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Year 6 Maths Adding & Subtracting Fractions

Add fractions - Related denominators

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Theory

To add fractions with related denominators, first rename one fraction so both have the same denominator, then add the numerators and keep the denominator. This Year 6 maths page shows how to add, simplify the answer, and write answers bigger than one as mixed numerals.

Fractions can only be added directly when the parts are the same size, which means the same denominator.

With related denominators, the larger denominator is the common denominator. For \(\dfrac{1}{5} + \dfrac{7}{10}\), rename \(\dfrac{1}{5} = \dfrac{2}{10}\).

Then add the numerators only: \(\dfrac{2}{10} + \dfrac{7}{10} = \dfrac{9}{10}\). The denominator stays the same.

Simplify when you can: \(\dfrac{8}{12} = \dfrac{2}{3}\). If the answer is more than one, write it as a mixed numeral: \(\dfrac{11}{8} = 1\dfrac{3}{8}\).

One fifth plus seven tenths Four bars the same length: 1 of 5 parts shaded; 2 of 10 parts shaded (the same length); 7 of 10 parts shaded; and 9 of 10 parts shaded for the total. 15 210 710 910
\(\dfrac{1}{5} = \dfrac{2}{10}\), and \(\dfrac{2}{10} + \dfrac{7}{10} = \dfrac{9}{10}\).
Eleven eighths as bars Two bars each split into 8 equal parts. The first bar is fully shaded and the second has 3 parts shaded: 11 eighths, or 1 and 3 eighths. 1 whole 38
\(\dfrac{5}{8} + \dfrac{6}{8} = \dfrac{11}{8} = 1\dfrac{3}{8}\).

Adding with related denominators:

\[ \dfrac{a}{n} + \dfrac{b}{n} = \dfrac{a + b}{n} \]
( a / n )+( b / n )=( a + b )/ n
\[ \dfrac{1}{5} + \dfrac{7}{10} = \dfrac{2}{10} + \dfrac{7}{10} = \dfrac{9}{10} \]
1/5+7/10=2/10+7/10=9/10
Never add the denominators. \(\dfrac{2}{10} + \dfrac{7}{10}\) is \(\dfrac{9}{10}\), not \(\dfrac{9}{20}\).
Check the size. If both fractions are more than a half, the answer must be more than one.

How to add fractions with related denominators

  1. Choose the larger denominator as the common denominator.
  2. Rename the other fraction (multiply top and bottom by the same number).
  3. Add the numerators; keep the denominator.
  4. Simplify, and change an improper fraction to a mixed numeral.
Example 1 — Fifths and tenths
Work out \(\dfrac{1}{5} + \dfrac{7}{10}\).
Solution
\(\dfrac{2}{10} + \dfrac{7}{10}\)\(=\)\(\dfrac{9}{10}\)
2/10+7/10=9/10

\(\dfrac{9}{10}\).

Example 2 — Sixths and twelfths
Work out \(\dfrac{1}{6} + \dfrac{5}{12}\).
Solution
\(\dfrac{2}{12} + \dfrac{5}{12}\)\(=\)\(\dfrac{7}{12}\)
2/12+5/12=7/12

\(\dfrac{7}{12}\).

Example 3 — Thirds and ninths
Work out \(\dfrac{1}{3} + \dfrac{4}{9}\).
Solution
\(\dfrac{3}{9} + \dfrac{4}{9}\)\(=\)\(\dfrac{7}{9}\)
3/9+4/9=7/9

\(\dfrac{7}{9}\).

Example 4 — Simplify
Work out \(\dfrac{1}{4} + \dfrac{5}{12}\) in simplest form.
Solution
\(\dfrac{3}{12} + \dfrac{5}{12}\)\(=\)\(\dfrac{8}{12}\)
\(\dfrac{8}{12}\)\(=\)\(\dfrac{2}{3}\)
3/12+5/12=8/12=2/3

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

Example 5 — Mixed numeral
Work out \(\dfrac{4}{5} + \dfrac{7}{10}\) as a mixed numeral in simplest form.
Solution
\(\dfrac{8}{10} + \dfrac{7}{10}\)\(=\)\(\dfrac{15}{10}\)
\(\dfrac{15}{10}\)\(=\)\(1\dfrac{1}{2}\)
8/10+7/10=15/10=11/2

\(1\dfrac{1}{2}\).

Example 6 — Eighths
Work out \(\dfrac{5}{8} + \dfrac{3}{4}\).
Solution
\(\dfrac{5}{8} + \dfrac{6}{8}\)\(=\)\(\dfrac{11}{8}\)
\(\dfrac{11}{8}\)\(=\)\(1\dfrac{3}{8}\)
5/8+6/8=11/8=13/8

\(1\dfrac{3}{8}\) (second diagram).

Example 7 — Walking
Ava walks \(\dfrac{3}{10}\) km to the shop and then \(\dfrac{1}{2}\) km to the park. How far does she walk?
Solution
\(\dfrac{3}{10} + \dfrac{5}{10}\)\(=\)\(\dfrac{8}{10}\)
\(\dfrac{8}{10}\)\(=\)\(\dfrac{4}{5}\)
3/10+5/10=8/10=4/5

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

Example 8 — Three fractions
A garden is \(\dfrac{1}{6}\) herbs, \(\dfrac{1}{3}\) tomatoes and \(\dfrac{1}{12}\) chillies. What fraction is planted?
Solution
\(\dfrac{2}{12} + \dfrac{4}{12} + \dfrac{1}{12}\)\(=\)\(\dfrac{7}{12}\)
2/12+4/12+1/12=7/12

\(\dfrac{7}{12}\) of the garden.

Example 9 — Find the mistake
Max wrote \(\dfrac{3}{4} + \dfrac{1}{12} = \dfrac{4}{16}\). What is right?
Solution
\(\dfrac{9}{12} + \dfrac{1}{12}\)\(=\)\(\dfrac{10}{12}\)
\(\dfrac{10}{12}\)\(=\)\(\dfrac{5}{6}\)
9/12+1/12=10/12=5/6

Max added the tops and the bottoms. Rename \(\dfrac{3}{4} = \dfrac{9}{12}\) first: the answer is \(\dfrac{5}{6}\).

Common pitfalls

Adding the denominators. The denominator names the size of the parts; it does not change when you add.
Not renaming first. \(\dfrac{1}{4} + \dfrac{5}{12}\) is not \(\dfrac{6}{12}\): rename \(\dfrac{1}{4}\) as \(\dfrac{3}{12}\).
Leaving an improper answer. Write \(\dfrac{11}{8}\) as \(1\dfrac{3}{8}\) when asked for a mixed numeral.
Not simplifying. \(\dfrac{8}{12}\) simplifies to \(\dfrac{2}{3}\).

Frequently asked questions

Can I add fractions with different denominators straight away?

No. Rename them so the denominators are the same first.

Which denominator do I use?

With related denominators, use the larger one.

Do I add the denominators?

No. Add the numerators and keep the denominator.

What if the answer is more than one?

Write it as a mixed numeral: \(\dfrac{15}{10} = 1\dfrac{1}{2}\).

What is 1/5 + 7/10?

\(\dfrac{9}{10}\).

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