How do you add fractions with different denominators?

Why expand at all

A third and a quarter are parts of different sizes, and they cannot be counted together — any more than three minutes and four metres can be added into one meaningful number.

Expanding makes both into parts of the same size. A third becomes four twelfths and a quarter becomes three twelfths, and now both are measured in the same unit and can be counted.

That is also why the denominator in the answer is the common denominator. We did not add it — it is simply the unit both parts are measured in once they have been expanded.

Finding the common denominator

The product of the two denominators always works, and it is the safe route when nothing smaller is obvious. The only cost is that the numbers grow and you have to reduce at the end.

The smallest one is quick to find: run through the multiples of the larger denominator and check which is also divisible by the smaller. If the larger already divides by the smaller, it is the common denominator itself.

Mixed numbers and a sanity check

With mixed numbers you can add the wholes separately from the fractions. If the fractions together pass a whole, carry that whole across into the whole-number part.

In subtraction the top fraction is sometimes smaller than the bottom one. Then you borrow one whole and add it to the fraction: one whole is a full denominator of parts, so two and a quarter becomes one and five quarters.

At the end, check the answer is sensible. Two fractions each close to a half should total around one whole, and an answer far from that points to a slip in the expanding.

Worked examples

1. Calculate 1 over 3 plus 1 over 4

   1. Common denominator: 3 times 4, which is 12
   2. Expand: a third is 4 over 12, a quarter is 3 over 12
   3. Add numerators: 4 plus 3

   Answer: 7 over 12

2. Calculate 5 over 6 minus 1 over 2

   1. 6 divides by 2, so the common denominator is 6
   2. Expand only the second: a half is 3 over 6
   3. Subtract numerators: 5 minus 3 is 2
   4. Reduce: 2 over 6 is a third

   Answer: A third

3. Calculate 2 and a quarter minus 1 and a half

   1. Common denominator for the fractions is 4: a quarter and two quarters
   2. A quarter is less than two quarters, so borrow a whole from the 2
   3. This gives 1 and five quarters, minus 1 and two quarters
   4. Subtract wholes and subtract fractions

   Answer: Three quarters

Common mistakes

Adding numerators and denominators without expanding

Parts of different sizes cannot be counted together. A third plus a quarter is not two sevenths — two sevenths is smaller than a third on its own, and that check exposes it at once.

Expanding the denominator without the numerator

Expanding means multiplying both by the same number. Changing the denominator alone turns it into a different fraction before you have even added.

Forgetting to borrow a whole in subtraction

When the top fraction is smaller than the bottom one you cannot subtract directly. Borrow one whole, convert it into parts, and add it to the fraction — exactly like borrowing with whole numbers.

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 and subtract fractions with the same denominator?