## Percentage Increase and Decrease

Here you are given a start number and you have to increase or decrease that number by a certain percentage.

There are two ways to solve these type of problems.

The first way is normally how you would do it in your head or on paper. For example: Fred earns £110 per week and gets a 5% increase, how much will his new weekly wage be? In your head you would work out 5% of £110 and add it to the original sum. 5% of £110 is £5.50, add this to £110 to give the answer of £115.50.

Alternatively, if you were using a calculator you would add the percentage increase to 100% to give 105%, convert this to a decimal to give 1.05 and then multiply it by the original wage, 1.05 x £110 = £115.50.

### Percentage Profit or Loss

Is the percentage profit or loss of the **cost price.**

So if the cost price is £10 and the sales price is £20, then the profit is £10 (£20 — £10) and the percentage profit = £10/£10 x 100% = 100%

In words:-

Percentage Profit (or loss) = actual profit (or loss)/cost price X 100%

### Repeated Percentage Change

This is most often used in finance. Usually when you invest money, interest is calculated using compound interest. This is where the interest is paid on the initial investment plus on any interest already earned. Sounds more complicated than it is! Best to look at an actual example:

John invests £1000 in a savers account which earns 5% interest annually compound interest. Assuming he makes no withdrawals, how much will he have in his account;

a) After 1 year

b) After 2 years

**Answer**

a) £1000 x 5% = £50, so after one year he’ll have £1,000 + £50 = £1,050

b) After 2 years he’ll have £1,050 (from the first year) + 5% interest on £1,050=

£1,050 + (£1,050 x 0.05) = £1,050 + £52.50* = £1,102.50

*1% of £1,050 = £10.50, so 5% of £1,050 = 5 x £10.50 = £52.50

### Reverse Percentages

This is another case where the two different methods, one without a calculator and one with. Best explained with an example.

A shirt is on sale at £60 and the ticket says that it has been reduced by 20%. What was the original price?

If you don’t have a calculator, follow these two steps:

a) Work out what percentage the new price represents of the original price:-

In this case 100% — 20% = 80%

b) Multiply the revised price by 100/(your answer from (a))

In this case £60 x 100/80

At this point could divide by 80 and multiple by 100 BUT it’s often a lot simpler to use your knowledge of fractions:-

so £60 x 100/80 = £60 x 10/8 = £60 x 5/4 = £60 x 1.25 (we know that 0.25 = a quarter)

so £60 x 1.25 = £60 + a quarter of £60 = £60 + £15 = £75.

If you do have a calculator follow these steps (note the first step is the same as the “without a calculator method” shown above):

a) Work out what percentage the new price represents of the original price:-

100% — 20% = 80%

b) Divide the answer by 100 to get a “multiplier”:-

80/100 = 0.8

c) Divide the new price by the multiplier to get the original price:-

£60/0.8 = £75