Fractions: revision notes and practice

What you will learn

• Use equivalent fractions and simplify fully.
• Calculate with proper fractions, improper fractions and mixed numbers.
• Explain why adding and multiplying need different methods.
• Use fractions of quantities and check the size of an answer.

Key rules

Add or subtract

Write both fractions with a common denominator. Add or subtract the numerators and keep that denominator.

Multiply or divide

To multiply, multiply the numerators and the denominators. To divide by a non-zero fraction, multiply by its reciprocal.

Simplify the result

Divide numerator and denominator by their highest common factor. Convert mixed numbers to improper fractions before calculating.

Worked examples

Example 1: Calculate 2/3 + 1/4. Give a fraction in simplest form.

1. Use denominator 12: 2/3 = 8/12 and 1/4 = 3/12.
2. Add the numerators: (8 + 3)/12 = 11/12.

11/12

Example 2: Calculate 3/5 ÷ 9/10. Give a fraction in simplest form.

1. Turn the second fraction over: 3/5 × 10/9.
2. Multiply to get 30/45.
3. Divide top and bottom by 15.

2/3

A practice question

Simplify 128/88. Give your answer in its simplest form.

1. Understand the two parts of a fraction

A fraction such as 3/5 represents three of five equal parts of a whole. The numerator is the number on top; the denominator is the number underneath. It also means the division 3 ÷ 5. A denominator cannot be zero, because division by zero is undefined.

The equal-parts condition matters. Three pieces out of five are not necessarily 3/5 of a cake if the pieces have different sizes. Fractions can also describe numbers greater than one: 7/5 is seven fifths, or one whole and another two fifths.

2. Make equivalent fractions and simplify

Multiplying the numerator and denominator by the same non-zero number leaves the value unchanged. For example, 2/3 = 8/12 because both parts have been multiplied by 4. You are multiplying the original fraction by 4/4, which equals 1.

Simplifying reverses this process. In 18/24, the highest common factor of 18 and 24 is 6, so divide both by 6 to obtain 3/4. The fraction is simplest when the numerator and denominator have no common positive factor greater than 1. Reducing one part without reducing the other changes the value.

3. Convert mixed and improper forms

A mixed number combines whole units and a fraction. To convert 2 3/4, remember that each whole contains four quarters: two wholes contain eight quarters, and another three give 11/4. Multiply the whole number by the denominator, then add the numerator.

To convert 17/5 back, divide 17 by 5: the quotient is 3 with remainder 2, so the mixed number is 3 2/5. Convert mixed numbers to improper fractions before multiplying or dividing. With a negative mixed number such as −2 1/3, the minus applies to the whole value: it equals −7/3.

Positive mixed number w n/d = (wd + n)/d

4. Add and subtract using a common unit

You can add counts only when they count equal-sized pieces. Thirds and quarters need to be rewritten as a common unit. For 2/3 + 1/4, use twelfths: 8/12 + 3/12 = 11/12. Keep the denominator and add the numerators.

A lowest common multiple often gives the most convenient denominator, but any common multiple works. For 5/6 − 1/4, denominator 12 gives 10/12 − 3/12 = 7/12. Never add or subtract the denominators: they describe the piece size, which stays fixed after the conversion.

5. Multiply and cancel factors

Multiplication does not need a common denominator. Multiply the numerators together and multiply the denominators together. Thus 2/3 × 3/5 = 6/15 = 2/5. This can be interpreted as taking two thirds of three fifths.

You may cancel common factors before multiplying to keep numbers smaller. In 5/6 × 9/10, divide 5 and 10 by 5, and 9 and 6 by 3; the remaining product is 1 × 3 over 2 × 2, or 3/4. Cancellation works on factors in a product, not on separate terms joined by addition or subtraction.

(a/b) × (c/d) = ac/bd, for b ≠ 0 and d ≠ 0

6. Divide by multiplying by the reciprocal

To divide by a non-zero fraction, multiply by its reciprocal: turn the divisor over. For 3/5 ÷ 9/10, calculate 3/5 × 10/9 = 30/45 = 2/3. Only the second fraction is inverted, because it is the divisor.

The method works because multiplying the divisor by its reciprocal makes 1. For example, dividing 3/4 by 1/2 asks how many halves fit into three quarters; the answer is 1.5, or 3/2. Dividing by a positive number less than 1 increases a positive starting quantity, so an answer larger than 3/4 is sensible here.

(a/b) ÷ (c/d) = (a/b) × (d/c), with b, c and d non-zero

7. Use fractions of amounts and compare values

To find a fraction of a quantity, divide by the denominator and multiply by the numerator. Three fifths of 40 is 40 ÷ 5 × 3 = 24. This is equivalent to 40 × 3/5. In a context, keep the quantity's unit in the answer.

To compare fractions, use equivalent fractions with the same denominator or convert accurately to decimals. Comparing 5/8 and 2/3 with denominator 24 gives 15/24 and 16/24, so 2/3 is larger. A larger denominator alone does not mean a larger fraction; the numerator and denominator work together.

8. Apply the order of operations and check

Brackets and the fraction bar group calculations. In (1/2 + 1/3) ÷ 5/6, first add inside the brackets to get 5/6, then divide by 5/6 to get 1. Without those brackets, the division would be done before the addition and the result would be different.

Estimate before accepting an answer. Adding two positive proper fractions must give something larger than either one, while multiplying them gives something smaller than either. Keep exact fractions during working, simplify at the end, and convert to a mixed number or decimal only when the question requests that form.

Check your understanding

How to earn the marks

• Use a common denominator for addition and subtraction only.
• Convert mixed numbers before multiplication or division.
• Show any cancellation and give a fully simplified answer.
• Check whether the result should be above or below the starting quantity.

A common mistake

Do not add denominators. For instance, 1/2 + 1/2 = 1, not 2/4.

Course context and official specifications

This guide covers selected skills. Some extensions are Higher content; choose your official tier separately from practice difficulty in the app. These are original questions and explanations.

• AQA GCSE Mathematics — N2, N8, N10

  Fraction arithmetic is available at both tiers. This bank covers selected calculation skills, not all exact-number content.

• Edexcel GCSE Mathematics — N2, N8, N10

  Fraction arithmetic is available at both tiers. This bank covers selected calculation skills, not all exact-number content.

• OCR GCSE Mathematics — 2.01

  Fraction arithmetic is available at both tiers. This bank covers selected calculation skills, not all exact-number content.

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