To simplify a ratio, divide every part by their highest common factor (HCF). Keep dividing until no whole number greater than 1 divides into all parts evenly. A simplified ratio shows the same proportion using the smallest possible whole numbers.
What is a ratio and why simplify it?
A ratio compares two or more quantities of the same kind. Writing 12 : 8 and 3 : 2 describe the exact same relationship — three parts to every two parts — but 3 : 2 is easier to read and work with. Simplifying a ratio is like cancelling a fraction: you are not changing the proportion, just expressing it more neatly.
How do you find the HCF of two numbers?
The highest common factor is the largest number that divides exactly into both values. Two quick methods:
Method 1 — list factors:
Factors of 12: 1, 2, 3, 4, 6, 12
Factors of 18: 1, 2, 3, 6, 9, 18
Highest common factor = 6
Method 2 — divide repeatedly by prime factors:
12 = 2 × 2 × 3
18 = 2 × 3 × 3
Common prime factors: 2 × 3 = 6
How do you simplify a ratio step by step?
Worked example 1 — simplify 24 : 36:
- Find the HCF of 24 and 36.
24 = 2³ × 3; 36 = 2² × 3². HCF = 2² × 3 = 12. - Divide both parts by 12: 24 ÷ 12 = 2; 36 ÷ 12 = 3.
- Simplified ratio = 2 : 3.
- Check: no whole number greater than 1 divides into both 2 and 3. ✓
Worked example 2 — simplify 45 : 30:
- HCF of 45 and 30: 45 = 3² × 5; 30 = 2 × 3 × 5. HCF = 3 × 5 = 15.
- 45 ÷ 15 = 3; 30 ÷ 15 = 2.
- Simplified ratio = 3 : 2.
What if the ratio has different units?
Before simplifying, convert both quantities to the same unit. You cannot compare centimetres with metres directly.
Worked example — simplify 50 cm : 2 m:
- Convert 2 m to centimetres: 2 m = 200 cm.
- Ratio becomes 50 : 200.
- HCF of 50 and 200 = 50.
- 50 ÷ 50 = 1; 200 ÷ 50 = 4.
- Simplified ratio = 1 : 4.
| Original ratio | Common unit | HCF | Simplified |
|---|---|---|---|
| 50 cm : 2 m | 50 : 200 cm | 50 | 1 : 4 |
| 400 g : 1 kg | 400 : 1000 g | 200 | 2 : 5 |
| 15 min : 1 hr | 15 : 60 min | 15 | 1 : 4 |
How do you simplify a three-part ratio?
Divide all three parts by their HCF — the largest number that divides into all three.
Worked example — simplify 6 : 9 : 15:
- Factors of 6: 1, 2, 3, 6. Factors of 9: 1, 3, 9. Factors of 15: 1, 3, 5, 15.
- HCF = 3.
- 6 ÷ 3 = 2; 9 ÷ 3 = 3; 15 ÷ 3 = 5.
- Simplified ratio = 2 : 3 : 5.
What if the ratio contains decimals or fractions?
Multiply all parts by the same number to make them whole numbers first, then simplify.
Worked example — simplify 0.4 : 1.2:
- Multiply both parts by 10: 4 : 12.
- HCF of 4 and 12 = 4.
- 4 ÷ 4 = 1; 12 ÷ 4 = 3.
- Simplified ratio = 1 : 3.
Frequently asked questions
How do I know when a ratio is fully simplified?
A ratio is in its simplest form when the only common factor of all its parts is 1. Ask yourself: is there any whole number greater than 1 that divides evenly into every part? If not, the ratio is fully simplified.
Can a simplified ratio ever contain a zero?
No. A ratio compares quantities that both exist, so neither part should be zero. A ratio of 0 : 5 would mean none of something compared to some of something, which is not a meaningful ratio in KS3 contexts.
What is the difference between simplifying a ratio and simplifying a fraction?
The process is identical — divide both values by their HCF. The notation differs: a fraction is written as 3/4 while a ratio is written as 3 : 4. In both cases you are expressing the same proportion with the smallest possible whole numbers.
How do ratios connect to dividing quantities?
Once you have a simplified ratio, you can use it to divide a total into parts. For example, a ratio of 2 : 3 splits a total into 5 equal shares (2 + 3), with one person getting 2/5 and the other 3/5. This skill appears in "sharing in a given ratio" questions at KS3 and GCSE.
For Socratic ratio and proportion practice, see aitutors.me.