# Brush up your maths: Percentages

## Percentages

A percentage is a fraction out of 100. For example, 10% is the same as 10/100. So taking 10% of something is the same as multiplying it by the fraction
10
–––
100

If we take 10% of 450 we get
10
–––   ×   450 = 45
100

It is the same as multiplying by 10 then dividing by 100, shifting the decimal point two places to the left: 450 × 10 = 4500, then shift the decimal point to give 45.0.

In the same way, 17½% of 450 is 450 × 17½ ÷ 100 = 7875.0 ÷ 100 = 78.75.

### Exercise: percentages

What is 20% of 150?

What is 30% of 764.86 to two decimal places?

## Converting percentages to decimals and decimals to percentages

To change a decimal to a percentage, multiply it by 100. For example 0.05 is equivalent to 5%, because 0.05 × 100 = 5. In the same way, 0.175 is equivalent to 17.5% or 17½%, because 0.175 × 100 = 17.5.

To change a percentage to a decimal, divide it by 100. For example 20% is equivalent to 0.20, because 20 ÷ 100 = 0.20.

### Exercise: percentages and decimals

What is 2½% as a decimal?

What is 0.15 as a percentage?

## Percentage increase and decrease

If we increase something by a percentage, we calculate that percentage and add it to the original number. For example, £1000 in a building society offering 5% interest will grow to £1050 over a year. We can do this by 5% of 1000 = 1000 × 5 ÷ 100 = 5000 ÷ 100 = 50. We then add 50 to 1000: 1000 + 50 = 1050.

We can also do this by changing the percentage to a decimal, 5% = 0.05, and adding this to 1.00, 1.00 + 0.05 = 1.05. We then multiply by this number: 1000 × 1.05 = 1050.

If we decrease something by a percentage, we calculate that percentage and subtract it from the original number. For example, £1000 income minus 40% tax will leave £600. We can do this by 40% of 1000 = 1000 × 40 ÷ 100 = 40000 ÷ 100 = 400. We then subtract 400 from 1000: 1000 – 400 = 600.

We can also do this by changing the percentage to a decimal, 40% = 0.40, and subtracting this from 1.00, 1.00 – 0.40 = 0.60. We then multiply by this number: 1000 × 0.60 = 600.

### Exercise: percentage increase and decrease

How much would a plumber's bill for £452 before tax be after 17½% VAT was added?

How much would a restaurant bill for £62.50 be after a 10% loyalty card bonus was subtracted?

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