Ratio: Simplifying, Sharing & Combining
Ratio: Simplifying, Sharing & Combining
A ratio compares how much of one thing there is compared to another, using the same units.
Simplifying Ratios
Divide all parts by their highest common factor (HCF).
Worked Example: Simplify 24 : 36.
- HCF of 24 and 36 is 12
- 24 ÷ 12 : 36 ÷ 12 = 2 : 3
With different units: Convert to the same unit first.
Worked Example: Write 40 minutes : 2 hours as a ratio in simplest form.
- 2 hours = 120 minutes
- 40 : 120 = 1 : 3
With decimals or fractions: Multiply to make whole numbers.
Worked Example: Simplify 1.5 : 2.5.
- Multiply by 2: 3 : 5 — already in simplest form: 3 : 5
Writing as 1 : n or n : 1
Divide both parts by one of the parts.
Worked Example: Write 5 : 8 in the form 1 : n.
- Divide by 5: 1 : 8/5 = 1 : 1.6
Sharing in a Given Ratio
Method: Find the total number of parts, work out the value of one part, then multiply.
Worked Example: Share £480 in the ratio 3 : 5.
- Total parts = 3 + 5 = 8
- One part = £480 ÷ 8 = £60
- First share = 3 × £60 = £180
- Second share = 5 × £60 = £300
- Check: £180 + £300 = £480 ✓
Three-way ratio: The same method works.
Worked Example: Three siblings share an inheritance of £12,000 in the ratio 2 : 3 : 5.
- Total parts = 2 + 3 + 5 = 10
- One part = £1,200
- Shares: £2,400, £3,600, £6,000
Given One Share, Find the Others
Worked Example: Ali and Ben share sweets in the ratio 4 : 7. Ben gets 35 sweets. How many does Ali get?
- Ben's 7 parts = 35 sweets → 1 part = 5 sweets
- Ali gets 4 × 5 = 20 sweets
The Difference Between Shares
Worked Example: Two amounts are in the ratio 3 : 8. The difference between them is 90. Find both amounts.
- Difference in parts = 8 − 3 = 5
- 5 parts = 90 → 1 part = 18
- Amounts: 3 × 18 = 54 and 8 × 18 = 144
Combining Ratios
When two ratios share a common quantity, combine them into a single three-part ratio.
Worked Example: A : B = 2 : 3 and B : C = 4 : 5. Find A : B : C.
- B appears as 3 in the first ratio and 4 in the second
- Make B the same: multiply the first ratio by 4 and the second by 3
- A : B = 8 : 12 and B : C = 12 : 15
- Combined: A : B : C = 8 : 12 : 15
Recipe and Best-Buy Problems
Worked Example: A recipe for 12 biscuits uses 200g flour. How much flour for 30 biscuits?
- Scale factor = 30 ÷ 12 = 2.5
- Flour = 200 × 2.5 = 500g
Best buy: Find the price per unit (or amount per penny) for each option.
Worked Example: 400g of cereal costs £2.80. 650g costs £4.20. Which is better value?
- Small: £2.80 ÷ 400 = 0.7p per gram
- Large: £4.20 ÷ 650 = 0.646p per gram
- The large box is better value (lower price per gram)
Exam Tips
- Always check your shares add up to the total
- When combining ratios, make the shared quantity the same in both ratios before merging
- Read carefully whether the question gives you the total, one share, or the difference
- Units must match before you simplify a ratio