← LearnYear 6 Maths · NumberFree placement check
Year 6AC9M6N05

Adding and Subtracting Fractions

Solve problems involving addition and subtraction of fractions with the same or related denominators.

You can only add fractions when the pieces are the same size.

Builds onYr 5 · Equivalent fractionsYr 5 · Factors and multiples

Why this works

Think about what a fraction actually is before worrying about the method.

25\tfrac{2}{5} means 2 pieces, each one-fifth in size. The denominator names the piece size; the numerator counts them.
So 25+15\tfrac{2}{5} + \tfrac{1}{5} is easy — 2 fifths plus 1 fifth is 3 fifths. Same piece size, so you just count.
25+15=35\tfrac{2}{5} + \tfrac{1}{5} = \tfrac{3}{5}
But 12+13\tfrac{1}{2} + \tfrac{1}{3} makes no sense as it stands. You are trying to add halves to thirds — different-sized pieces. It is like adding 2 apples to 3 oranges and calling it 5 apples.
The fix is to rename both fractions so the pieces match. Sixths work for both, because 6 is a multiple of 2 and of 3.
12=3613=26\tfrac{1}{2} = \tfrac{3}{6} \qquad \tfrac{1}{3} = \tfrac{2}{6}
Now the pieces are the same size, so you can count them.
36+26=56\tfrac{3}{6} + \tfrac{2}{6} = \tfrac{5}{6}

That is the entire idea. You never add denominators — the denominator tells you the piece size, and the piece size does not change when you add. Only the count changes.

Worked example

e.g. 34+58\tfrac{3}{4} + \tfrac{5}{8}
  1. Look at the denominators. Are they already the same?4 and 8 — not the same4 \text{ and } 8 \text{ — not the same}
  2. Find a common denominator. Ask: is one a multiple of the other? Here 88 is a multiple of 44, so use 88.common denominator=8\text{common denominator} = 8
  3. Rename any fraction that needs it. Multiply top and bottom by the same number.34=3×24×2=68\tfrac{3}{4} = \tfrac{3 \times 2}{4 \times 2} = \tfrac{6}{8}
  4. Add the top numbers only. Leave the bottom alone.68+58=118\tfrac{6}{8} + \tfrac{5}{8} = \tfrac{11}{8}
  5. Simplify. Turn a top-heavy fraction into a mixed number.118=138\tfrac{11}{8} = 1\tfrac{3}{8}

Common mistakes

Wrong12+13=25\tfrac{1}{2} + \tfrac{1}{3} = \tfrac{2}{5}
Right12+13=36+26=56\tfrac{1}{2} + \tfrac{1}{3} = \tfrac{3}{6} + \tfrac{2}{6} = \tfrac{5}{6}
Adding the denominators. This is the single most common fraction error there is. Sanity-check it: 12\tfrac{1}{2} alone is already bigger than 25\tfrac{2}{5}, so the answer cannot possibly be right.
Wrong34+58=88=1\tfrac{3}{4} + \tfrac{5}{8} = \tfrac{8}{8} = 1
Right68+58=118=138\tfrac{6}{8} + \tfrac{5}{8} = \tfrac{11}{8} = 1\tfrac{3}{8}
Adding the numerators before making the denominators match. Rename first, always.
Wrong23=2×23=43\tfrac{2}{3} = \tfrac{2 \times 2}{3} = \tfrac{4}{3}
Right23=2×23×2=46\tfrac{2}{3} = \tfrac{2 \times 2}{3 \times 2} = \tfrac{4}{6}
Multiplying only the top when renaming. Whatever you do to the bottom you must also do to the top, or you have changed the value.
WrongLeaving 118 as the final answer\text{Leaving } \tfrac{11}{8} \text{ as the final answer}
Right1381\tfrac{3}{8}
Not wrong exactly, but at your level a top-heavy fraction should usually be written as a mixed number. Check what the question asks for.

Practice

FluencyGet quick and accurate at the method.
1
15+25\tfrac{1}{5} + \tfrac{2}{5}
Answer35\tfrac{3}{5}
2
38+14\tfrac{3}{8} + \tfrac{1}{4}
Answer58\tfrac{5}{8}
3
12+16\tfrac{1}{2} + \tfrac{1}{6}
Answer23\tfrac{2}{3}
4
56−13\tfrac{5}{6} - \tfrac{1}{3}
Answer12\tfrac{1}{2}
5
23+19\tfrac{2}{3} + \tfrac{1}{9}
Answer79\tfrac{7}{9}
6
710−15\tfrac{7}{10} - \tfrac{1}{5}
Answer12\tfrac{1}{2}
7
34+78\tfrac{3}{4} + \tfrac{7}{8}
Answer1581\tfrac{5}{8}
8
512+14\tfrac{5}{12} + \tfrac{1}{4}
Answer23\tfrac{2}{3}
9
910−25\tfrac{9}{10} - \tfrac{2}{5}
Answer12\tfrac{1}{2}
10
23+56\tfrac{2}{3} + \tfrac{5}{6}
Answer1121\tfrac{1}{2}
ReasoningExplain why. Say it in your own words.
1
Without calculating, explain why 12+13\tfrac{1}{2} + \tfrac{1}{3} cannot equal 25\tfrac{2}{5}.
Answer25\tfrac{2}{5} is less than 12\tfrac{1}{2}, but you started with 12\tfrac{1}{2} and added something to it. An answer smaller than what you started with must be wrong. This estimate-first check catches the denominator-adding error every time.
2
Why do you never add the denominators when adding fractions?
AnswerThe denominator names the size of each piece, not a quantity. Adding two fifths to one fifth gives three pieces that are still fifths — the piece size has not changed. Only the count changes, so only the numerator changes.
3
To add 14+16\tfrac{1}{4} + \tfrac{1}{6}, Kai uses 24 and Mia uses 12. Are both correct? Which is better?
AnswerBoth are correct — 24 and 12 are each multiples of 4 and 6. Mia's is better because 12 is the lowest common denominator, so her numbers stay smaller and she has less simplifying to do at the end.
4
Is 34+12\tfrac{3}{4} + \tfrac{1}{2} bigger or smaller than 1? Answer without working it out.
AnswerBigger. 34\tfrac{3}{4} is already close to 1, and you are adding another half on top, so the total must be more than 1. Estimating first tells you roughly what to expect and warns you if your working goes wrong.
AppliedThe same maths, inside a real question.
1
Ana walked 34\tfrac{3}{4} km before lunch and 58\tfrac{5}{8} km after. How far did she walk altogether?
Answer1381\tfrac{3}{8} km. Rename 34\tfrac{3}{4} as 68\tfrac{6}{8}, then 68+58=118=138\tfrac{6}{8} + \tfrac{5}{8} = \tfrac{11}{8} = 1\tfrac{3}{8}.
2
A recipe needs 23\tfrac{2}{3} cup of flour. Ben has already added 16\tfrac{1}{6} cup. How much more does he need?
Answer12\tfrac{1}{2} cup. Rename 23\tfrac{2}{3} as 46\tfrac{4}{6}, then 46−16=36=12\tfrac{4}{6} - \tfrac{1}{6} = \tfrac{3}{6} = \tfrac{1}{2}.
3
A water tank is 56\tfrac{5}{6} full. 13\tfrac{1}{3} of the tank is used. What fraction is left?
Answer12\tfrac{1}{2}. Rename 13\tfrac{1}{3} as 26\tfrac{2}{6}, then 56−26=36=12\tfrac{5}{6} - \tfrac{2}{6} = \tfrac{3}{6} = \tfrac{1}{2}.
4
Priya spends 14\tfrac{1}{4} of her pocket money on a book and 38\tfrac{3}{8} on a game. What fraction has she spent, and what fraction is left?
AnswerShe has spent 58\tfrac{5}{8} and has 38\tfrac{3}{8} left. 14=28\tfrac{1}{4} = \tfrac{2}{8}, so 28+38=58\tfrac{2}{8} + \tfrac{3}{8} = \tfrac{5}{8}; and 1−58=381 - \tfrac{5}{8} = \tfrac{3}{8}.
Next step
Practise Adding and Subtracting Fractions with instant marking
A free 10-minute placement check finds which Year 6 topics to work on first, then Summit builds a weekly plan around them.
See where my child is →