Fractions give exact answers. To find a fraction of an amount, divide by the denominator and multiply by the numerator. Multiply fractions by multiplying the numerators and the denominators; divide by flipping the second fraction and multiplying. To add or subtract, first write both fractions with a common denominator, and always simplify the answer.
Fraction rules
- Fraction of an amount: ÷ denominator, × numerator
- Multiply: numerator × numerator over denominator × denominator
- Divide: keep the first, flip the second, multiply
- Add and subtract: use a common denominator, then combine the numerators
- Mixed numbers: convert to improper fractions before multiplying or dividing
Finding a fraction of an amount
Divide the amount by the denominator to find one part, then multiply by the numerator. For 3/4 of 28: 28 ÷ 4 = 7, then 7 × 3 = 21. The same method works for quantities such as grams, metres or money.
Multiplying and dividing fractions
To multiply, multiply the numerators together and the denominators together, cancelling common factors first to keep the numbers small: 2/5 × 15/16 = 30/80 = 3/8. To divide, keep the first fraction, flip the second and multiply. 3/4 ÷ 5/8 = 3/4 × 8/5 = 24/20 = 6/5.
Adding and subtracting fractions
Fractions can only be added or subtracted once they have the same denominator. Find a common denominator, usually the lowest common multiple of the two denominators, convert each fraction, then add or subtract the numerators. The denominator stays the same: 7/8 − 1/3 = 21/24 − 8/24 = 13/24.
Mixed numbers
Convert mixed numbers to improper fractions before multiplying or dividing: 2⅓ = 7/3. For adding and subtracting, you can either convert to improper fractions or deal with the whole numbers and fractions separately. Give the final answer as a mixed number if the question uses them.
Fractions in problem solving
Many exam questions chain fraction steps together. If 2/5 of a class walk and 1/3 of the rest cycle, first find the fraction left after walking, 3/5, then take 1/3 of that, which is 1/5 of the class. Write down each fraction of the whole as you go, so the final step is clear.
Simplifying and exact answers
Simplify by dividing the numerator and denominator by their highest common factor. An exact answer means leaving the fraction as it is rather than converting it to a rounded decimal: 1/3, not 0.33.
Worked examples
Example 1. Work out 3/4 of 28.
- Divide by the denominator: 28 ÷ 4 = 7.
- Multiply by the numerator: 7 × 3.
Answer: 21.
Example 2. Work out 7/8 − 1/3.
- Common denominator 24: 7/8 = 21/24 and 1/3 = 8/24.
- Subtract the numerators: 21 − 8 = 13.
Answer: 13/24.
Example 3. Work out 2/5 × 15/16, giving your answer in its simplest form.
- Cancel first: 5 into 15 gives 3, and 2 into 16 gives 8.
- Multiply: 1/1 × 3/8.
Answer: 3/8.
Example 4. Work out 3/4 ÷ 5/8.
- Keep, flip, multiply: 3/4 × 8/5 = 24/20.
- Simplify by dividing by 4.
Answer: 6/5, or 1⅕.
Example 5. Work out 2⅓ + 1¾.
- Whole numbers: 2 + 1 = 3.
- Fractions: 1/3 + 3/4 = 4/12 + 9/12 = 13/12 = 1 1/12.
Answer: 4 1/12.
Example 6. A recipe uses 5/8 of a 72 g bar of chocolate. How many grams is that?
- 72 ÷ 8 = 9.
- 9 × 5.
Answer: 45 g.
Common mistakes
- Adding the denominators. 1/3 + 1/4 is not 2/7; use a common denominator.
- Flipping the wrong fraction when dividing. Keep the first fraction and flip the second.
- Multiplying mixed numbers directly. Convert to improper fractions first.
- Not simplifying. Divide the numerator and denominator by their highest common factor.
- Rounding when an exact answer is asked. Leave the answer as a fraction.
How fractions are examined
Fraction arithmetic is a core non-calculator skill at both tiers of the 1MA1 GCSE and the 4MA1 iGCSE. You will be asked to:
- find a fraction of an amount, often in a context
- add, subtract, multiply or divide fractions and mixed numbers
- give an answer in its simplest form
- solve a multi-step problem involving fractions of quantities
Work through the fractions worksheet, which has full answers, and use the GCSE number revision engine for unlimited practice.
Fractions connect to fractions, decimals and percentages, and the common denominator uses the LCM.
Frequently asked questions
How do you add fractions with different denominators?
Rewrite both fractions with a common denominator, then add the numerators and keep the denominator.
How do you divide fractions?
Keep the first fraction, flip the second upside down, and multiply.
How do you find a fraction of an amount?
Divide the amount by the denominator, then multiply by the numerator.