Ratio: revision notes and practice
A ratio compares amounts. The total number of ratio parts is the key to sharing a whole amount.
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What you will learn
- Interpret part-to-part and part-to-whole comparisons.
- Simplify ratios using consistent units.
- Share a total or recover an amount from a known share.
- Scale quantities in direct proportion and state the assumption.
Key rules
Simplify a ratio
Put quantities in the same unit, then divide every part by a common factor. For example, 6:9 = 2:3.
Share an amount
Add the ratio parts. Divide the total amount by this sum to find one part, then multiply for each share.
Scale a direct proportion
When quantities are directly proportional, they use the same scale factor. This assumes a constant amount per item or person.
Worked examples
Example 1: Share £84 between Alex and Bea in the ratio 2:5.
- Total parts = 2 + 5 = 7.
- One part = £84 ÷ 7 = £12.
- Alex gets 2 × £12; Bea gets 5 × £12.
Alex: £24.00; Bea: £60.00
Example 2: A recipe uses 300 g of flour for 4 equal servings. How much is needed for 7 servings?
- Assume the amount per serving stays constant.
- One serving needs 300 ÷ 4 = 75 g.
- Seven servings need 75 × 7 = 525 g.
525 g
A practice question
Simplify the ratio 55:99. Give the smallest whole-number ratio.
Check the answer and explanation
5:9
The highest common factor of all terms is 11. Divide each term by it to get 5:9.
1. Read a ratio in the stated order
A ratio compares quantities multiplicatively. If red counters to blue counters is 2:3, there are two red parts for every three blue parts. This could mean 2 red and 3 blue, 8 red and 12 blue, or any positive common multiple. It does not specify an actual total on its own.
Order matters: red:blue = 2:3 is different from blue:red = 2:3. Label each part before calculating. A bar model with two equal boxes for red and three for blue makes the comparison visible without assuming how many counters each box represents.
2. Simplify after converting units
Equivalent ratios multiply or divide every part by the same non-zero factor. The ratio 18:30 simplifies to 3:5 when both parts are divided by their highest common factor, 6. Changing only one part changes the comparison.
Convert compatible measurements to the same unit first. For 150 cm:2 m, write 150:200, then simplify to 3:4. For a ratio containing decimals, multiplying all parts by 10 or 100 can create integers before simplification. Thus 1.5:2.5 becomes 15:25 and then 3:5.
3. Connect part-to-part ratios to fractions
In a 2:3 red-to-blue ratio, there are five parts altogether. Red is 2/5 of the total and blue is 3/5. The fraction 2/3 instead compares the red amount with the blue amount. Both are useful, but they answer different questions.
Always identify whether a statement compares part with part or part with whole. If boys:girls = 4:5, boys:all students = 4:9, not 4:5. Ratios with more than two parts work the same way: in 1:2:3, the middle quantity is 2/6, or 1/3, of the whole.
For a:b, the first share is a/(a + b) of the total.
4. Share a total by finding one part
Add the ratio numbers to find the total number of equal parts. Then divide the whole amount by that total and multiply by each ratio number. To share £96 in the ratio 3:5, there are eight parts, each worth £12; the shares are £36 and £60.
Check that the shares add to the original total and have the correct relative size. The second share should be larger because 5 is larger than 3. Sharing £96 by simply dividing by 3 and by 5 separately would produce amounts that do not represent eight equal parts.
5. Start from a known share or difference
Sometimes one share is given instead of the total. If Alex:Bea = 2:5 and Bea receives £35, five parts represent £35. One part is £7, Alex receives £14, and the total is £49. Divide by the ratio number corresponding to the stated share.
If the difference is given, use the difference in ratio parts. With a 2:5 ratio and a difference of £18, the three-part difference represents £18, so one part is £6. The shares are £12 and £30. Drawing the matching bars helps explain why the difference uses three parts rather than seven.
6. Scale direct proportions
In direct proportion, corresponding quantities use the same multiplier and their ratio stays constant. If 300 g of flour makes four equal servings, each serving requires 75 g. Seven servings therefore need 7 × 75 = 525 g, assuming the recipe scales consistently.
A unitary method finds one unit first, then scales to the required number. A scale-factor method uses 7/4 directly: 300 × 7/4 = 525. These routes are equivalent. Direct proportion is not guaranteed in every situation; a fixed delivery charge, for example, can stop total cost being proportional to item count.
7. Compare mixtures and value fairly
Two mixtures may have different totals but the same ratio. A drink made with concentrate:water = 1:4 contains concentrate as 1/5 of the total. Scaling both ingredients by the same factor preserves the concentration. Adding only water or only concentrate changes the ratio.
For value comparisons, put quantities on a common basis. If 300 g costs £1.20 and 500 g costs £1.75, the prices per 100 g are £0.40 and £0.35. The second is cheaper per equal quantity. Compare like units and check whether the question asks for unit value, total cost, or a particular ratio.
8. Model changing amounts carefully
When an amount is added or removed, the old ratio does not automatically stay true. Suppose red:blue = 2:3 and there are 10 blue counters. One part is 10/3, so the implied red amount is 20/3; that cannot be a whole number of counters. The given information would be inconsistent for indivisible counters.
Choose a scale factor compatible with the context, then update actual quantities before forming a new ratio. If red:blue = 2:3 with 10 red and 15 blue, adding 5 red gives 15:15 = 1:1. Keep people or counters whole when required, and check both the arithmetic and the meaning of the result.
Check your understanding
Simplify 24:36.
2:3, dividing both parts by 12.
Share £90 in the ratio 2:3.
£36 and £54; five parts make £90, so one part is £18.
In a 3:7 ratio, the second amount is 42. Find the first.
18, because seven parts equal 42 and one part equals 6.
Simplify 0.75 m:50 cm.
3:2, from 75 cm:50 cm.
How to earn the marks
- Write labels in the same order as the ratio.
- Convert units before simplifying.
- State what one part represents, especially for a known share or difference.
- Check the final total, relative sizes and any whole-number requirement.
A common mistake
In a 2:5 ratio, the first share is 2/7 of the total, not 2/5.
Course context and official specifications
This guide covers selected skills. Some extensions are Higher content; choose your official tier separately from practice difficulty in the app. These are original questions and explanations.
- AQA GCSE Mathematics — R4, R5, R8
Simplifying ratios, sharing and using ratios are available at both tiers. Full direct and inverse proportion coverage is not included.
- Edexcel GCSE Mathematics — R4, R5, R8
Simplifying ratios, sharing and using ratios are available at both tiers. Full direct and inverse proportion coverage is not included.
- OCR GCSE Mathematics — 5.01
Simplifying ratios, sharing and using ratios are available at both tiers. Full direct and inverse proportion coverage is not included.
Keep your next step small
Try five questions, read the feedback, and revisit one thing you found difficult. The app tracks a 20-answer practice goal separately from accuracy.
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