Percentages

Spec reference: Number - Fractions, decimals and percentages
Key idea: Calculate percentages of amounts, percentage change, and work with reverse percentages.

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Percentages of amounts

Method 1 - Using a multiplier:

Convert the percentage to a decimal, then multiply.

Example

Find 35% of 240.

$35\mathrm{\%}=0.35$

$0.35\times 240=\boxed{{\displaystyle 84}}$

Method 2 - Build from 10%:

Example

Find 15% of 80.

$10\mathrm{\%}=8$

$5\mathrm{\%}=4$

$15\mathrm{\%}=8+4=\boxed{{\displaystyle 12}}$

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Percentage increase and decrease

$$\text{Multiplier for increase}=1+\frac{\text{percentage}}{100}$$$$\text{Multiplier for decrease}=1-\frac{\text{percentage}}{100}$$

Example

Increase £360 by 15%.

Multiplier: $1+0.15=1.15$

$360\times 1.15=\boxed{{\displaystyle \pounds 414}}$

Example

Decrease 520 by 30%.

Multiplier: $1-0.30=0.70$

$520\times 0.70=\boxed{{\displaystyle 364}}$

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Percentage change

$$\text{Percentage change}=\frac{\text{change}}{\text{original}}\times 100$$

Example

A jacket costs £80 and is reduced to £52. Find the percentage decrease.

$$\text{Change}=80-52=28$$$$\text{Percentage decrease}=\frac{28}{80}\times 100=\boxed{{\displaystyle 35\mathrm{\%}}}$$

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Reverse percentages (finding the original)

If you know the value after a percentage change, divide by the multiplier.

Example

After a 20% increase, a price is £156. Find the original price.

Multiplier: $1.20$

$$\text{Original}=156÷1.20=\boxed{{\displaystyle \pounds 130}}$$

Common mistake

Do not just find 20% of 156 and subtract. You must divide by the multiplier.

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Repeated percentage change

Apply the multiplier multiple times.

Example

A car costs £12,000. It depreciates by 15% each year. Find its value after 3 years.

Multiplier: $0.85$

$$12000\times {0.85}^3=12000\times 0.614125=\boxed{{\displaystyle \pounds 7,369.50}}$$

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Expressing one quantity as a percentage of another

$$\text{Percentage}=\frac{\text{part}}{\text{whole}}\times 100$$

Example

Express 45 as a percentage of 120.

$$\frac{45}{120}\times 100=\boxed{{\displaystyle 37.5\mathrm{\%}}}$$

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Exam tips

Watch out for

• Reverse percentages: always divide by the multiplier, never work backwards by taking a % off the final value
• "Percentage of" means multiply
• Always use the original value as the denominator in percentage change questions

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Test Yourself

Question 1 of 5

Find 35% of 280