Fractions: Adding and Subtracting
Adding and subtracting fractions means combining two or more fractional amounts into a single fraction, and because the denominator shows how many equal parts a whole has been split into, fractions with different denominators must first be rewritten with a common denominator before only the numerators are added or subtracted. Mixed numbers, made up of a whole number and a fraction together, are usually converted to improper, top-heavy, fractions first, calculated, then converted back if a mixed-number answer is required.
Before you start
Make sure you're comfortable with these topics first:
Method
- To add or subtract fractions with the same denominator, keep the denominator the same and add or subtract the numerators only.
- If the denominators are different, first find a common denominator, ideally the lowest common multiple (LCM) of the two denominators, so the numbers stay as small as possible.
- Convert each fraction to an equivalent fraction with the common denominator, multiplying the numerator by the same number used to scale the denominator.
- Add or subtract the numerators of the converted fractions, keeping the common denominator unchanged.
- If a question involves mixed numbers, convert each one to an improper fraction first, by multiplying the whole number by the denominator and adding the numerator.
- Once the calculation is finished, simplify the answer fully by dividing the numerator and denominator by their highest common factor, and convert an improper fraction back to a mixed number if the question asks for one.
- When subtracting mixed numbers, converting both to improper fractions before subtracting avoids having to 'borrow' from the whole-number part.
Worked example
Work out 2 and 3/5 minus 1 and 5/6, giving your answer as a fraction in its simplest form.
- Convert both mixed numbers to improper fractions: 2 and 3/5 = (2 x 5 + 3)/5 = 13/5, and 1 and 5/6 = (1 x 6 + 5)/6 = 11/6.
- Find the lowest common denominator of 5 and 6, which is 30.
- Convert each fraction to thirtieths: 13/5 = 78/30, and 11/6 = 55/30.
- Subtract the numerators: 78/30 - 55/30 = 23/30.
- Check whether 23/30 simplifies: the highest common factor of 23 and 30 is 1, so 23/30 is already in its simplest form.
Practice questions
Type your answer and press Check to be marked straight away, or reveal the answer and mark yourself.
Q1Work out 2/7 + 3/7.Show answer
Answer: 5/7
Q2Work out 3/4 - 1/6, giving your answer in its simplest form.Show answer
Answer: 7/12
Q3Work out 5/8 + 1/3, giving your answer in its simplest form.Show answer
Answer: 23/24
Q4Work out 4 and 1/4 + 2 and 1/3, giving your answer as a mixed number.Show answer
Answer: 6 and 7/12
Q5Work out 5 and 1/2 - 2 and 3/4, giving your answer as a mixed number.Show answer
Answer: 2 and 3/4
Q6A recipe uses 2/3 of a bag of flour for a cake and 1/5 of a bag for a batch of scones. What fraction of the bag of flour is used in total?Show answer
Answer: 13/15 of the bag
Q7Nadia has 7/8 of a pizza left. She eats 1/4 of the whole pizza. What fraction of the whole pizza is left after she eats?Show answer
Answer: 5/8 of the pizza
Q8Explain why 1/3 + 1/4 is not equal to 2/7.Show answer
Answer: Fractions can only be added by first finding a common denominator; adding the numerators (1 + 1 = 2) and the denominators (3 + 4 = 7) separately, as in 2/7, is not a valid method, since 1/3 + 1/4 is actually 4/12 + 3/12 = 7/12.
Exam-style questions
Written in the style of a KS3 Maths exam paper, with a full mark scheme.
Work out 3 and 5/6 - 1 and 7/10, giving your answer as a mixed number in its simplest form.
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Jamal spends 2/5 of his savings on a games console and 1/4 of his savings on a controller. (a) Work out what fraction of his savings Jamal has spent in total. (b) Work out what fraction of his savings Jamal has left.
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Work out 1 and 1/2 + 2/3 - 5/6, giving your answer as a single fraction in its simplest form.
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Free printable worksheet
Want more practice on paper? Download the fractions: adding and subtracting worksheet pack - 8 pages of exam-style questions with a full mark scheme. One email opens every download in this browser for 14 days - no account, no card. Print it for personal and classroom use.
This topic is chapter 9 of KS3 Maths Workbook 1, the whole course as one free printable PDF.
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