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Addition of Fractions

Addition of fractions involves first and foremost, checking if the denominators of the given fractions are the same or different.

When you are given two fractions, you can add them together provided they are split in the same equal parts, i.e. the denominator of the two fractions should be the same.

If the denominators are different, then we have to get them to a common denominator before adding the two fractions.

We discuss both these scenarios of addition of fractions below:

 

Addition of fractions with like denominators

When we have to add fractions with the same/common denominators,

  1. add the numerator parts of the two fractions, and write the same common denominator.
  2. if the fractions are in mixed numeral form, add the whole number parts, and then the fraction parts with the same denominator together.
  3. if the fractions are in improper form, change them to mixed numerals first, and follow the above step to add them.

 

Let us look at examples for all three scenarios.

  1. 1/5  +  2/5  =  3/5
  2.  16/5  +  3/5
    =  31/5  +  3/5
    = 3 + (1/5 + 3/5)
    =  34/5
    =  19/5
  3.  33/7 + 42/7
    =  (3 + 4) + (3/7 + 2/7)
    = 7 + ({3~+~2}/7)
    =  7 + 5/7  = 75/7

 

Addition of fractions with unlike denominators

When two fractions with different denominators are to be added, we obey the following steps:

  1. make the two fractions equivalent with the same denominator i.e. express the fractions with the lowest common denominator (LCD)
  2. now add the two fractions with the common denominators

 

Let’s look at a few examples –

  1.  1/3  +  1/2Least common denominator of 3 and 2 = 3 x 2 = 6Now express both fractions with denominator of 6

    = ({1/3}~*~{2/2}) + ({1/2}~*~{2/2})
    2/6  +  3/6
    5/6

  2.  1/2  +  1/4LCD of 2 and 4 is 4, because 4 is a multiple of 2.= ({1/2}~*~{2/2}) + ({1/4}~*~{1/1})
    =  2/4  +  1/4
    =  3/4
  3.  5/3  +  2/6LCD of 3 and 6 is 6, because 6 is a multiple of 3.= ({5/3}~*~{2/2}) + 2/6
    =  10/6  +  2/6
    =  12/6  = 2
  4.  2/6  +  3/4LCD of 3 and 6 is 6 (since 6 = 3 x 2 and 4 = 2 x 2, so the common multiples =  3 x 2 x 2 = 12)= ({2/6}~*~{2/2}) + ({3/4}~*~{3/3})
    =  4/12  +  9/12
    =  13/12
  5.  12/5  +  21/3
    = (1 + 2) + (2/5  +  1/3)
    = 3 + (2/5  +  1/3)
    = 3 +  ({2/5}~*~{3/3}  +  {1/3}~*~{5/5})   (LCD of 3 and 5 = 15)
    = 3 + (6/15  +  5/15)
    = 3 + 11/15
    =  311/15
  6.  61/2  +  22/6
    = (6 + 2) + (1/2  +  2/6)
    = 8 +  ({1/2}~*~{3/3}  +  2/6)   (LCD of 2 and 6 = 6)
    = 8 + (3/6  +  2/6)
    = 8 + 5/6
    =  85/6