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This lesson covers DfE content statements L2.4, L2.7 and L2.8 — understanding and comparing fractions, carrying out all four operations with fractions (including mixed numbers), and expressing one number as a fraction of another.
A fraction represents a part of a whole. The number on top is the numerator (how many parts you have) and the number on the bottom is the denominator (how many equal parts the whole is divided into).
| Term | Meaning | Example |
|---|---|---|
| Proper fraction | Numerator < denominator | 3/4 |
| Improper fraction | Numerator ≥ denominator | 7/4 |
| Mixed number | Whole number + proper fraction | 1 3/4 |
| Unit fraction | Numerator is 1 | 1/5 |
Two fractions are equivalent if they represent the same amount. You create equivalent fractions by multiplying or dividing both the numerator and denominator by the same number.
2/3 = 4/6 = 6/9 = 8/12 = 10/15 ...
Divide both numerator and denominator by their Highest Common Factor (HCF).
Scenario: A recipe uses 450g of flour from a 1,000g bag. What fraction has been used? Simplify your answer.
450/1000 — HCF of 450 and 1000 is 50 → 450 ÷ 50 = 9, 1000 ÷ 50 = 20 → 9/20
To compare or order fractions, convert them to the same denominator (the LCM of the denominators), then compare the numerators.
Scenario: Three shops offer discounts: Shop A offers 2/5 off, Shop B offers 3/8 off, and Shop C offers 1/3 off. Which discount is the largest?
Find the LCM of 5, 8, and 3 = 120
| Fraction | Equivalent with denominator 120 |
|---|---|
| 2/5 | 48/120 |
| 3/8 | 45/120 |
| 1/3 | 40/120 |
Answer: 2/5 (48/120) is the largest discount.
Exam Tip: When comparing fractions, find a common denominator. Do not try to "eyeball" which fraction is bigger — even experienced mathematicians can be fooled.
Mixed number → Improper fraction: Multiply the whole number by the denominator, add the numerator, keep the same denominator.
3 2/5 → (3 × 5 + 2)/5 = 17/5
Improper fraction → Mixed number: Divide the numerator by the denominator. The quotient is the whole number, the remainder is the new numerator.
23/4 → 23 ÷ 4 = 5 remainder 3 → 5 3/4
Step 1: Find a common denominator. Step 2: Convert each fraction. Step 3: Add or subtract the numerators. Keep the denominator. Step 4: Simplify if possible.
Scenario: A care worker does 3 1/2 hours in the morning and 2 3/4 hours in the afternoon. What is the total?
3 1/2 + 2 3/4
Convert to improper fractions: 7/2 + 11/4
Common denominator is 4: 14/4 + 11/4 = 25/4 = 6 1/4 hours
Scenario: A baker has 5 1/3 kg of sugar and uses 2 5/6 kg. How much is left?
5 1/3 − 2 5/6
Convert to improper fractions: 16/3 − 17/6
Common denominator is 6: 32/6 − 17/6 = 15/6 = 2 1/2 kg
Rule: Multiply the numerators together and multiply the denominators together. Simplify the result.
With mixed numbers: Convert to improper fractions first.
Scenario: A recipe for 4 people uses 2/3 cup of rice. You are cooking for 6 people, so you need 1 1/2 times the recipe. How much rice do you need?
2/3 × 1 1/2 = 2/3 × 3/2 = 6/6 = 1 cup
Scenario: A building firm quotes £4,800 for a job. They require 3/8 as a deposit. How much is the deposit?
3/8 of £4,800 = 3/8 × 4800 = 3 × 600 = £1,800
(Tip: divide by the denominator first, then multiply by the numerator. 4800 ÷ 8 = 600, then 600 × 3 = 1800.)
Exam Tip: "Of" in maths means multiply. So "3/8 of £4,800" means 3/8 × £4,800.
Rule: Keep the first fraction, change ÷ to ×, and flip (reciprocal) the second fraction.
Scenario: A curtain maker has 7 1/2 metres of fabric. Each curtain requires 2 1/2 metres. How many curtains can be made?
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