Number

Percentages

Find a percentage of an amount, percentage change, reverse percentages, and profit/loss

Explanation & Worked Examples

What a Percentage Is

A percentage means "out of 100". So 25% means 25 out of 100, which is the same as the fraction 25/100 and the decimal 0.25.

PercentageFractionDecimal
10%1/100.1
25%1/40.25
50%1/20.5
75%3/40.75
100%11.0
  • Percentage → decimal: divide by 100 (45% = 0.45).
  • Decimal → percentage: multiply by 100 (0.2 = 20%).
Key idea: "of" means multiply. 25% of 60 = 0.25 × 60. Convert the percentage to a decimal or fraction first.

Finding a Percentage of an Amount

Two reliable methods — pick whichever makes the numbers easy.

Decimal method
Turn the % into a decimal, then multiply.
30% of 90 = 0.3 × 90 = 27
Building-block method
Find 10% (÷10) and 1% (÷100), then combine.
10% of 90 = 9, so 30% = 3 × 9 = 27
Example — 15% of 240
110% of 240 = 24
25% is half of 10% = 12
315% = 24 + 12 = 36
What percentage is one number of another? Use (part ÷ whole) × 100. e.g. 15 out of 60 = 15/60 × 100 = 25%.

Percentage Increase & Decrease

You can either find the change and add/subtract it, or use a multiplier in one step — the multiplier method is faster and essential for the harder questions.

Increase by 20% → multiply by 1.20  (100% + 20% = 120% = 1.2)
Decrease by 20% → multiply by 0.80  (100% − 20% = 80% = 0.8)
Example 1 — Increase

Increase 200 by 10%.

1Multiplier = 1.10
2200 × 1.10 = 220
Example 2 — Decrease (a sale price)

A coat costs £40. In a 15% off sale, what is the new price?

1Multiplier = 1 − 0.15 = 0.85
2£40 × 0.85 = £34
Tip: for adding VAT at 20%, just multiply by 1.2. £25 × 1.2 = £30.

Reverse Percentages

Here you are told the amount after a change and must find the original. The trap is to apply the percentage to the new figure — instead, divide by the multiplier.

Original = New amount ÷ multiplier
Example 1 — After an increase

After a 20% increase, a price is £60. What was the original?

1A 20% increase uses multiplier 1.2
2Original = 60 ÷ 1.2 = £50
3Check: £50 × 1.2 = £60 ✓
Example 2 — After a discount

After a 25% discount, an item costs £75. What was the original price?

1A 25% discount uses multiplier 0.75
2Original = 75 ÷ 0.75 = £100
Common mistake: 25% of £75 is £18.75, giving £93.75 — that is wrong. You must divide by 0.75, not take 25% of the new price.

Profit, Loss & Compound Change

Profit and loss percentage

Profit (or loss) % = (profit or loss ÷ cost price) × 100

Always divide by the original cost price, not the selling price.

Example 1 — Percentage profit

A shop buys an item for £40 and sells it for £50.

1Profit = 50 − 40 = £10
2Profit % = 10 ÷ 40 × 100 = 25%

Successive (compound) changes

Apply each change in turn by multiplying — do not just add the percentages.

Example 2 — Compound interest

£500 is invested at 10% compound interest per year. What is it worth after 2 years?

1Each year multiply by 1.10
2500 × 1.10 × 1.10 = 500 × 1.21 = £605
Example 3 — Up then down

A price of £100 rises by 10%, then falls by 10%. What is the final price?

1100 × 1.10 = 110
2110 × 0.90 = £99 — not £100!
Why £99? The 10% fall is taken off the bigger £110, so it removes more than the rise added. A rise and fall by the same percentage never return you to the start.
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