Percentages

This lesson covers DfE content statements L2.5 and L2.6 — finding percentages of amounts, expressing one number as a percentage of another, percentage increase and decrease, and finding the original value after a percentage change (reverse percentages).

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What Is a Percentage?

A percentage is a fraction with a denominator of 100. The word comes from the Latin per centum — "per hundred."

25% means 25 out of 100, or 25/100, or 0.25.

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Finding a Percentage of an Amount

There are several methods. Choose whichever is easiest for the numbers involved.

Method 1 — Convert to a Decimal

Divide the percentage by 100 and multiply.

Scenario: A washing machine costs £650. There is a 15% discount. How much do you save?

15% of £650 = 0.15 × 650 = £97.50

Method 2 — The "Building Blocks" Method (Non-Calculator)

Find 10% by dividing by 10, and 1% by dividing by 100. Then build up to the percentage you need.

Scenario: Find 35% of £240 without a calculator.

Building block	Calculation	Amount
10%	240 ÷ 10	£24
30%	£24 × 3	£72
5%	£24 ÷ 2	£12
35%	£72 + £12	£84

Method 3 — Fraction Equivalents

Use known fraction-percentage equivalents for quick mental calculations.

Percentage	Fraction	To calculate...
50%	1/2	÷ 2
25%	1/4	÷ 4
75%	3/4	÷ 4, × 3
20%	1/5	÷ 5
10%	1/10	÷ 10
1%	1/100	÷ 100
33.3...%	1/3	÷ 3
12.5%	1/8	÷ 8

Exam Tip: The building blocks method (Method 2) is the safest approach on the non-calculator paper. Always show your working — start with 10% and build from there.

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Expressing One Number as a Percentage of Another

Formula: (Part ÷ Whole) × 100

Worked Example — Test Score

Scenario: You scored 42 out of 60 on a workplace assessment. What is your percentage score?

(42 ÷ 60) × 100 = 0.7 × 100 = 70%

Worked Example — Quality Control

Scenario: A factory produced 1,500 items. 45 were defective. What percentage were defective?

(45 ÷ 1,500) × 100 = 0.03 × 100 = 3%

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Percentage Increase and Decrease

Method 1 — Find the Amount, Then Add or Subtract

Scenario (Increase): A landlord increases rent of £850 by 4%. What is the new rent?

4% of £850 = 0.04 × 850 = £34 New rent = £850 + £34 = £884

Scenario (Decrease): A coat originally costs £120 and is reduced by 30%. What is the sale price?

30% of £120 = 0.30 × 120 = £36 Sale price = £120 − £36 = £84

Method 2 — The Multiplier Method (Faster)

For an increase, the multiplier is 1 + (percentage ÷ 100). For a decrease, the multiplier is 1 − (percentage ÷ 100).

Change	Multiplier
+5%	× 1.05
+12%	× 1.12
+20%	× 1.20
−10%	× 0.90
−25%	× 0.75
−33%	× 0.67

Worked Example — VAT

Scenario: A plumber quotes £360 before VAT. VAT is 20%. What is the total including VAT?

Multiplier for +20% = 1.20 £360 × 1.20 = £432

Worked Example — Depreciation

Scenario: A company van was bought for £24,000. It depreciates by 15% per year. What is it worth after one year?

Multiplier for −15% = 0.85 £24,000 × 0.85 = £20,400

Exam Tip: The multiplier method is quicker and less error-prone. Use it whenever you have a calculator. On the non-calculator paper, use the building blocks method instead.

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Percentage Change

Formula: Percentage change = (Change ÷ Original) × 100

Worked Example — House Price

Scenario: A house was bought for £180,000 and sold for £207,000. What is the percentage increase?

Change = £207,000 − £180,000 = £27,000 Percentage increase = (27,000 ÷ 180,000) × 100 = 15%

Worked Example — Weight Loss

Scenario: A person weighed 96 kg and now weighs 84 kg. What is the percentage decrease?

Change = 96 − 84 = 12 kg Percentage decrease = (12 ÷ 96) × 100 = 12.5%

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Reverse Percentages (Finding the Original Value)

When you know the value after a percentage change, you can work backwards to find the original.