Adding fractions

Last updated: 11 Nov 2008

Addition is easy when the denominators are the same. You add the numerators and keep the denominator the same.

Example of Addition (1)

mathematical expression

Both fractions are expressed in fifths and so the denominator is 5 in both fractions. In this case, simply add the two numerators 1 and 3 to give you 4. The denominator stays the same, so the sum is mathematical expression.

Sadly, it is much more common to find that the denominators are different. In these cases you have to reorganize the fractions so that they have the same denominator. This is called a common denominator.

If the two denominators have no common factors or are variables, then you simply multiply them together to find the common denominator. You then have to multiply the numerators by the same value that you multiplied the denominator by so that the fractions still have the same value. Now you have two fractions with the same denominator you can add them up as before.

Example of Addition (2)

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The denominators are 3 and 4, which are different. Therefore we need to find a common denominator. Since the denominators 3 and 4 have no common factors, multiply them together mathematical expression.

Then multiply the numerators to give us the same fractions but expressed in twelfths i.e. with the same denominator.

mathematical expression

mathematical expression

Now we can add them.

mathematical expression

If the two denominators do have a common factor then it is usually best to find the lowest common denominator you can, which will be smaller than that given by multiplying the two denominators together.

Example of Addition (3)

mathematical expression

First find a common denominator. We could multiply the denominators 8 and 12 together to give us a common denominator of 96. But, since they have a common factor of 4 we can work with the lowest common denominator which is 24 and will make our calculations a little simpler.

Then multiply the numerators.

mathematical expression

mathematical expression

Now we can add them.

mathematical expression

Next page: Subtracting fractions

Comments (7):

  1. In your addition example (2) there is an error. You have written 2/3 + 1/4 = 8/12 + 3/12 = 11+3 over 12 = 11/12. The mistake is 11+3 over 12 should be 8+3 over 12 equalling 11/12.

    Thoroughly enjoying the refresher course.

    Anonymous on 15 Aug 2007 (permalink)
  2. I've fixed this now.

    Thanks very much for pointing it out and best of luck with your prep.

    joel on 16 Aug 2007 (permalink)
  3. In Your example 3 - How to you come to the lowest common denominator of 24 ?

    hrishikeshgmat on 2 Dec 2008 (permalink)
  4. Generally I take the larger of the two numbers i.e. 12.

    Is the smaller number a factor 12? No, 8 is not a factor of 12.

    OK, multiply the larger number by 2 giving us 24.

    Is the smaller number a factor of 24? Yes, 8 is a factor of 12 and so 24 is the lowest common denominator of 8 and 12.

    In the case where multiplying by two does not give you the lowest common denominator then you can keep going i.e. multiply by 3 and check, multiply by 4 and check, etc.

    joel on 3 Dec 2008 (permalink)
  5. In adding fractions with with problem 1/8 + 5/12,you give this example
    1/8=1/8x3/3
    5/12=5/12x2/2

    how did you arive with the 3/3
    and the 2/2
    I understand the process except for this part

    nts21001 on 18 May 2009 (permalink)
  6. I multiply by these numbers to make all the denominators the same (so that the fractions can be added)

    mathematical expression

    So multiplying a number by any of these doesn't actually change the value, and 2/2 and 3/3 were picked so that the denominators would all be 24.

    joel on 23 May 2009 (permalink)
  7. Thumbs up... this is the fastest method i have ever seen

    alola on 8 Dec 2009 (permalink)

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