Number and ratio · GCSE Maths
Fractions, decimals and percentages
Convert fluently between fractions, decimals and percentages, then use multipliers for increase, decrease and reverse percentage.
Fractions, decimals and percentages are three costumes of the same number. Change the costume; do not change the value.
The important bits
What you need to know
- 1
To convert a fraction to a decimal, divide the numerator by the denominator. To convert a decimal to a percentage, multiply by 100.
- 2
A percentage is a fraction with denominator 100. 37.5% means 37.5/100, which simplifies to 3/8.
- 3
Equivalent fractions are found by multiplying or dividing the numerator and denominator by the same non-zero number.
- 4
To find a percentage of an amount, use a multiplier: 23% of 80 is 0.23 × 80. Increase by 23% with 1.23; decrease by 23% with 0.77.
- 5
Reverse percentage starts from the new amount. If a price is 80% of the original after a 20% decrease, divide by 0.8, not by 0.2.
- 6
Compound change repeats the multiplier. After three years at 4% growth, multiply by 1.04³ rather than adding 12%.
- 7
Write mixed numbers as improper fractions before multiplying or dividing, then convert back if the question asks for a mixed number.
- 8
A recurring decimal such as 0.1̇6̇ can be turned into a fraction by setting x equal to the decimal, multiplying to shift the repeat, and subtracting.
Go deeper
Multipliers stop the “of” mistakes
Students lose marks when they treat percentage as a separate topic from decimals. It is not. Finding 17% of 240 is multiplication by 0.17. Increasing 240 by 17% is multiplication by 1.17, because you keep the original 1 and add 0.17. Decreasing by 17% is multiplication by 0.83, because 1 − 0.17 = 0.83. The same idea scales: a 2.5% pay rise uses 1.025. Once the multiplier is written, the calculator work is ordinary. The thinking sits in choosing 1.17 rather than 0.17, and in writing that choice on the page so the method mark survives even if a button is pressed twice.
Go deeper
Reverse percentage is division, not a second decrease
If a jumper costs £48 after a 20% sale, £48 is 80% of the original, not 20% less in a second calculation. Original × 0.8 = 48, so original = 48 ÷ 0.8 = £60. The common error is to find 20% of 48 and add it back, which invents a different jumper. Always name what you have: “this is 80% of the start.” Then divide by the multiplier that produced it. The same structure works for VAT and for “after a 12% increase.” If the new value is 112% of the old, divide by 1.12. The story of the question tells you whether you multiply or divide; the multiplier tells you by how much.
Go deeper
Keep fractions exact until the last line
Where a question allows a fraction, keep it. 2/3 of 90 is cleaner as (2 × 90) ÷ 3 than as 0.666… × 90. Cancelling before you multiply reduces arithmetic: 3/8 × 4/9 becomes 1/8 × 4/3 after cancelling a factor of 3. Division of fractions is multiply by the reciprocal: 2/5 ÷ 3/4 = 2/5 × 4/3. On non-calculator papers this is not a style preference; it is how you avoid rounding that the mark scheme will not accept. Convert to a decimal or a percentage only when the question asks for one, or when comparison of size is clearer in that form.
See the idea in action
A coat is reduced by 15% to £68. Find the original price. The sale price is 85% of the original, so original × 0.85 = 68. Original = 68 ÷ 0.85 = £80. Check: 15% of £80 is £12, and £80 − £12 = £68.
Exam technique
Turn knowledge into marks
Write the multiplier before you press any keys. Examiners award method for 68 ÷ 0.85 even if the decimal is mistyped; they cannot award it for a lone 80 with no story.
Common mistakes
Do not give these marks away
- 01
Finding a percentage of the new amount and adding it back on reverse-percentage questions.
- 02
Using 0.4 for a 4% change instead of 0.04, or using 1.4 for a 4% increase.
- 03
Rounding a fraction such as 2/3 to 0.67 too early and then missing the exact mark.
A jacket is reduced by 20% to £64. What was the original price?
A£76.80
B£80
C£84
D£51.20
Show the answer
£80. £64 is 80% of the original, so divide by 0.8: 64 ÷ 0.8 = 80. Adding 20% of 64 back on would be the classic reverse-percentage error.
Quick questions
If this is the bit you searched
When should I use a multiplier instead of finding 10% and 1%?
Use a multiplier whenever the percentage is awkward (17%, 2.5%, 12.5%) or when the change is repeated. Building from 10% is fine for simple non-calculator work, but the multiplier is the method that scales.
How do I convert 3/8 to a percentage without a calculator?
Divide 3 by 8: 8 into 30 is 3 remainder 6, 8 into 60 is 7 remainder 4, 8 into 40 is 5, so 0.375, then multiply by 100 to get 37.5%.