R3: Fractions of Amounts
Express one quantity as a fraction of another
📋 Key Concepts
Definition: To express one quantity as a fraction of another, write the first quantity over the second, then simplify.
Fraction = Part ÷ Whole
Write as: Part/Whole, then simplify
Write as: Part/Whole, then simplify
Steps
- Write the part as the numerator
- Write the whole as the denominator
- Simplify the fraction if possible
📝 Finding a Fraction of an Amount
Method: Divide by the denominator, then multiply by the numerator.
Example 1
Find 3/5 of £60.
Solution:
Step 1: Divide by 5 → 60 ÷ 5 = 12
Step 2: Multiply by 3 → 12 × 3 = 36
Answer: £36
Example 2
Find 2/3 of 270 grams.
Solution:
Step 1: 270 ÷ 3 = 90
Step 2: 90 × 2 = 180
Answer: 180 g
📝 Expressing as a Fraction
Method: Write part/whole, then simplify.
Example 3
Express 12 as a fraction of 30.
Solution:
Fraction = 12/30
Simplify: 12/30 = 2/5 (divide both by 6)
Example 4
In a class of 32 students, 12 are boys. Express the number of boys as a fraction of the class.
Solution:
Fraction = 12/32
Simplify: 12/32 = 3/8 (divide both by 4)
📝 Fractions from Ratios
From Ratio to Fraction: Add the parts of the ratio to find the whole, then express as required.
Example 5
A bag contains red and blue counters in the ratio 3:5. What fraction of the counters are red?
Solution:
Total parts = 3 + 5 = 8
Fraction that are red = 3/8
Example 6
A prize of £200 is shared in the ratio 2:3. What fraction does the larger share receive?
Solution:
Total parts = 2 + 3 = 5
Larger share = 3 parts out of 5
Fraction = 3/5
📝 Working Backwards
Method: If you know a fraction of an amount, divide by the fraction to find the whole.
Example 7
3/8 of a number is 24. Find the original number.
Solution:
Method 1: 24 ÷ 3 × 8 = 64
Method 2: 24 ÷ (3/8) = 24 × (8/3) = 64
Example 8
2/5 of a class is 14 students. How many students are in the class?
Solution:
Total = 14 ÷ 2 × 5 = 35 students
🧮 Worked Examples — Exam Style
Fractions of amounts come in two shapes: 'fraction OF amount' (divide then multiply) and 'fraction IS amount' (unitary method backwards). Learn both directions.
A class of 32 students: 3/8 walk to school. How many walk?
1/8 of 32 = 32 ÷ 8 = 4. Then 3/8 = 3 × 4 = 12 students.
Order matters: divide by the bottom FIRST, then multiply by the top — the numbers stay small and friendly.
M1 for 32 ÷ 8, A1 for 12. Writing 32 × 3/8 as one step is fine but show the ÷8 line to bank the method mark.
2/5 of a train journey is 14 miles. How long is the whole journey?
2/5 = 14 miles → 1/5 = 14 ÷ 2 = 7 miles → whole journey (5/5) = 7 × 5 = 35 miles.
Check: 2/5 of 35 = 35 ÷ 5 × 2 = 14 ✓
M1 for 14 ÷ 2 (unitary method), A1 for 35. Reverse 'fraction of' questions appear every year — recognise the phrase '… is 14'.
Exam wording to expect: "Find … of …", "What fraction of … is …?", or a reverse: "… of the total is 24, how many altogether?"
❓ Practice Questions
Q1: Find 5/8 of 240.
Q2: Express 18 as a fraction of 45. Simplify your answer.
Q3: A team won 15 games out of 25 played. What fraction of games did they win?
Q4: Apples and oranges are in the ratio 4:7. What fraction are apples?
Q5: 3/4 of a number is 54. What is the number?
Q6: Find 7/10 of £90
Q7: 5/6 of a number is 45. Find the number.
✅ Answers
- 150 (240 ÷ 8 × 5)
- 2/5
- 3/5
- 4/11
- 72 (54 ÷ 3 × 4)
🧮 Worked Examples — Harder Exam Style
A 480 g bag of sweets is shared in the ratio 2 : 3 : 5 between Ali, Ben and Cara. Ben eats 2/3 of his share. How many grams does Ben have left?
Ben’s share: 3 parts of 10 → 480 ÷ 10 = 48, then 48 × 3 = 144 g
Ben eats: 2/3 of 144 = 144 ÷ 3 × 2 = 96 g
Left: 144 − 96 = 48 g
M1 for the ratio share (144), M1 for 2/3 of 144, A1 for 48. Multi-step fraction-of-amount chains like this are worth 4–5 marks at Foundation.
📋 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. 240 ÷ 8 = 30, then 30 × 5 = 150
Q2. 18/45 → ÷9 top and bottom → 2/5
Q3. 15/25 → ÷5 → 3/5
Q4. Total parts 4 + 7 = 11; apples are 4 parts → 4/11. The denominator is ALL the parts, not just the other item.
Q5. 3/4 of the number = 54 → 1/4 = 54 ÷ 3 = 18 → whole = 18 × 4 = 72. Check: 72 ÷ 4 × 3 = 54 ✓
Q6. 90 ÷ 10 = 9, 9 × 7 = £63
Q7. 1/6 = 45 ÷ 5 = 9 → number = 9 × 6 = 54
📌 Key Facts to Memorise
- Fraction OF amount: divide by the denominator (bottom), multiply by the numerator (top)
- Reverse: "3/4 of the number is 54" → 1/4 = 54 ÷ 3, then × 4
- Fraction of a ratio group: part ÷ total parts (4 : 7 → apples are 4/11)
- Simplify before multiplying when you can — smaller numbers, fewer slips
- Check direction: "of" shrinks, "is … of" rebuilds
🗺️ In the Exam
Fractions of amounts: 2–3 marks, every paper, both calculator and non-calculator. The reverse version is the AO2 twist worth the extra mark.
Links to: ratio (R4), percentages of amounts (R9/N12), probability fractions (P).
🧭 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
- "Of" means multiply in fraction calculations
- Always simplify fractions to their lowest terms
- When finding fractions of amounts: divide by denominator, multiply by numerator
- For ratio to fraction: part ÷ (sum of all parts)
- Working backwards: divide by the fraction or use inverse operations
🧠 Problem-Solving Strategies
Problem-Solving
To find a fraction of an amount, divide by the denominator then multiply by the numerator. For compound fractions (e.g. ⅔ of ¾ of), work left to right. Always simplify your final answer where possible.
Multi-Step Problem
In a school, ¾ of the students are in KS4. Of those, ⅖ study Maths in the top set. If there are 840 students in total, how many KS4 students are NOT in the top Maths set?
KS4 students = ¾ × 840 = 630. Top set Maths = ⅖ × 630 = 252. Not in top set = 630 − 252 = 378 students.
⚠️ Common Errors
Watch Out!
1. Wrong: ⅖ of 60 = 60 ÷ 5 × 2 = 240 Correct: 60 ÷ 5 = 12, then 12 × 2 = 24
2. Wrong: Finding ⅕ of 60 then subtracting from 60 to get ⅘: 60 − 12 = 58 Correct: ⅘ of 60 = 4 × 12 = 48, or 60 − 12 = 48 (recalculate properly)
3. Wrong: ¾ of ½ = 7/8 Correct: ¾ × ½ = 3/8 (multiply fractions, don't add)
✍️ 6-Mark Exam Question
Extended Answer
6 marks: A charity raises £36,000. ⅜ of the money goes to Country A. Of the remainder, ⅖ goes to Country B. The rest is split equally between Countries C and D. How much more does Country A receive than Country C? Show all working.
Country A = ⅜ × 36,000 = 13,500. Remainder = 36,000 − 13,500 = 22,500. Country B = ⅖ × 22,500 = 9,000. Remainder for C and D = 22,500 − 9,000 = 13,500. Country C = 13,500 ÷ 2 = 6,750. Difference = 13,500 − 6,750 = £6,750.
Mark scheme: M1 fraction of amount for A, A1 £13,500, M1 remainder and fraction for B, A1 £9,000, M1 splitting C and D, A1 difference = £6,750
📊 AO3: Reason & Interpret
Reasoning and Interpretation
A shop has 180 tins of paint. In the first week it sells ⅔ of them. In the second week it sells ¾ of the remaining tins.
(a) How many tins are left after two weeks?
(b) The shop manager says "We sold more tins in the first week than the second." Is this true? Show calculations.
(c) In week 3, the shop wants to sell ALL remaining tins at ⅔ of the original price. If the original price was £12, what is the new price per tin?
Answers: (a) Week 1: ⅔ × 180 = 120 sold, 60 left. Week 2: ¾ × 60 = 45 sold, 15 left. (b) Yes — week 1: 120, week 2: 45. (c) ⅔ × 12 = £8 per tin.
📝 Exam Questions by Topic
🎬 Video Resources
Ready to ace your GCSE Mathematics exams?
Get the best revision books and guides to boost your grades.