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Year 5 Maths Adding & Subtracting Fractions

Add fractions - Same denominator

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Theory

When fractions have the same denominator, the parts are the same size, so you just count the parts: \(\dfrac{2}{7} + \dfrac{3}{7} = \dfrac{5}{7}\). Add the numerators and keep the denominator. This Year 5 maths page also shows how to simplify the answer and what to do when the total is \(1\) or more.

\(\dfrac{2}{7}\) is \(2\) sevenths and \(\dfrac{3}{7}\) is \(3\) sevenths. Together they are \(5\) sevenths: \(\dfrac{5}{7}\).

The denominator stays the same, because adding does not change the size of the parts. Only the number of parts changes.

If the answer has a common factor, write it in simplest form: \(\dfrac{1}{9} + \dfrac{5}{9} = \dfrac{6}{9} = \dfrac{2}{3}\).

If the numerator equals the denominator, the answer is \(1\). If it is bigger, write a mixed numeral: \(\dfrac{4}{7} + \dfrac{5}{7} = \dfrac{9}{7} = 1\dfrac{2}{7}\).

Two sevenths plus three sevenths Three bars, each split into 7 equal parts. The first has 2 parts shaded, the second 3 parts and the third 5 parts, showing 2 sevenths plus 3 sevenths equals 5 sevenths. 27 37 57
\(\dfrac{2}{7} + \dfrac{3}{7} = \dfrac{5}{7}\).
Nine sevenths as bars Two bars, each split into 7 equal parts. The first is fully shaded and the second has 2 parts shaded: 4 sevenths plus 5 sevenths is 9 sevenths, or 1 and 2 sevenths. 1 whole 27
\(\dfrac{4}{7} + \dfrac{5}{7} = \dfrac{9}{7} = 1\dfrac{2}{7}\).

Add the numerators; keep the denominator:

\[ \dfrac{2}{7} + \dfrac{3}{7} = \dfrac{2 + 3}{7} = \dfrac{5}{7} \]
2/7+3/7=5/7
\[ \dfrac{4}{10} + \dfrac{6}{10} = \dfrac{10}{10} = 1 \]
4/10+6/10=1
\[ \dfrac{5}{8} + \dfrac{5}{8} = \dfrac{10}{8} = 1\dfrac{2}{8} = 1\dfrac{1}{4} \]
5/8+5/8=11/4
Never add the denominators. Sevenths plus sevenths are still sevenths. \(\dfrac{2}{7} + \dfrac{3}{7}\) is not \(\dfrac{5}{14}\).
Tidy the answer. Simplify, and change an improper fraction to a whole number or mixed numeral.

How to add fractions with the same denominator

  1. Check the denominators are the same.
  2. Add the numerators.
  3. Keep the denominator.
  4. Simplify, or change to a mixed numeral if the answer is more than \(1\).
Example 1 — Add the parts
Work out \(\dfrac{4}{9} + \dfrac{1}{9}\).
Solution
\(\dfrac{4}{9} + \dfrac{1}{9}\)\(=\)\(\dfrac{5}{9}\)
4/9+1/9=5/9

\(\dfrac{5}{9}\).

Example 2 — Simplify
Work out \(\dfrac{1}{9} + \dfrac{5}{9}\) in simplest form.
Solution
\(\dfrac{1}{9} + \dfrac{5}{9}\)\(=\)\(\dfrac{6}{9}\)
\(\dfrac{6}{9}\)\(=\)\(\dfrac{2}{3}\)
1/9+5/9=2/3

\(\dfrac{2}{3}\).

Example 3 — Makes one
Work out \(\dfrac{4}{10} + \dfrac{6}{10}\).
Solution
\(\dfrac{4}{10} + \dfrac{6}{10}\)\(=\)\(\dfrac{10}{10}\)
\(\dfrac{10}{10}\)\(=\)\(1\)
4/10+6/10=1

\(1\) whole.

Example 4 — Three fractions
Work out \(\dfrac{1}{10} + \dfrac{3}{10} + \dfrac{4}{10}\) in simplest form.
Solution
\(\dfrac{1}{10} + \dfrac{3}{10} + \dfrac{4}{10}\)\(=\)\(\dfrac{8}{10}\)
\(\dfrac{8}{10}\)\(=\)\(\dfrac{4}{5}\)
1/10+3/10+4/10=4/5

\(\dfrac{4}{5}\).

Example 5 — More than one
Work out \(\dfrac{5}{6} + \dfrac{3}{6}\). Give the answer as a mixed numeral in simplest form.
Solution
\(\dfrac{5}{6} + \dfrac{3}{6}\)\(=\)\(\dfrac{8}{6}\)
\(\dfrac{8}{6}\)\(=\)\(1\dfrac{2}{6} = 1\dfrac{1}{3}\)
5/6+3/6=11/3

\(1\dfrac{1}{3}\).

Example 6 — Word problem
A plant grew \(\dfrac{5}{8}\) cm one week and \(\dfrac{5}{8}\) cm the next. How much did it grow altogether?
Solution
\(\dfrac{5}{8} + \dfrac{5}{8}\)\(=\)\(\dfrac{10}{8}\)
\(\dfrac{10}{8}\)\(=\)\(1\dfrac{2}{8} = 1\dfrac{1}{4}\)
5/8+5/8=11/4

\(1\dfrac{1}{4}\) cm.

Example 7 — Twelfths
Ari paints \(\dfrac{7}{12}\) of a fence and Bea paints \(\dfrac{1}{12}\). What fraction have they painted, in simplest form?
Solution
\(\dfrac{7}{12} + \dfrac{1}{12}\)\(=\)\(\dfrac{8}{12}\)
\(\dfrac{8}{12}\)\(=\)\(\dfrac{2}{3}\)
7/12+1/12=2/3

\(\dfrac{2}{3}\) of the fence.

Example 8 — Find the mistake
Jay wrote \(\dfrac{2}{7} + \dfrac{3}{7} = \dfrac{5}{14}\). Explain the mistake.
Solution
\(\dfrac{2}{7} + \dfrac{3}{7}\)\(=\)\(\dfrac{5}{7}\)
2/7+3/7=5/7

He added the denominators too. The parts are still sevenths, so the answer is \(\dfrac{5}{7}\).

Common pitfalls

Adding the denominators. \(\dfrac{2}{7} + \dfrac{3}{7} = \dfrac{5}{7}\), not \(\dfrac{5}{14}\).
Multiplying instead of adding. Add the numerators; do not multiply them.
Leaving an answer like 8/6. Change it to a mixed numeral and simplify: \(1\dfrac{1}{3}\).
Forgetting that n/n is 1. \(\dfrac{10}{10}\) is one whole.

Frequently asked questions

How do you add fractions with the same denominator?

Add the numerators and keep the denominator: \(\dfrac{4}{9} + \dfrac{1}{9} = \dfrac{5}{9}\).

Why do you not add the denominators?

The denominator tells the size of the parts. Adding more parts of the same size does not change their size.

What if the answer can be simplified?

Divide the numerator and denominator by a common factor: \(\dfrac{6}{9} = \dfrac{2}{3}\).

What if the answer is more than 1?

Write it as a mixed numeral: \(\dfrac{9}{7} = 1\dfrac{2}{7}\).

Can you add three fractions?

Yes. Add all the numerators: \(\dfrac{1}{10} + \dfrac{3}{10} + \dfrac{4}{10} = \dfrac{8}{10}\).

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