To simplify a ratio, find the largest common factor of all parts, then divide every part by it.
Example 1: Simplify \(36 : 48\)
Step 1
Largest common factor of 36 and 48 = 12
Step 2
\(36 \div 12 : 48 \div 12 = \mathbf{3:4}\)
✓ Quick Check 1 — Divide by the HCF
Question 1 of 10
Two Methods — Same Answer
You can simplify by dividing by the HCF in one step, or by dividing by small factors repeatedly. Both routes give the same simplified ratio.
💡 Tip: Both methods work. If you can spot the largest common factor straight away, Method B is faster. If not, keep dividing by small primes (2, 3, 5…) until you can't go further.
Converting Units First
Both quantities must be in the same unit before simplifying. Always convert first — never simplify with mixed units.
Example 2: Write the ratio \(75\text{ cm} : 2\text{ m}\) in its simplest form.
Step 1
Convert to same units: \(2\text{ m} = 200\text{ cm}\)
Step 2
\(75 : 200\) — HCF(75, 200) = 25
Step 3
\(75\div25 : 200\div25 = \mathbf{3:8}\)
Example 3: Simplify \(45\text{ min} : 3\text{ hours}\)
Step 1
Convert hours to minutes: \(3\text{ h} = 180\text{ min}\)
Step 2
\(45:180\) — HCF = 45 → \(\mathbf{1:4}\)
3-Part Ratios and the Form 1:n
To write a ratio as \(1:n\), divide both sides by the first number.
Example 4: Write \(4:7\) in the form \(1:n\)
Step 1
Divide both by 4: \(1 : \frac{7}{4} = 1 : 1.75\)
💡 Map scales are always given as \(1:n\) because 1 unit on the map represents \(n\) units in real life.
✓ Quick Check 2 — Converting Units & Three-Part Ratios
Question 1 of 10
Simplifying Ratios with Decimals and Fractions
If a ratio has decimals or fractions, clear them first, then simplify as normal using the HCF.
Example 5: Simplify 1.5 : 2.5
Step 1
Multiply both parts by 10 to clear the decimals: \(1.5 \times 10 : 2.5 \times 10 = 15:25\)
💡 How much to multiply by: 1 decimal place → ×10. 2 decimal places → ×100. This always works, whatever the numbers are (unlike shortcuts such as ×2, which only work for specific decimals like .5 or .25).
Example 7: Simplify \(\frac{1}{2} : \frac{1}{3}\)
Step 1
Find the LCM of the denominators 2 and 3: LCM = 6.
Step 2
Multiply both parts by 6: \(\frac{1}{2} \times 6 : \frac{1}{3} \times 6 = 3:2\)
Step 3
HCF(3,2) = 1, so \(\mathbf{3:2}\) is already fully simplified.
Common mistake: Don't try to simplify a decimal or fraction ratio directly (e.g. dividing 1.5 by 2.5) — clear the decimal/fraction to whole numbers first, then simplify.
✓ Quick Check 3 — Mixed Simplification Practice
Question 1 of 10
Practice: Extra Questions
Simplify each ratio fully. Write answers in the form a:b (e.g. 2:3).
#
Question
Your answer
Result
1
Simplify 4:6
2
Simplify 10:15
3
Simplify 8:12
4
Simplify 15:25
5
Simplify 6:9:12
6
Simplify 20:30
7
Simplify 9:15
8
Simplify 6:9
9
Simplify 15:10
10
Simplify 20:30:50
11
Simplify 500 m : 2 km (convert to the same unit first)
12
Simplify 250 g : 1 kg (convert to the same unit first)
13
Simplify 45:60
14
Simplify 1.5:2.5
15
Simplify 12:16:20
16
Simplify 30 minutes : 2 hours (convert to the same unit first)
17
Simplify 0.8:1.2
⚠ Things to Watch Out For
Only dividing one side: To simplify 6:10, divide BOTH sides by 2 → 3:5. Dividing only one side changes the relationship between the quantities.
Dividing by a factor but not the largest common factor: 12:18 ÷ 2 = 6:9 (not fully simplified). ÷ 3 again → 2:3. Or find the largest common factor = 6 and do it in one step.
Converting units before simplifying: 500m : 2km → convert to same units first: 500m : 2000m → 1:4. Ratios must use the same unit.
Three-part ratios: 4:6:8 → divide all three by 2 → 2:3:4. All parts must be divided by the same number.
Leaving a decimal in the ratio: 1.5:3 is not fully simplified. Multiply both by 2 → 3:6 → then simplify → 1:2.