Ratio and the number skills behind it
A ratio compares two or more quantities, written with a colon: 3:4 means "for every 3 of one thing, there are 4 of the other". Almost every ratio question relies on Number skills already covered elsewhere on this site — simplifying uses the HCF, and finding equivalent ratios uses the same idea as equivalent fractions. This page brings those skills together specifically for ratio problems.
Simplifying a ratio
Worked Example 1
Simplify the ratio 18:24 to its simplest form.
1
Find the HCF of 18 and 24: HCF = 6
2
Divide both parts of the ratio by 6: 18 ÷ 6 = 3, and 24 ÷ 6 = 4
Answer3:4
Sharing an amount in a given ratio
Worked Example 2
Share £120 in the ratio 2:3:5.
1
Add the parts of the ratio together: 2 + 3 + 5 = 10 parts in total
2
Find the value of 1 part: £120 ÷ 10 = £12
3
Multiply £12 by each share: 2 × £12 = £24, 3 × £12 = £36, 5 × £12 = £60
Answer£24 : £36 : £60
Each bar segment is the same width per part — the whole bar (10 parts) represents £120.
The unitary method
The unitary method means finding the value of one part or unit first, then scaling up or down to find any other amount you need.
Worked Example 3
A recipe uses flour and sugar in the ratio 5:2. If 15kg of flour is used, how much sugar is needed?
1
The flour corresponds to 5 parts, and this equals 15kg, so find the value of 1 part: 15 ÷ 5 = 3kg
2
Sugar is 2 parts: 2 × 3kg = 6kg
Answer6kg
Ratios and fractions
Worked Example 4
A garden is split into flowerbed and lawn in the ratio 3:4. What fraction of the garden is lawn?
1
Add the parts to find the total: 3 + 4 = 7 parts
2
Lawn is 4 parts out of the 7 total parts: 4/7
Answer4/7
Practice questions
Work through each question before checking the answers.
Foundation (Grade 3–5)
Q1Simplify the ratio 15:20.Foundation
Q2Share £80 in the ratio 1:3.Foundation
Q3Simplify the ratio 12:18:30.Foundation
Q4The ratio of red to blue counters is 2:5. There are 10 red counters. How many blue counters are there?Foundation
Q5A ribbon is cut in the ratio 3:5. Write each part as a fraction of the whole ribbon.Foundation
Higher (Grade 5–7)
Q6Share £150 in the ratio 2:3:5.Higher
Q7Simplify the ratio 0.4:0.6.Higher
Q8A concrete mix uses cement, sand and gravel in the ratio 1:2:3. How much sand is needed to make 120kg of concrete?Higher
Q9Two numbers are in the ratio 3:7 and their sum is 90. Find the two numbers.Higher
Q10Simplify the ratio 2/3 : 3/4.Higher
Higher — Hard (Grade 8–9)
Q11A map has a scale of 1:25,000. Find the real distance, in km, represented by 4cm on the map.Grade 8–9
Q12The ratio of boys to girls in a class is 3:4. There are 6 more girls than boys. How many students are in the class in total?Grade 8–9
Q13A sum of money is shared between A, B and C in the ratio 2:3:4. B receives £45 more than A. Find the total sum of money.Grade 8–9
Q14Two similar shapes have side lengths in the ratio 2:5. Find the ratio of their areas.Grade 8–9
Q15A recipe for 6 people uses 300g of rice. Using the unitary method, find how much rice is needed for 15 people.Grade 8–9
Answers
Foundation (Q1–Q5)
Q13:4
Q2£20 : £60
Q32:3:5
Q425 blue counters
Q53/8 and 5/8
Higher (Q6–Q10)
Q6£30 : £45 : £75
Q72:3(×10 gives 4:6, then simplify)
Q840kg(6 parts total, 1 part = 20kg, sand = 2 parts)
Q927 and 63(10 parts total, 1 part = 9)
Q108:9(multiply both sides by 12, the LCM of 3 and 4)
Higher — Hard (Q11–Q15)
Q111km(4 × 25,000 = 100,000cm = 1000m = 1km)
Q1242 students(the 1-part difference = 6, so 7 parts total = 42)
Q13£405(the 1-part difference between B and A = £45, so 9 parts total = £405)
Q144:25(area ratio = length ratio squared: 2² : 5²)
Q15750g(1 person = 50g, so 15 people = 750g)
Common mistakes
Common Mistake 1
Dividing the total by the wrong number of parts
When sharing in the ratio 2:3:5, there are 10 total parts (2+3+5), not 3 (the number of shares). Always add the ratio numbers together first.
Common Mistake 2
Writing a ratio as a fraction of the wrong total
For a ratio of 3:4, "3 as a fraction of the whole" is 3/7 (using the total of 7 parts), not 3/4 — the ratio 3:4 itself is not the same as the fraction 3/4.
Common Mistake 3
Not fully simplifying a ratio
12:18:30 simplifies fully to 2:3:5 by dividing by the HCF of all three numbers (6) — dividing by a smaller common factor, like 2, leaves the ratio only partly simplified (6:9:15).
Common Mistake 4
Using the difference in amounts instead of the difference in parts
When told "B receives £45 more than A", the £45 corresponds to the difference in parts between B and A's ratio numbers — not to a single part or the whole amount.
Exam tips
💡 Exam Tip 1
Always find the value of 1 part first
Whatever the ratio question, working out what a single part is worth is almost always the key first step — everything else follows by simple multiplication.
💡 Exam Tip 2
Check your shares add back up to the total
After sharing an amount in a ratio, add your answers together — they must equal the original total. This instantly catches arithmetic slips.
💡 Exam Tip 3
Draw a bar model for tricky ratio problems
Sketching a bar split into the correct number of equal parts makes "difference" questions (like Q12 and Q13) much easier to visualise and solve correctly.
💡 Exam Tip 4
Remember area and volume scale differently to length
For similar shapes, if the length ratio is a:b, the area ratio is a²:b², and the volume ratio is a³:b³ — a very common exam trap is using the length ratio directly for area or volume.
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