How to Add Fractions with Different Denominators

The addition of fractions with different (unlike) denominators cannot be done directly, and requires the following steps:

  • Find the Lowest Common Multiple (lcm) of both denominators.
  • Once the lcm has been determined, scale up the numerators and denominators of both fractions such that both fractions have the lcm as their new denominators.
  • Once this is done, the numerators can then simply be added together.
  • Remember to reduce the resulting fraction to its lowest form, if necessary.

For example, to add the fractions one and a half to two and two-thirds, and to find the sum:

 1  1
2
+
 2  2
3

Step 1 :
Convert both fractions to vulgar fractions, by multiplying the whole number by the denominator and adding the result to the numerator:

=
 1  1
2
+
 2  2
3
=
  (1 x 2 + 1)
2
+
  (2 x 3 + 2)
3
=
  3
2
+
  8
3

Step 2:
Determine a common denominator for both fractions, by finding the Lowest Common Multiple (lcm) of the denominators (2 and 3).
In this example, since both 2 and 3 are primes, this means that there are no common factors, so the lcm is simply 2 x 3 = 6. So we need to 'scale up' both fractions to 6:

=
  3
2
+
  8
3
=
  (3 x 3) + (8 x 2)
2 x 3
=
  9 + 16
6
=
  25
6

Step 3:
As a final step for all types of fraction arithmetic, remember to ensure that the result is reduced to its simplest form (that is, check whether the numerator and denominator can be divided out by a common factor, and to extract any whole numbers if the result is an improper fraction).

In this example, the result 25/6 is an improper fraction, and the numerator needs to be divided through by the denominator:

=
  25
6
=
 4  1
6

So the sum of

 1  1
2
+
 2  2
3

is

=
 4  1
6

You can check the answer to this fraction addition, and other fraction problems by using our Fraction Calculator!