Math Vale

🍰 Fraction Isles · Fractions & Proportion

Adding & Subtracting Fractions

Add and subtract fractions and mixed numbers by rewriting them with a common denominator.

In short

  • The denominator names the size of the pieces, so it is never added or subtracted — only the numerators are combined.
  • Unlike denominators must be rewritten over a common denominator first, scaling numerator and denominator by the same factor.
  • The least common denominator is the least common multiple of the denominators; any common multiple works but leaves larger numbers.
  • Mixed-number subtraction sometimes needs regrouping: borrow one whole and write it as d/d before subtracting.
  • Estimate with benchmarks before you start, and simplify and sanity-check the size of the answer at the end.

You can only add things of the same size

Adding fractions is counting, and counting only works when the things being counted are alike.

2/7 + 3/7 is "two sevenths plus three sevenths". They are the same size of piece, so you end up with five of them: 5/7.

The denominator did not change, and it must not. It is not a count — it is the *name* of the piece, like "metre" or "gold coin". You would never say 2 metres + 3 metres = 5 double-metres.

That single idea explains the whole skill:

  • Same denominators: add or subtract the numerators, keep the denominator, then simplify.
  • Different denominators: change them until they match, then do the above.

Subtraction works identically: 5/9 - 2/9 = 3/9 = 1/3.

Unlike denominators

Halves and thirds cannot be counted together, so rewrite both as pieces of one shared size — a common denominator.

1/2 + 1/3: sixths work for both, because 6 is a multiple of 2 and of 3.

1/2 = 3/6 and 1/3 = 2/6, so 1/2 + 1/3 = 3/6 + 2/6 = 5/6.

The least common denominator is the least common multiple of the denominators. Any common multiple works; the least one just keeps the arithmetic small. When the denominators share no factor, their product is already the least: for 4 and 7 it is 28.

Remember to scale the numerator by the same factor as the denominator. Turning 1/2 into "?/6" means multiplying top and bottom by 3, not writing 1/6.

A fraction bar cut once into halves and once into thirds, then overlaid, shows why sixths are the first size that fits both.

Mixed numbers and regrouping

A mixed number like 4 1/3 is a whole part plus a fraction part, and there are two safe routes.

Route 1 — convert to improper fractions. 4 1/3 = 13/3, 2 1/2 = 5/2. Then work over sixths: 26/6 + 15/6 = 41/6 = 6 5/6. This route never goes wrong, so it is the one to use when you are unsure.

Route 2 — keep them separate. Add the wholes, add the fractions, then tidy up: 4 + 2 = 6 and 1/3 + 1/2 = 5/6, giving 6 5/6.

Route 2 needs care for subtraction. In 5 1/4 - 2 3/4 the fraction being taken away is larger, so you must regroup: borrow one whole from the 5 and write it as 4/4.

5 1/4 = 4 + 4/4 + 1/4 = 4 5/4, and now 4 5/4 - 2 3/4 = 2 2/4 = 2 1/2.

What you must never do is flip the fraction subtraction round to keep it positive.

Estimating, and checking your answer

Before calculating, decide roughly what the answer should be. Benchmarks make this quick.

7/8 + 9/10: both are nearly 1, so the answer must be a bit less than 2. If you get 16/18, which is under 1, you know something has gone wrong — almost certainly the denominators were added.

Useful checks after the fact:

  • Is it simplified? 3/9 should be written 1/3.
  • Does the size make sense? A sum of two positive fractions must be bigger than each of them; a difference must be smaller than the one you started with.
  • Reverse it. If 5/6 - 1/3 = 1/2, then 1/2 + 1/3 should give 5/6 again.

In word problems the whole thing counts as 1, and 1 can be written as 6/6, 12/12 or whatever denominator you need — which is exactly how you work out how much is left.

Worked examples

Example 1

5/6 - 3/8 = ?

  1. The pieces differ, so find a common denominator. The least common multiple of 6 and 8 is 24.
  2. 5/6 = 20/24, because 6 x 4 = 24 and 5 x 4 = 20.
  3. 3/8 = 9/24, because 8 x 3 = 24 and 3 x 3 = 9.
  4. Subtract the numerators only: 20 - 9 = 11, so the answer is 11/24.
  5. 11 and 24 share no factor, so 11/24 is in lowest terms. Estimate check: 5/6 is a bit over 3/4 and 3/8 a bit under 1/2, so an answer near 1/2 is sensible.

Example 2

6 1/5 - 2 3/4 = ?

  1. Common denominator for fifths and quarters: 20. So 6 1/5 = 6 4/20 and 2 3/4 = 2 15/20.
  2. 4/20 is smaller than 15/20, so regroup: borrow 1 whole from the 6 and write it as 20/20.
  3. 6 4/20 becomes 5 24/20.
  4. Now subtract each part: 5 - 2 = 3 wholes, and 24/20 - 15/20 = 9/20.
  5. The answer is 3 9/20. Check by adding back: 3 9/20 + 2 15/20 = 5 24/20 = 6 4/20 = 6 1/5.

Example 3

A flask is full. Ines drinks 1/3 of it, then 2/5 of it. What fraction is left?

  1. The whole flask counts as 1. First find how much was drunk altogether.
  2. Common denominator of 3 and 5 is 15: 1/3 = 5/15 and 2/5 = 6/15.
  3. Drunk in total: 5/15 + 6/15 = 11/15.
  4. Write the whole flask as 15/15 and subtract: 15/15 - 11/15 = 4/15.
  5. 4/15 of the flask is left, which is a little over a quarter — sensible, since roughly three quarters was drunk.

Practice problems, with solutions

Three problems of increasing difficulty, each with the full working. In the game these are generated fresh every time; these three are fixed so this page always shows the same ones.

Problem 1

Difficulty 1 of 5

4/6 + 1/6 = ?

Answer: 5/6

  1. The pieces already match, so add the numerators only: 4 + 1 = 5.
  2. 4/6 + 1/6 = 5/6.
  3. 5 and 6 share no factor, so 5/6 is the final answer.

Problem 2

Difficulty 3 of 5

5/6 + 4/9 = ?

Answer: 1 5/18

  1. Least common denominator of 6 and 9: 18.
  2. 5/6 = 15/18 and 4/9 = 8/18.
  3. 15/18 + 8/18 = 23/18.
  4. 5/6 + 4/9 = 23/18 (that is 1 5/18).

Problem 3

Difficulty 4 of 5

6 2/9 - 1 3/4 = ?

Answer: 4 17/36

  1. Write both as improper fractions: 6 2/9 = 56/9 and 1 3/4 = 7/4.
  2. Common denominator 36: 224/36 - 63/36.
  3. = 161/36
  4. 6 2/9 - 1 3/4 = 4 17/36.

Common mistakes

  • Adding the denominators as well as the numerators: writing 1/2 + 1/3 = 2/5.
  • Changing the denominator to the common one but carrying the old numerator across, so 1/2 becomes 1/6.
  • Flipping a fraction subtraction round to avoid a negative instead of regrouping a whole.
  • Adding only the whole numbers of two mixed numbers and forgetting the fraction parts.
  • Leaving an answer unsimplified, or leaving an improper fraction where a mixed number was wanted.

What you should be able to do

  • Add and subtract fractions with like denominators.
  • Find a common denominator and add or subtract unlike fractions.
  • Add and subtract mixed numbers, regrouping when needed.
  • Simplify the result and check it against an estimate.

Where this fits in the curriculum

Common Core

  • 4.NF.B.3.A

    Grade 4 — Understand adding and subtracting fractions with like denominators as joining and separating parts of the same whole.

  • 5.NF.A.1

    Grade 5 — Add and subtract fractions with unlike denominators, including mixed numbers, using equivalent fractions.

  • 5.NF.A.2

    Grade 5 — Solve word problems with fractions, and check the answer against a benchmark estimate.

Ontario

  • G5.B2.5

    Grade 5 — Add and subtract fractions with like denominators in various contexts.

  • G6.B2.5

    Grade 6 — Add and subtract fractions with like and unlike denominators, using appropriate strategies.

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