R5: Dividing in a Ratio
Divide a given quantity into two parts in a given part:part or part:whole ratio
📋 Key Concepts
Definition: When dividing in a ratio, we split a total amount into parts that are proportional to the given ratio.
Steps for Part:Part Ratios
- Add the parts of the ratio to find the total number of parts
- Divide the total amount by the total parts to find one part
- Multiply by each number in the ratio
One part = Total Amount ÷ Total Parts
Each share = One part × Ratio number
Each share = One part × Ratio number
📝 Dividing in a Ratio
Example 1
Share £60 in the ratio 2:3.
Solution:
Step 1: Total parts = 2 + 3 = 5
Step 2: One part = £60 ÷ 5 = £12
Step 3: First share = 2 × £12 = £24
Step 4: Second share = 3 × £12 = £36
Answer: £24 and £36
Check: £24 + £36 = £60 ✓
Example 2
Divide 120 kg in the ratio 3:4:5.
Solution:
Step 1: Total parts = 3 + 4 + 5 = 12
Step 2: One part = 120 ÷ 12 = 10 kg
Step 3: First share = 3 × 10 = 30 kg
Step 4: Second share = 4 × 10 = 40 kg
Step 5: Third share = 5 × 10 = 50 kg
Answer: 30 kg, 40 kg, 50 kg
📝 Part:Whole Ratios
Part:Whole Ratio: The ratio gives one part and the total directly. For example, in a ratio of 2:5, if 2 is the part, then 5 is the whole.
Example 3
John spends 3/8 of his money on books. He has £160. How much does he spend on books?
Solution:
This is a part:whole situation. The ratio of books:total is 3:8.
Total parts = 8
One part = £160 ÷ 8 = £20
Books = 3 × £20 = £60
Answer: £60
📝 Finding the Total
Working Backwards: If you know one share, you can find the total.
Example 4
A sum of money is shared in the ratio 4:7. The larger share is £56. Find the total amount.
Solution:
The larger share is 7 parts = £56
One part = £56 ÷ 7 = £8
Total parts = 4 + 7 = 11
Total = 11 × £8 = £88
Answer: £88
📝 Difference Problems
Example 5
Two numbers are in the ratio 5:8. The difference between them is 36. Find the numbers.
Solution:
Difference in parts = 8 - 5 = 3 parts
3 parts = 36
One part = 36 ÷ 3 = 12
First number = 5 × 12 = 60
Second number = 8 × 12 = 96
Answer: 60 and 96
🧮 Worked Examples — Exam Style
Sharing in a ratio: total parts → one part → multiply out. Difference questions: parts difference ↔ amount difference.
Share £88 in the ratio 3 : 5.
8 parts → £88 ÷ 8 = £11 per part
3 × 11 = £33 and 5 × 11 = £55
M1 for £11 per part, A1 for both amounts. The check (£33 + £55 = £88) is worth writing.
Two lengths are in ratio 4 : 9. The longer is 45 cm. Find the shorter.
9 parts = 45 → 1 part = 5 → 4 × 5 = 20 cm
M1 for 45 ÷ 9, A1 for 20. When ONE share is given (not the total), match that ratio number first.
Exam wording to expect: "Share/divide … in the ratio …", "The smaller share is …", "Their difference is …".
❓ Practice Questions
Q1: Share £90 in the ratio 2:7.
Q2: Divide 210 grams in the ratio 2:3:5.
Q3: A prize is shared in the ratio 3:4. The smaller share is £75. Find the total prize.
Q4: Two numbers are in the ratio 7:11. Their difference is 24. Find the numbers.
Q5: A farm has sheep and cows in the ratio 5:3. There are 120 animals altogether. How many are sheep?
Q6: £360 is shared between three people in the ratio 1 : 2 : 6. Find the largest share.
Q7: Two amounts are in ratio 5 : 8. Their total is £78. Find the difference between them.
✅ Answers
- £20 and £70
- 42 g, 63 g, 105 g
- £175
- 42 and 66
- 75 sheep
📋 Model Answers — Full Working
Every mark comes from the working, not just the answer. These model answers show the lines a mark scheme expects to see.
Q1. 9 parts → £10 each → £20 and £70
Q2. 10 parts → 21 g each → 42 g, 63 g, 105 g
Q3. 3 parts = £75 → 1 part = £25 → 7 parts = £175 → total £175
Q4. Difference 4 parts = 24 → 1 part = 6 → 42 and 66
Q5. 8 parts = 120 → 15 each → sheep = 5 × 15 = 75
Q6. 9 parts → £40 each → largest = 6 × 40 = £240
Q7. 13 parts = £78 → £6 each → difference = 3 parts = £18
📌 Key Facts to Memorise
- Sharing: total ÷ total parts = one part; multiply by each ratio number
- Given ONE share: divide by ITS ratio number to get one part
- Given the DIFFERENCE: difference of ratio numbers ↔ amount difference
- Total = sum of all shares — always check it adds back
- Ratio 1 : 2 : 6 — the numbers are PARTS, not amounts
🗺️ In the Exam
Ratio division: 3–5 marks on every paper — money shares, ingredients, teams. The one-share and difference versions are the AO2 twists.
Links to: simplifying ratio (R4), fractions of amounts (R3), proportion (R7).
🧭 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 find the total number of parts first
- Check your answer by adding the shares - they should equal the total
- For difference problems: difference in parts = difference in values
- Label your working clearly to avoid confusion
- If the ratio has decimals or fractions, simplify first
🧠 Problem-Solving Strategies
Problem-Solving
To divide an amount in a given ratio: add the ratio parts to find the total, divide the amount by the total to find one part, then multiply by each ratio number. For multi-step problems, check whether you need the whole amount or just a share.
Multi-Step Problem
£2400 is divided between A, B and C in the ratio 3 : 4 : 5. C then gives 20% of her share to A. How much does A now have?
Total parts = 3 + 4 + 5 = 12. One part = 2400 ÷ 12 = 200. A = 3 × 200 = 600, C = 5 × 200 = 1000. C gives A: 20% of 1000 = 200. A now has 600 + 200 = £800.
⚠️ Common Errors
Watch Out!
1. Wrong: Dividing £2400 by 3, 4 and 5 individually: £800, £600, £480 Correct: One part = £2400 ÷ 12 = £200, then A=£600, B=£800, C=£1000
2. Wrong: For ratio 2:3, dividing by 2 parts and giving A half Correct: Total parts = 5, so A gets 2/5 and B gets 3/5
3. Wrong: Dividing 100 in ratio 1:4 and getting 25 and 75 Correct: Total = 5 parts. 1 part = 20, so 20 and 80
✍️ 6-Mark Exam Question
Extended Answer
6 marks: Profits of £18,000 are shared between three partners in the ratio of their investments. Ali invested £24,000, Beth invested £36,000 and Carlos invested £60,000. (a) Find each person's share. (b) Carlos reinvests 30% of his profit. How much does he reinvest? (c) Beth says "I get exactly double what Ali gets." Is she correct? Justify your answer.
(a) Ratio = 24000 : 36000 : 60000 = 2 : 3 : 5. Total parts = 10. One part = 18000 ÷ 10 = 1800. Ali = 2 × 1800 = £3600. Beth = 3 × 1800 = £5400. Carlos = 5 × 1800 = £9000. (b) 30% of 9000 = 0.3 × 9000 = £2700. (c) Beth = £5400, Ali = £3600. 5400 ÷ 3600 = 1.5, not 2. Beth is wrong — she gets 1.5 times Ali's share.
Mark scheme: M1 simplifying ratio, A1 all three shares, M1 percentage calculation, A1 £2700, M1 comparison method, A1 correct conclusion with justification
📊 AO3: Reason & Interpret
Reasoning and Interpretation
A bag contains red, blue and green counters in the ratio 5 : 3 : 2. There are 40 red counters.
(a) How many blue and green counters are there?
(b) 10 more blue counters are added. What is the new ratio of red : blue : green?
(c) A student says "The fraction of red counters is 5/10 so it's exactly half." Is this always true? Explain.
Answers: (a) One part = 40 ÷ 5 = 8. Blue = 3 × 8 = 24, Green = 2 × 8 = 16. (b) New blue = 24 + 10 = 34. New ratio = 40:34:16 = 20:17:8. (c) Yes, 5 out of 10 parts = ½, so exactly half are red. This is always true for this ratio regardless of the total number.
📝 Exam Questions by Topic
🎬 Video Resources
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