Ratio and Proportion
A ratio compares two or more quantities by showing how many times bigger one is than another, written with a colon such as 3 : 2, while proportion describes how quantities scale together. 11+ papers test simplifying ratios, sharing amounts in a given ratio, and direct proportion problems such as best-value and scaling recipes.
Method
- Simplify a ratio by dividing every part by their highest common factor, just as you would simplify a fraction.
- To share an amount in a given ratio, add the parts to find the total number of parts, divide the amount by this total to find the value of one part, then multiply by each part of the ratio.
- For direct proportion problems such as recipes, best value or exchange rates, find the value of a single unit first, then scale it up or down to the amount you need.
- When a ratio compares two groups and you are given a difference rather than a total, find the value of one 'gap' part first, then scale to find the total or individual amounts.
- To combine two ratios that share a common quantity, such as A:B and B:C, scale both ratios so the shared term matches, then merge them into a single three-part ratio.
- Check your final ratio or shared amounts by adding the parts back together to confirm they match the original total.
Worked example
56 pounds is shared between Ben and Cara in the ratio 3 : 5. Work out how much more Cara receives than Ben.
- Add the parts of the ratio to find the total number of parts: 3 + 5 = 8.
- Divide the total amount by the number of parts to find the value of one part: 56 / 8 = 7 pounds.
- Find Ben's share by multiplying his number of parts by the unit value: 3 x 7 = 21 pounds.
- Find Cara's share by multiplying her number of parts by the unit value: 5 x 7 = 35 pounds.
- Subtract Ben's share from Cara's share to find the difference: 35 - 21 = 14 pounds.
Practice questions
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Exam-style questions
Written in the style of a 11+ exam paper, with a full mark scheme.
The ratio of cats to dogs at an animal shelter is 3 : 7. Calculate what percentage of the animals at the shelter are dogs.
In a choir, the ratio of sopranos to altos is 5 : 3. There are 16 more sopranos than altos. Calculate the total number of members in the choir.
A school has 240 pupils. The ratio of pupils who take the bus, walk or cycle to school is 5 : 3 : 2. The number of pupils who cycle is expected to increase by 50%, while the total number of pupils remains 240 and the ratio of bus to walk, among the remaining pupils, stays the same as before. Calculate the new ratio of bus to cycle to walk, giving your answer in its simplest form.
Free printable worksheet
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