Number and ratio · GCSE Maths
Ratio and proportion
Share amounts in a ratio, scale recipes, and distinguish direct proportion from inverse proportion with a unitary method.
A ratio splits a total into parts. Add the parts first; never share using only one side of the colon.
The important bits
What you need to know
- 1
A ratio a:b means a parts to b parts. The total number of parts is a + b. Each part is the total quantity divided by a + b.
- 2
Simplify a ratio by dividing all parts by the same number, as you would a fraction. 12:18 simplifies to 2:3.
- 3
To share £70 in the ratio 2:5, there are 7 parts, so one part is £10. The shares are £20 and £50.
- 4
If you are given one share rather than the total, find the value of one part from that share, then scale the other side.
- 5
Direct proportion: as one quantity doubles, so does the other. The unitary method finds the value of one, then multiplies.
- 6
Inverse proportion: as one quantity doubles, the other halves. Total work or total distance often stays constant.
- 7
Map scales, recipes and exchange rates are proportion. Write a single line that shows the scale factor clearly.
- 8
Best-buy questions compare a consistent unit: price per 100 g, or number of grams per pound, not mixed units.
Go deeper
Add the parts before you share
The most reliable ratio method is parts, not percentages. For 3:5, there are eight equal parts. If the total is 96, one part is 12, so the shares are 36 and 60. If instead you are told that the larger share is 60, then five parts equal 60, one part is 12, and the smaller share is 36. The total then follows as 96. Students who jump from 3:5 to “3/5 of the total” have described the wrong fraction: 3/5 would be three parts out of five, not three out of eight. Write “3 + 5 = 8 parts” every time until it is automatic. That single line prevents the most common ratio error in the paper.
Go deeper
Unitary method makes proportion honest
Direct proportion questions collapse when students scale the wrong way. If 8 pens cost £6.40, one pen costs £0.80, so 5 pens cost £4.00. Finding one first, then multiplying, is slower to write than a single scale factor, but it is harder to reverse. For recipes, find the multiplier from the given quantity to the required quantity: 12 biscuits from a 9-biscuit recipe is a scale factor of 12/9 = 4/3. Multiply every ingredient by 4/3. Inverse proportion needs a different picture. If 6 workers take 10 days, the work is 60 worker-days. Four workers take 60 ÷ 4 = 15 days. Name the conserved quantity — worker-days, litre-hours, journey-distance — before you divide.
See the idea in action
Share 84 in the ratio 3:4. Total parts = 7. One part = 84 ÷ 7 = 12. The shares are 3 × 12 = 36 and 4 × 12 = 48. Check: 36 + 48 = 84 and 36:48 simplifies to 3:4.
Exam technique
Turn knowledge into marks
If the question gives one part of the ratio rather than the total, do not add the parts to that number. Find the value of one part from the given share, then build the rest.
Common mistakes
Do not give these marks away
- 01
Treating 2:3 as 2/3 of the total instead of 2/5 and 3/5.
- 02
Scaling a recipe by adding on the extra biscuits instead of multiplying every ingredient by the same factor.
- 03
Using direct proportion on a question where time falls as the number of people rises.
£90 is shared in the ratio 2:7. How much is the smaller share?
A£20
B£25.71
C£70
D£18
Show the answer
£20. There are 9 parts, so one part is £10. The smaller share is 2 parts: £20. £70 is the larger share.
Quick questions
If this is the bit you searched
What is the difference between a ratio and a fraction?
A ratio compares parts with each other. A fraction compares a part with the whole. In 2:3, the first quantity is 2/5 of the total, not 2/3 of the total.
How do I know if a question is inverse proportion?
Ask whether the product should stay constant. More people, fewer days; faster speed, less time. If doubling one quantity halves the other, it is inverse.