Simplify the ratio 10 : 5.
Divide both numbers by 5.
10 : 5 → divide by 5 → 2 : 1.
Simplifying Ratios
Simplifying ratios involves dividing each part by a common factor to express the relationship in its simplest form. This builds on ratio and uses ideas from factors and multiples to make comparisons clearer and easier to work with.
Overview
Simplifying a ratio means writing it in its smallest whole-number form.
You do this by dividing every part of the ratio by the same number.
\( 8:12 = 2:3 \)
The simplified ratio still means exactly the same thing.
It is just written more neatly and more efficiently.
What you should understand after this topic
- Find the highest common factor of a ratio
- Divide all parts correctly
- Recognise when a ratio is fully simplified
- Simplify 2-part and 3-part ratios
- Deal with units before simplifying
Key Definitions
Simplify
Write something in its smallest equivalent form.
Ratio
A comparison between quantities.
Equivalent Ratio
A ratio with the same value but written differently.
Highest Common Factor
The largest number that divides all parts exactly.
Whole-number Form
A simplified ratio is usually written using whole numbers.
Common Unit
The same measurement unit used before simplifying.
Key Rules
Divide all parts
Every part of the ratio must be divided by the same number.
Use the HCF
This gives the simplest whole-number form in one step.
Keep the order the same
Only the size changes, not the order.
Convert units first
Do not simplify cm and m together without converting.
Quick Pattern Check
Simple example
\(10:15 = 2:3\)
Three-part ratio
\(6:9:12 = 2:3:4\)
Already simplest
\(3:5\) cannot be simplified further.
Units first
\(2m:50cm = 200:50 = 4:1\)
How to Solve
What does simplifying a ratio mean?
Simplifying a ratio means writing it in the smallest possible whole numbers while keeping the same comparison.
\( 4:6 = 2:3 \)
The relationship stays the same, but the numbers are smaller.
Step-by-step method
- Find the highest common factor (HCF) using factors and multiples.
- Divide every part of the ratio by the HCF.
- Check no further simplification is possible.
Finding the HCF
\( 12:18 \)
Factors of 12: 1, 2, 3, 4, 6, 12.
Factors of 18: 1, 2, 3, 6, 9, 18.
HCF = 6.
Applying the method
Divide each part by 6.
\(12 \div 6 = 2\), \(18 \div 6 = 3\).
So \(12:18 = 2:3\).
Ratios with units
All parts must be in the same unit before simplifying using units and conversions.
Convert first, then simplify.
Exam tip: Always check units before dividing.
When is a ratio fully simplified?
If the only common factor is 1, the ratio is fully simplified.
Example: \(4:7\) cannot be simplified further.
Exam thinking
Always simplify ratios fully unless told otherwise.
Check units before simplifying.
Exam tip: Many errors come from dividing only one part instead of all parts.
Example Questions
Edexcel
Exam-style questions inspired by Edexcel GCSE Mathematics, focusing on simplifying ratios using common factors.
Edexcel
Simplify the ratio \( 12:18 \).
Edexcel
Simplify the ratio \( 20:35 \).
Edexcel
Simplify the ratio \( 45:60 \).
Edexcel
Simplify the ratio \( 2.5:5 \).
Edexcel
Simplify the ratio \( 0.8:1.2 \).
AQA
Exam-style questions based on the AQA GCSE Mathematics specification, focusing on simplifying ratios involving units and decimals.
AQA
Simplify the ratio \( 18:27 \).
AQA
Simplify the ratio \( 0.6:1.2 \).
AQA
Simplify the ratio \( 150\text{ cm} : 2\text{ m} \).
AQA
Simplify the ratio \( £3 : 75\text{ p} \).
AQA
Simplify the ratio \( 3.5:0.5 \).
OCR
Exam-style questions aligned with OCR GCSE Mathematics, emphasising unit conversion, reasoning, and problem-solving.
OCR
Simplify the ratio \( 84:126 \).
OCR
Simplify the ratio \( 3\text{ kg} : 750\text{ g} \).
OCR
Simplify the ratio \( 2\text{ hours} : 30\text{ minutes} \).
OCR
Simplify the ratio \( 1.2\text{ m} : 30\text{ cm} \).
OCR
Explain why quantities must be expressed in the same units before simplifying a ratio.
Exam Checklist
Step 1
Make sure all values are in the same unit.
Step 2
Find the highest common factor of all parts.
Step 3
Divide every part by that factor.
Step 4
Check whether the result can be simplified any further.
Most common exam mistakes
HCF mistake
Using a factor that is not the highest common factor.
Unit mistake
Trying to simplify mixed units without converting first.
Missing part
Dividing two parts of a 3-part ratio but forgetting the third.
Not fully simplified
Leaving the answer partly simplified instead of fully simplified.
Common Mistakes
These are common mistakes students make when simplifying ratios in GCSE Maths.
Dividing only one part
Incorrect
A student simplifies one number but not the other.
Correct
You must divide all parts of the ratio by the same number to keep the proportion correct.
Not using the highest common factor
Incorrect
A student simplifies but does not fully reduce the ratio.
Correct
Always divide by the highest common factor (HCF) to write the ratio in its simplest form.
Changing the order of the ratio
Incorrect
A student reverses the ratio while simplifying.
Correct
The order must stay the same. Simplifying does not change the meaning or order of the ratio.
Simplifying before converting units
Incorrect
A student simplifies values with different units.
Correct
Convert all quantities into the same unit before simplifying the ratio.
Stopping too early
Incorrect
A student stops simplifying when further reduction is possible.
Correct
Continue simplifying until no common factors remain.
Try It Yourself
Practise simplifying ratios into their simplest form.
Foundation
Foundation Practice
Simplify ratios using common factors.
Simplify the ratio 12 : 8.
Find the highest common factor.
12 : 8 → divide by 4 → 3 : 2.
Which is the simplest form of 9 : 3?
Divide both parts by 3.
9 : 3 → divide by 3 → 3 : 1.
Simplify 15 : 25.
Divide both by 5.
15 : 25 → divide by 5 → 3 : 5.
Simplify the ratio 18 : 12.
Divide both by 6.
18 : 12 → divide by 6 → 3 : 2.
Simplify 20 : 30.
Divide both by 10.
20 : 30 → divide by 10 → 2 : 3.
A student simplifies 8 : 12 to 4 : 6. What should they do next?
4 : 6 can still be simplified.
4 : 6 → divide by 2 → 2 : 3. They must simplify fully.
Simplify 14 : 21.
Divide both by 7.
14 : 21 → divide by 7 → 2 : 3.
Which ratio is already in its simplest form?
Check if both numbers have a common factor.
7 and 5 have no common factors, so 7 : 5 is simplest.
Simplify 16 : 24.
Divide both by 8.
16 : 24 → divide by 8 → 2 : 3.
Higher
Higher Practice
Simplify ratios with decimals, fractions and units.
Simplify 0.4 : 0.6.
Multiply both by 10 first.
0.4 : 0.6 → 4 : 6 → 2 : 3.
Simplify 1.2 : 0.8.
Multiply both by 10.
1.2 : 0.8 → 12 : 8 → 3 : 2.
Simplify 3/5 : 6/10.
Convert both fractions.
3/5 = 6/10, so ratio is 1 : 1.
Simplify 0.25 : 0.5.
Multiply both by 100.
0.25 : 0.5 → 25 : 50 → 1 : 2.
Simplify 200 g : 1 kg.
Convert kg to grams.
1 kg = 1000 g → 200 : 1000 → 1 : 5.
Simplify 45 minutes : 2 hours.
Convert hours to minutes.
2 hours = 120 minutes → 45 : 120 → 3 : 8.
Simplify 0.75 : 0.5.
Multiply both by 100.
0.75 : 0.5 → 75 : 50 → 3 : 2.
Simplify 5/6 : 15/18.
Simplify both fractions first.
5/6 = 15/18, so ratio is 1 : 1.
A student simplifies 0.3 : 0.9 to 3 : 9. What is the correct final answer?
Simplify further.
3 : 9 → divide by 3 → 1 : 3.
Simplify 150 : 0.5.
Remove the decimal first.
150 : 0.5 → multiply both by 2 → 300 : 1.
Games
Practise this topic with interactive games.
Games coming soon.