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
❓ 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?
✅ Answers
- 150 (240 ÷ 8 × 5)
- 2/5
- 3/5
- 4/11
- 72 (54 ÷ 3 × 4)
🎯 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.
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