which interest may be paid: simple and compound.

1. Simple interest means that for each period, interest is calculated on the original amount. The interest will therefore be exactly the same in each period. This is, as it sounds, extremely simple to work out, but it is also, unfortunately, not the type of interest most commonly used.
2. In a compound interest system, interest is charged or earned on the amount present each period. That will include the interest gained from the previous period, which is added on to the original amount. This is the system used by banks and lenders to calculate interest on loans and savings accounts.

Some worked examples should help to show you how compound interest works in practice.

Worked Example 1

Holly places \$100 in a bank account paying 2% per year. After five years, how much is in the bank account, if the bank pays compound interest?

 Interest Amount Initial deposit \$100.00 At the end of Year 1 2% of \$100 = \$2 \$102.00 At the end of Year 2 2% of \$102 = \$2.04 \$104.04 At the end of Year 3 2% of \$104.04 = \$2.08 \$106.12 At the end of Year 4 2% of \$106.12 = \$2.12 \$108.24 At the end of Year 5 2% of \$108.24 = \$2.16 \$110.40

After 5 years Holly’s bank account contains \$110.40; she has earned \$10.40 in interest payments.

Note that if Holly had received only simple interest on her savings, she would have earned only \$10.00 in interest payments. She would have made \$2 a year - interest only on the original amount invested.

Worked Example 2

John has spent £50 on his credit card. The ‘loan’ is interest-free until the first bill is due, after which he needs to make a minimum payment of £5 each month.

Interest will be charged on any outstanding amount at 18% per month. John chooses to repay the money at £10 per month.

Assuming that he does not buy anything else using the same credit card, how long will it take before he does not owe anything, and how much will he finally have paid?

 Period End of month 1 Initial bill £50.00 First payment on account -£10.00 Outstanding amount £40.00 Interest at 18% £7.20 End of month 2 Second bill £47.20 Second payment on account -£10.00 Outstanding amount £37.20 Interest at 18% +£6.70 End of month 3 Third bill £43.90 Third payment on account -£10.00 Outstanding amount £33.90 Interest at 18% +£6.10 End of month 4 Fourth bill £40.00 Fourth payment on account -£10.00 Outstanding amount £30.00 Interest at 18% +£5.40 End of month 5 Fifth bill £35.40 Fifth payment on account -£10.00 Outstanding amount £25.40 Interest at 18% +£4.57 End of month 6 Sixth bill £29.97 Sixth payment on account -£10.00 Outstanding amount £19.97 Interest at 18% +£3.59 End of month 7 Seventh bill £23.56 Seventh payment on account -£10.00 Outstanding amount £13.56 Interest at 18% £2.44 End of month 8 Eighth bill £16.00 Eighth payment on account -£10.00 Outstanding amount £6.00 Interest at 18% £1.08 End of month 9 Ninth bill £7.08 Ninth payment on account -£7.08 Outstanding amount 0.00

John will therefore take 9 months to pay off his bill, and he will make eight payments of £10, plus one of £7.08, paying £87.08 in total. His £50 will therefore have cost him a further £37.08, or 74% of the original loan, before he has paid it off.

This example should help you to see why interest quickly mounts up on a loan.

You can also see that John’s early payments mostly repaid the interest, not the capital. Also note that if John had paid only the £5 minimum payment each month, he would not have paid off any of the capital and that the amount of money owed would simply keep growing, he would never pay it off.

For more about these ideas, see our page on Savings and Loans.