How do you add and subtract fractions with the same denominator?

Counting parts

Two fifths plus two fifths is four fifths, exactly as two apples plus two apples is four apples. The word fifths is the name of the part, and counting does not change it.

So the rule is short: add the numerators, keep the denominator. Subtraction is the same — subtract the numerators, keep the denominator.

Why denominators are not added

The denominator is not a quantity but a unit. It says what size of part we are talking about, not how many there are, so there is nothing in it to add.

If you add denominators too, you get an answer smaller than the things you added. A half plus a half would come out as two quarters, which is a half — and two halves are plainly a whole. Any child can run that check in their head.

What to do with the result

If the result can be reduced, reduce it. Four eighths is a half, and that is usually the form wanted.

If the numerator comes out larger than the denominator, the result is more than one whole. You can leave it as an improper fraction or convert it to a mixed number, depending on what the question asks.

Worked examples

1. Calculate 2 over 7 plus 3 over 7

   1. The denominators match, so add numerators
   2. 2 plus 3 is 5
   3. The denominator stays 7

   Answer: 5 over 7

2. Calculate 5 over 8 minus 1 over 8, and reduce

   1. Subtract numerators: 5 minus 1 is 4
   2. The denominator stays 8, giving 4 over 8
   3. Reduce by 4

   Answer: A half

3. Calculate 3 over 4 plus 3 over 4

   1. Add numerators: 3 plus 3 is 6
   2. This gives 6 over 4, an improper fraction
   3. Convert: 4 fits into 6 once with remainder 2
   4. Reduce the remainder: 2 over 4 is a half

   Answer: One and a half

Common mistakes

Adding the denominators too

The denominator names the part, not a quantity. A half plus a half is a whole, whereas adding denominators would give a half — an answer smaller than either addend, which is impossible for positive numbers.

Forgetting to reduce

The answer is right but not in the form asked for. After every addition, ask whether the numerator and denominator share a factor.

Leaving an improper fraction when a mixed number was wanted

Both forms are correct, but the question decides. If it asks how many wholes, you have to convert.

More in Fractions

• What do the numerator and the denominator of a fraction mean?
• What are equivalent fractions and how do you reduce one?
• What is an improper fraction and how do you convert it to a mixed number?
• How do you compare two fractions and put them in order?
• How do you add fractions with different denominators?