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This lesson covers all the fraction skills required for AQA GCSE Mathematics: simplifying, converting between improper fractions and mixed numbers, and performing all four operations — addition, subtraction, multiplication and division. Fractions appear throughout the GCSE course, so mastering these techniques is essential.
| Term | Definition | Example |
|---|---|---|
| Numerator | The top number of a fraction | In 3/4, the numerator is 3 |
| Denominator | The bottom number of a fraction | In 3/4, the denominator is 4 |
| Proper fraction | Numerator is less than the denominator | 3/4, 2/5 |
| Improper fraction | Numerator is greater than or equal to the denominator | 7/3, 9/4 |
| Mixed number | A whole number plus a proper fraction | 2 and 1/3, 4 and 3/4 |
| Equivalent fractions | Fractions that represent the same value | 1/2 = 2/4 = 3/6 |
To simplify (or cancel) a fraction, divide the numerator and denominator by their highest common factor (HCF).
Simplify 24/36.
24/36 = (24 / 12) / (36 / 12) = 2/3
Exam Tip: Always check whether your answer can be simplified further. An unsimplified fraction will usually lose a mark.
Divide the numerator by the denominator:
Convert 17/5 to a mixed number.
17 divided by 5 = 3 remainder 2
Answer: 3 and 2/5
Multiply the whole number by the denominator, then add the numerator.
Convert 4 and 3/7 to an improper fraction.
(4 x 7) + 3 = 28 + 3 = 31
Answer: 31/7
To add or subtract fractions, the denominators must be the same.
Calculate 2/3 + 3/5.
Step 1: LCM of 3 and 5 = 15.
Step 2: Convert:
Step 3: Add numerators: 10/15 + 9/15 = 19/15
Step 4: Convert to mixed number: 1 and 4/15
Calculate 3 and 1/4 - 1 and 2/3.
Step 1: Convert to improper fractions:
Step 2: LCM of 4 and 3 = 12.
Step 3: Convert:
Step 4: Subtract: 39/12 - 20/12 = 19/12
Step 5: Convert: 1 and 7/12
Exam Tip: When working with mixed numbers, convert to improper fractions first. This avoids errors when borrowing across the whole number.
To multiply fractions:
Calculate 2/3 x 4/5.
(2 x 4) / (3 x 5) = 8/15
Answer: 8/15 (already in its simplest form)
Calculate 1 and 2/3 x 2 and 1/4.
Step 1: Convert: 1 and 2/3 = 5/3 and 2 and 1/4 = 9/4
Step 2: Multiply: (5 x 9) / (3 x 4) = 45/12
Step 3: Simplify: 45/12 = 15/4 = 3 and 3/4
Exam Tip: You can "cross-cancel" before multiplying to keep numbers small. For example, in (4/9) x (3/8), you can cancel the 4 and 8 (both divide by 4) and the 3 and 9 (both divide by 3) to get (1/3) x (1/2) = 1/6.
To divide by a fraction, flip the second fraction (find its reciprocal) and multiply.
graph LR
A["a/b divided by c/d"] --> B["Flip the second fraction"]
B --> C["a/b x d/c"]
C --> D["Multiply as normal"]
The reciprocal of a fraction is found by swapping the numerator and denominator:
Calculate 3/4 divided by 2/5.
Step 1: Flip the second fraction: 2/5 becomes 5/2.
Step 2: Multiply: 3/4 x 5/2 = 15/8
Step 3: Convert: 1 and 7/8
Calculate 2 and 1/2 divided by 1 and 1/3.
Step 1: Convert: 2 and 1/2 = 5/2 and 1 and 1/3 = 4/3
Step 2: Flip: 4/3 becomes 3/4
Step 3: Multiply: 5/2 x 3/4 = 15/8
Step 4: Convert: 1 and 7/8
To find a fraction of an amount, divide by the denominator and multiply by the numerator.
Find 3/5 of 240.
To compare or order fractions, convert them to equivalent fractions with the same denominator, then compare numerators.
Put these fractions in ascending order: 3/4, 2/3, 5/6, 7/12
Step 1: LCM of 4, 3, 6, 12 = 12
Step 2: Convert all to twelfths:
Step 3: Order by numerator: 7/12, 8/12, 9/12, 10/12
Answer: 7/12, 2/3, 3/4, 5/6
Exam Tip: When ordering fractions, always show your equivalent fractions — the examiner needs to see your working for method marks.
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