Adding and Subtracting Fractions With the Same Denominator
When the bottom numbers already match, the work is easy. The whole topic rests on one idea: you are counting pieces, and the pieces do not change size.
The one idea
Say the fractions out loud instead of reading the digits. 38 is "three eighths". 28 is "two eighths".
Three eighths plus two eighths is five eighths, in the same way that three apples plus two apples is five apples. Nobody would answer "five apple-apples", and nobody should answer 516 either.
The rule
When the denominators match, add or subtract the numerators and keep the denominator exactly as it is.
The bottom number is the size of the pieces. You are not changing the pieces, only counting them.
Adding
So 38 + 28 = 58. The 8 stays put because the pieces stayed the same size.
Subtracting
Subtracting is the same picture read the other way. Start with what you have, take some pieces away, count what is left.
So 78 minus 38 = 48, which tidies to 12.
Tidying the answer
Getting 48 is correct. Leaving it there usually is not, because most mark schemes want the simplest form.
- If the top and bottom share a factor, simplify. 48 becomes 12.
- If the top ends up equal to the bottom, the answer is 1. 55 is one whole.
- If the top ends up bigger than the bottom, that is an improper fraction and it is a perfectly good answer, though some questions ask for it as a mixed number.
Worked examples
A straightforward sum
- Check the bottom numbers. Both are 9, so no rewriting is needed.
- Add the top numbers only. 2 + 5 = 7.
- Keep the denominator as 9.
- Can 79 be simplified? 7 is prime and does not divide into 9, so no.
An answer that needs tidying
- Denominators match at 12, so add the tops. 5 + 3 = 8, giving 812.
- Now check whether it simplifies. Both 8 and 12 are even, so it does.
- The highest common factor is 4. 8 ÷ 4 = 2 and 12 ÷ 4 = 3.
- Finished check: 2 and 3 share nothing but 1.
A subtraction word problem
- Both amounts are in tenths, so the pieces are the same size.
- Subtract the top numbers. 9 − 4 = 5.
- The denominator stays 10, giving 510 of a litre.
- Simplify: both divide by 5, giving 12.
Common mistakes
1. Adding the bottom numbers as well
Writing 38 + 28 = 516. Almost every child does this once. The quickest cure is the picture above: the bar never gets cut into sixteenths at any point. Sixteenths would be smaller pieces, so the answer would be smaller than what you started with, which cannot be right when you have just added something.
2. Leaving the answer unsimplified
Stopping at 812 instead of 23. The arithmetic was right, but the mark often is not. Make simplifying the automatic last step of every fraction question.
3. Using this rule when the denominators do not match
Writing 12 + 13 = 25. The pieces are different sizes, so they cannot be counted together yet. That case needs a common denominator first, which is the next topic.
Practice questions
Seven questions. One attempt each, and the explanation appears once the answer is in.
Check yourself
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For parents
If your child adds the bottom numbers, do not just correct it. Ask them whether the answer should be bigger or smaller than what they started with. They will say bigger, then look at their own answer and see it is smaller. Catching their own error that way makes the correction stick far better than being told.
What to learn next
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