Percentages

What is a Percentage?

A percentage (%) means per hundred — it is a way of expressing a number as a fraction of 100.

50% = 50 out of 100 = 1/2 = 0.5

Percentages appear everywhere: sales discounts, tax rates, test scores, statistics. Mastering them is one of the most practical maths skills you can develop.

Key Terms

• Percentage: a number expressed as a part of 100
• Multiplier: the decimal you multiply by to apply a percentage change (e.g. 1.15 for a 15% increase, 0.85 for a 15% decrease)
• Original value: the starting amount before any percentage change
• Percentage change: how much something has risen or fallen, expressed as a percentage

Converting Between Formats

Every percentage can also be written as a fraction and a decimal. Knowing all three forms makes calculations much faster.

Fraction to Percentage: divide numerator by denominator, then multiply by 100

• 3/4 = 0.75 × 100 = 75%

Percentage to Decimal: divide by 100

• 30% = 30 ÷ 100 = 0.30

Decimal to Percentage: multiply by 100

• 0.08 × 100 = 8%

Finding a Percentage of a Quantity

Method 1 — Decimal multiplier (fastest): What is 30% of 200? → 0.30 × 200 = 60

Method 2 — Build up from 10%: 10% of 200 = 20, so 30% = 3 × 20 = 60

Method 3 — Fraction method: 30% = 30/100, so (30/100) × 200 = 6000/100 = 60

All three methods give the same answer. Choose whichever feels most natural.

Percentage Increase and Decrease

Using the multiplier method is the most efficient approach for exam questions.

Increase: Multiply by (1 + rate)

• Increase £80 by 20%: multiplier = 1.20 → 80 × 1.20 = £96
• Increase 150 by 6%: multiplier = 1.06 → 150 × 1.06 = £159

Decrease: Multiply by (1 - rate)

• Decrease £80 by 15%: multiplier = 0.85 → 80 × 0.85 = £68
• Decrease 200 by 35%: multiplier = 0.65 → 200 × 0.65 = £130

Worked Example — Percentage Increase

A shop increases the price of a laptop from £640 to £720. Find the percentage increase.

• Change = 720 - 640 = £80
• Percentage increase = (80 ÷ 640) × 100 = 12.5%

Percentage Change Formula

Percentage change = (change ÷ original) × 100

• A positive result = percentage increase
• A negative result = percentage decrease

Example: A value drops from 80 to 60. Change = 60 - 80 = -20 Percentage change = (-20 ÷ 80) × 100 = -25% (a 25% decrease)

Finding the Original Value (Reverse Percentages)

When you know the result after a percentage change and need to find the original, divide by the multiplier.

After a 20% increase, a price is £96. Find the original.

• Multiplier used = 1.20
• Original = 96 ÷ 1.20 = £80

After a 15% discount, a coat costs £68. Find the original price.

• Multiplier used = 0.85
• Original = 68 ÷ 0.85 = £80

Common Mistakes to Avoid

• Adding 20% then subtracting 20% does NOT get back to the start. If you increase 100 by 20% you get 120. Decreasing 120 by 20% gives 96, not 100.
• 1% is not 0.1 — 1% = 0.01 as a decimal (0.1 is 10%).
• Do not find the percentage of the wrong value. For percentage change, always divide by the original value, not the new one.
• Percentages over 100% are valid. 150% = 1.5 as a decimal (more than the original).

Tips and Tricks

• Memorise common multipliers: 10% → ×0.1, 25% → ×0.25, 50% → ×0.5
• To find 17.5%, find 10% + 5% + 2.5% (halve each time)
• For reverse percentage: always divide by the multiplier, never subtract the percentage from the result
• Check answers make sense — after a discount, the price must be lower than the original