Writing and Simplifying Ratios
A ratio compares two or more quantities, showing the relationship between their sizes rather than a single value, written using a colon, such as 3 : 5, with the quantities kept in the exact order stated and measured in the same unit before it is written down. A ratio is simplified by dividing every part by their highest common factor until the parts share no factor other than 1, and is sometimes written in the form 1 : n for comparing scales or concentrations.
Before you start
Make sure you're comfortable with these topics first:
Method
- Write down the quantities being compared in the exact order stated in the question, separated by a colon; a ratio comparing 'cats to dogs' must keep cats first and dogs second throughout the question.
- Before simplifying, check every part of the ratio is in the SAME unit; if not, convert all parts to the smaller unit first so every number in the ratio represents the same kind of unit.
- Find the highest common factor (HCF) of all the parts of the ratio, then divide every part by this HCF to simplify it fully in one step.
- If you cannot spot the HCF straight away, simplify in stages instead, dividing all parts by any common factor you notice (such as 2 or 5) and repeating until no common factor remains.
- A ratio in its simplest form uses whole numbers with no common factor other than 1; check your final ratio cannot be divided down any further.
- To write a ratio in the form 1 : n (or n : 1), divide every part of the ratio by whichever part needs to become 1, keeping the other part as a decimal if it does not divide exactly.
- When a worded problem gives amounts rather than a ready-made ratio, write the amounts as a ratio in the stated order before simplifying.
Worked example
A youth club has 42 members in Year 8 and 63 members in Year 9. Write the ratio of Year 8 members to Year 9 members in its simplest form.
- Write the ratio in the order given, Year 8 to Year 9: 42 : 63.
- Find the highest common factor of 42 and 63: both divide exactly by 21 (42 = 21 x 2 and 63 = 21 x 3), so the HCF is 21.
- Divide both parts by 21: 42 / 21 = 2, and 63 / 21 = 3.
- Write the simplified ratio: 2 : 3.
Practice questions
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Q1A fruit bowl contains 16 apples and 24 pears. Write the ratio of apples to pears in its simplest form.Show answer
Answer: 2 : 3
Q2In a school choir, there are 18 boys and 30 girls. Write the ratio of boys to girls in its simplest form.Show answer
Answer: 3 : 5
Q3A cake recipe uses 900 g of flour and 1.2 kg of butter. Write the ratio of flour to butter in its simplest form, giving whole numbers.Show answer
Answer: 3 : 4 (convert 1.2 kg to 1200 g first, then simplify 900 : 1200)
Q4A gym session lasts 1 hour 20 minutes, of which 25 minutes is spent on cardio. Write the ratio of cardio time to total session time in its simplest form.Show answer
Answer: 5 : 16 (convert 1 hour 20 minutes to 80 minutes first, then simplify 25 : 80)
Q5A model train is built using a scale where 4 cm on the model represents 100 cm on the real train. Write this as a ratio in the form 1 : n.Show answer
Answer: 1 : 25
Q6A fruit punch recipe uses 300 ml of orange juice, 200 ml of pineapple juice and 500 ml of lemonade. Write the ratio of orange juice to pineapple juice to lemonade in its simplest form.Show answer
Answer: 3 : 2 : 5
Q7Explain why the ratio 20 : 32 is not written in its simplest form, and state its simplest form.Show answer
Answer: 20 and 32 share a common factor of 4 (20 = 4 x 5 and 32 = 4 x 8), so they can both be divided by 4; the simplest form is 5 : 8.
Q8Two classes take part in a fundraising challenge. Class 7X raises 60 pounds and Class 7Y raises 84 pounds. Write the ratio of the amount raised by 7X to the amount raised by 7Y in its simplest form.Show answer
Answer: 5 : 7
Exam-style questions
Written in the style of a KS3 Maths exam paper, with a full mark scheme.
A recipe for shortbread uses 500 g of butter and 1.25 kg of flour. (a) Convert 1.25 kg to grams. (b) Write the ratio of butter to flour in its simplest form.
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A type of concrete mix is made from cement, sand and gravel in the amounts 40 kg, 80 kg and 120 kg. Write the ratio of cement to sand to gravel in its simplest form.
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On a map, a distance of 8 cm represents an actual distance of 6 km. (a) Convert 6 km to cm (1 km = 100,000 cm). (b) Write the map's scale as a ratio in the form 1 : n.
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Free printable worksheet
Want more practice on paper? Download the writing and simplifying ratios worksheet pack - 5 pages of exam-style questions with a full mark scheme. One email opens every download in this browser for 14 days - no account, no card. Print it for personal and classroom use.
This topic is chapter 35 of KS3 Maths Workbook 1, the whole course as one free printable PDF.
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