R4: Simplifying Ratio
Use ratio notation; reduce ratios to simplest form
📋 Key Concepts
Definition: A ratio compares two or more quantities. The ratio a:b means 'a parts to b parts'.
Ratio Notation
- Written as a:b (for two quantities)
- Written as a:b:c (for three quantities)
- Order matters - 3:5 is different from 5:3
To simplify: Divide all parts by their highest common factor (HCF).
📝 Simplifying Ratios
Method: Find the HCF of all numbers in the ratio, then divide each by the HCF.
Example 1
Simplify the ratio 12:18
Solution:
HCF of 12 and 18 is 6
12 ÷ 6 = 2, 18 ÷ 6 = 3
Answer: 2:3
Example 2
Simplify the ratio 24:36:12
Solution:
HCF of 24, 36 and 12 is 12
24 ÷ 12 = 2, 36 ÷ 12 = 3, 12 ÷ 12 = 1
Answer: 2:3:1
📝 Ratios with Decimals and Fractions
Method: Convert to whole numbers first by multiplying, then simplify.
Example 3
Simplify the ratio 2.5:3.5
Solution:
Multiply by 10: 25:35
HCF of 25 and 35 is 5
25 ÷ 5 = 5, 35 ÷ 5 = 7
Answer: 5:7
Example 4
Simplify the ratio 3/4 : 1/2
Solution:
Convert to common denominator: 3/4 : 2/4
Multiply by 4: 3:2
Answer: 3:2
📝 Ratios with Units
Method: Convert to the same units first, then simplify.
Example 5
Simplify the ratio 50 cm : 2 m
Solution:
Convert: 2 m = 200 cm
Ratio: 50:200
HCF = 50
50 ÷ 50 = 1, 200 ÷ 50 = 4
Answer: 1:4
Example 6
Simplify the ratio 30 minutes : 2 hours
Solution:
Convert: 2 hours = 120 minutes
Ratio: 30:120
HCF = 30
Answer: 1:4
📝 Equivalent Ratios
Equivalent Ratios: Ratios that simplify to the same ratio. Multiply or divide all parts by the same number.
Example 7
Find two equivalent ratios to 4:7
Solution:
Multiply by 2: 8:14
Multiply by 3: 12:21
Answer: 8:14 and 12:21
🧮 Worked Examples — Exam Style
Ratio questions at Foundation are worth 3–5 marks and always reward one habit: find the value of ONE part first.
Share £63 between Amy and Ben in the ratio 4 : 5
Steps: 4 + 5 = 9 parts → £63 ÷ 9 = £7 per part
Amy: 4 × £7 = £28 Ben: 5 × £7 = £35
Check: £28 + £35 = £63 ✓
Answer: Amy £28, Ben £35
M1 for 63 ÷ 9, M1 for one correct share, A1 for both. The final check line is free insurance against slips.
A fruit bowl has apples and bananas in the ratio 3 : 7. There are 40 pieces of fruit. How many are bananas?
3 + 7 = 10 parts. 40 ÷ 10 = 4 per part.
Bananas: 7 × 4 = 28
Ratio method check: 3 × 4 = 12 apples, 12 + 28 = 40 ✓
M1 for 40 ÷ 10, A1 for 28. If asked for the fraction of bananas it would be 7/10 — read the question twice.
Exam wording to expect: "Share … in the ratio …", "How many … does … get?", or a part-empty table.
❓ Practice Questions
Q1: Simplify the ratio 16:24
Q2: Simplify the ratio 45:30:15
Q3: Simplify the ratio 1.5:2.5:4
Q4: Simplify the ratio 75 cm : 1.5 m
Q5: Write three ratios equivalent to 2:5
Q6: Simplify: 250 g : 0.5 kg
Q7: Share 48 sweets between two children in the ratio 5 : 3
Q6: Simplify: 250 g : 0.5 kg
Q7: Share 48 sweets between two children in the ratio 5 : 3
Q6: Simplify: 250 g : 0.5 kg
Q7: Share 48 sweets between two children in the ratio 5 : 3
✅ Answers
- 2:3
- 3:2:1
- 3:5:8 (multiply by 10 to get 15:25:40, then divide by 5)
- 1:2 (150 cm : 150 cm after conversion, or 75:150 = 1:2)
- 4:10, 6:15, 8:20 (any three multiples)
🧮 Worked Examples — Harder Exam Style
Golden syrup, flour and sugar are mixed in the ratio 2 : 5 : 3. A recipe uses 250 g of flour. How much golden syrup and sugar are needed?
Flour = 5 parts = 250 g → 1 part = 250 ÷ 5 = 50 g
Golden syrup = 2 parts = 2 × 50 = 100 g
Sugar = 3 parts = 3 × 50 = 150 g
M1 for 250 ÷ 5, A1 for both answers. Recipe-scaling wording ('A recipe uses…') is the standard ratio context.
📋 Model Answers — Full Working
Every mark in GCSE maths comes from the working, not just the answer. These model answers show the lines a mark scheme expects to see.
Q1. Highest common factor of 16 and 24 is 8: 16 ÷ 8 : 24 ÷ 8 = 2 : 3
Q2. HCF of 45, 30, 15 is 15: 3 : 2 : 1
Q3. Decimals: multiply by 10 → 15 : 25 : 40, then ÷ 5 → 3 : 5 : 8
Q4. Convert units FIRST: 1.5 m = 150 cm, so 75 : 150 → ÷75 → 1 : 2. The unit conversion is usually worth a mark — always show it.
Q5. Multiply BOTH parts by the same number: ×2 → 4 : 10, ×3 → 6 : 15, ×4 → 8 : 20 (any three).
Q6. 0.5 kg = 500 g, so 250 : 500 → 1 : 2
Q7. 5 + 3 = 8 parts, 48 ÷ 8 = 6 per part: 30 sweets and 18 sweets
📌 Key Facts to Memorise
- Simplify a ratio: divide every part by the same number (HCF gives simplest form)
- Ratio with units: convert to the SAME unit first
- Sharing: total parts → value of one part (total ÷ parts) → multiply
- Ratio to fraction: part ÷ TOTAL parts (e.g. 3 : 5 → 3/8)
- Equivalent ratios: multiply or divide BOTH parts by the same number
🗺️ In the Exam
Ratio appears on every Foundation paper, usually 3–4 marks: one "simplify", one "share", and often a recipe-scaling or best-buy twist in Paper 2/3 (calculator).
Links to: fractions of amounts (R3), proportion (R7), scale drawings (G).
🧭 Board Focus — AQA
Spec 8300. Three papers, each 80 marks and 1h30: Paper 1 non-calculator, Papers 2 and 3 calculator. Foundation/Higher tiers. Roughly a third of the marks sit under AO2 (reasoning) and AO3 (problem-solving), so show every line of working — method marks survive arithmetic slips.
Always check: specification details change — confirm paper structure and set texts against the current specification for your exam year.
🎯 Exam Tips
- Always convert to whole numbers first if there are decimals or fractions
- Convert to the same units before simplifying
- Find the HCF to simplify in one step
- The numbers in a simplified ratio should have no common factors
- Check your answer by multiplying back to see if you get the original
🧠 Problem-Solving Strategies
Problem-Solving
To simplify a ratio, divide all parts by their highest common factor (HCF). For ratios with decimals or fractions, multiply all parts to get whole numbers first, then simplify. Ratios with different units must be converted to the same unit before simplifying.
Multi-Step Problem
Simplify the ratio 0.75 : 1.25 : 0.5, then find the total number of parts and express each part as a fraction of the whole.
Multiply all by 100: 75 : 125 : 50. Divide by HCF of 25: 3 : 5 : 2. Total parts = 3 + 5 + 2 = 10. Fractions: 3/10, 5/10 = 1/2, 2/10 = 1/5.
⚠️ Common Errors
Watch Out!
1. Wrong: Simplifying 3:6:12 to 1:3 (dropping a term) Correct: 3:6:12 = 1:2:4 (divide ALL parts by 3)
2. Wrong: Simplifying 25cm : 1.5m by dividing by 25 to get 1:0.06 Correct: Convert first: 25cm : 150cm = 1:6
3. Wrong: Simplifying ¾ : ½ by subtracting to get ¼ : 0 Correct: Multiply both by 4: 3 : 2
✍️ 6-Mark Exam Question
Extended Answer
6 marks: Three friends A, B and C share a prize in the ratio 3 : 5 : 7. After tax, A receives £126, B receives £210 and C receives £294. (a) Verify the ratio is 3:5:7. (b) The tax rate was 10%. Show that the original total prize was £700. (c) Express B's original share as a fraction of the total.
(a) A:B = 126:210 = 3:5, B:C = 210:294 = 5:7, A:C = 126:294 = 3:7 ✓ (b) After tax each receives 90% of original. A's original = 126 ÷ 0.9 = 140. One part = 140 ÷ 3 ≈ 46.67. Total parts = 15. Original total = 15 × 46.67 = £700 ✓ (c) B's original = 5 × 46.67 = 233.33. Fraction = 233.33/700 = ⅓ (or 5/15 = 1/3).
Mark scheme: M1 dividing by common factor, A1 ratio verified, M1 reverse tax calculation, A1 total £700, M1 fraction method, A1 1/3
📊 AO3: Reason & Interpret
Reasoning and Interpretation
A recipe for 4 people uses flour, sugar and butter in the ratio 6 : 3 : 2 by weight. The total weight of dry ingredients (flour + sugar) is 450 g.
(a) How much butter is needed?
(b) A student says "Since the ratio 6:3:2 has 11 parts, I just divide 450 by 11." What is wrong with their reasoning?
(c) The recipe needs to serve 10 people. How much flour is required?
Answers: (a) Flour + sugar = 9 parts = 450 g, so 1 part = 50 g. Butter = 2 × 50 = 100 g. (b) 450 g is only the flour + sugar (9 parts), not all 11 parts. Dividing by 11 gives the wrong unit weight. (c) Scale factor = 10/4 = 2.5. Flour = 6 × 50 × 2.5 = 750 g.
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