14.1 Percentages

A free GCSE Maths Foundation lesson for Edexcel (9–1) from OBA School: fractions, decimals and percentages, percentage change, interest, tax and reverse percentages. Watch the video, read the explanation and worked examples, then try the revision questions. You can work through the full lesson, with interactive activities, practice quizzes and printable worksheets, in our free online course.

What you will learn

• Convert between fractions, decimals and percentages, and put them in order
• Find a percentage of an amount with and without a calculator
• Increase or decrease an amount by a percentage, and work out a percentage change, profit or loss
• Work out simple interest and find the original amount before a percentage change

Key words

• Per cent (%): Out of 100. 37% means 37/100 = 0.37.
• Multiplier: The number you multiply by to find a percentage or make a change. 15% of x is 0.15x; a 15% increase is 1.15x.
• Percentage change: The change as a percentage of the original amount.
• Simple interest: Interest that is the same amount every year, worked out on the original amount only.
• VAT: Value Added Tax, added to the price of many goods and services. In the UK the standard rate is 20%.

Fractions, decimals and percentages

To change a percentage to a decimal, divide by 100. To change a decimal to a percentage, multiply by 100. For a fraction, divide the top by the bottom to get a decimal first.

• Decimal; 1/2 0.5; 1/4 0.25; 3/4 0.75; 1/5 0.2; 1/10 0.1; 1/8 0.125; 1/3 0.333…
• Percentage; 1/2 50%; 1/4 25%; 3/4 75%; 1/5 20%; 1/10 10%; 1/8 12.5%; 1/3 33.3…%

Worked example: putting in order

Write in order, smallest first: 0.63, 5/8, 61%, 2/3.
As decimals: 0.63, 0.625, 0.61, 0.666…
Order: 0.61, 0.625, 0.63, 0.666…

Answer: 61%, 5/8, 0.63, 2/3

Percentage of an amount

Without a calculator, build up from 10% (÷ 10), 5% (half of 10%) and 1% (÷ 100). With a calculator, multiply by the decimal.

Worked example: 35% of £240

Non-calculator: 10% = £24, so 30% = £72. 5% = £12. 35% = £72 + £12 = £84.
Calculator: 0.35 × 240 = 84.

Answer: £84

Worked example: one amount as a percentage of another

Priya scores 42 out of 60 in a test. What percentage is this?
42/60 × 100 = 70.

Answer: 70%

Percentage increase and decrease

Either find the percentage and add or subtract it, or use a single multiplier.

• Increase by 20% (VAT); Multiplier 1.2; Example £85 + VAT: 1.2 × 85 = £102
• Decrease by 30% (sale); Multiplier 0.7; Example £85 in a 30% sale: 0.7 × 85 = £59.50
• Increase by 4.5%; Multiplier 1.045; Example £300 rent rise: 1.045 × 300 = £313.50

Key fact: Percentage change = change ÷ original × 100. Profit or loss is a change, so use the original (cost) price.

Worked example: profit

A trader buys a lamp for £40 and sells it for £52. Work out the percentage profit.
Profit = £52 − £40 = £12. 12/40 × 100 = 30.

Answer: 30% profit

Simple interest and income tax

Simple interest is the same every year, because it is always worked out on the original amount.

Worked example: simple interest

Tom invests £1500 at 4% per year simple interest for 3 years.
One year’s interest = 4% of £1500 = £60. Three years = 3 × £60 = £180.
Total after 3 years = £1500 + £180 = £1680.

Answer: £1680

Worked example: income tax (made-up rates)

In a game, players pay no tax on the first £10 000 they earn and 20% on the rest. Kit earns £26 000.
Taxed amount = £26 000 − £10 000 = £16 000. Tax = 0.2 × 16 000 = £3200.

Answer: £3200

Reverse percentages

If you know the amount after a percentage change, work backwards to the original. Do not take the percentage of the new amount.

Worked example: a sale price

In a 15% sale a coat costs £68. Find the price before the sale.
£68 is 100% − 15% = 85% of the original.
1% = 68 ÷ 85 = £0.80, so 100% = £80. (Or: 68 ÷ 0.85 = 80.)

Answer: £80

Common mistakes

• Dividing by the new amount when finding a percentage change. Always divide by the original.
• Reverse percentages: taking 15% of the sale price and adding it on. 68 + 15% of 68 = £78.20, not £80.
• Thinking a 10% rise followed by a 10% fall gets you back to the start. It does not.
• Confusing 0.5% (half of one per cent) with 50%.

Revision questions

Try these, then check your answers below each one.

Question 1. Work out 15% of £340.

Answer: 51 £

Working: 10% = £34, 5% = £17. 34 + 17 = £51.

Question 2. Which of these is the largest? (Choose from: 0.72, 7/10, 71%, 17/25.)

Answer: 0.72

Working: 7/10 = 0.7, 71% = 0.71, 17/25 = 0.68. The largest is 0.72.

Question 3. Ben scores 18 out of 40 in a quiz. What percentage is this?

Answer: 45 %

Working: 18/40 × 100 = 45%.

Related lessons

• Previous lesson: 13.6 More Tree Diagrams
• Next lesson: 14.2 Growth and Decay
• In this chapter: 14.3 Compound Measures
• In this chapter: 14.4 Distance, Speed and Time
• In this chapter: 14.5 Direct and Inverse Proportion

Continue in the full course

This lesson is section 14.1 (EGCMF1401) of our free GCSE Maths Foundation course. In the full course you get every video for this section, an animated interactive lesson, a self-marking practice quiz, a printable worksheet with a mark scheme, and your progress saved. Go to the full GCSE Maths Foundation course.