Operations and calculations

2. Addition

2.2. Adding Fractions

There are three steps to follow when adding fractions:

  1. Ensure the fractions have a common denominator (the numbers below the fraction bars must be the same).
  2. Add the numerators (the numbers above the fraction bars), placing the total over the denominator.
  3. Simplify the resulting fraction if possible.

If the fractions do not have a common denominator, they must first be adjusted, using multiplication or division, to use the least common denominator (the smallest multiple they have in common).

Note: To maintain the value of a fraction, you must use the same operation to adjust both the numerator (top number) and the denominator (bottom number).

Multiplication and division processes are covered in separate sections of this book.

Let’s work through some examples.


1/8 + 2/8 (one eighth plus two eighths)
  1. The denominators are the same (8), so we can go ahead and add the numerators.
  2. 1 + 2 = 3, which placed over the denominator gives 3/8.
  3. This cannot be simplified, so the answer is 3/8.
1/8 plus 2/8 is the same as 1+2/8, which equals 3/8

2/3 + 1/6 (two thirds plus one sixth)

  1. The denominators are not the same, so we need to find and use the least common denominator.

    If we multiply the denominator (3) on the first fraction by 2, this will make it 6, which is the same as the denominator on the second fraction.

    We'll go ahead and do that, remembering we also have to multiply the numerator by the same amount.

    2 x 2 = 4

    3 x 2 = 6

    This gives us an adjusted fraction of 4/6.

  2. 4/6 + 1/6 = 5/6
  3. This cannot be simplified so the answer is 5/6.
4/6 plus 1/6 is the same as 4+1/6 which is equal to 5/6

2/6 + 4/7 (two sixths plus four sevenths): Method 1

  1. The denominators are not the same, so we need to find the least common denominator, i.e. the smallest multiple they all have in common.

    In this method, we multiply the denominators to find the common denominator:

    6 x 7 = 42

    Then we multiply the numerator of each fraction by the original denominator of the other.

    2 x 7 = 14

    4 x 6 = 24

    Placing these over the common denominator gives us adjusted fractions of 14/42 and 24/42.

  2. 14/42 + 24/42 = 38/42
  3. This fraction can simplified to 19/21 by dividing both the numerator (38) and the denominator (42) by 2.
14/42 plus 24/42 is the same as 14+24/42 which equals 38/42 and simplified to 19/21

2/6 + 4/7 (two sixths plus four sevenths): Method 2

  1. The denominators are not the same, so we need to find the least common denominator, i.e. the smallest multiple they all have in common.

    In this method, we list the multiples of each of the denominators until we find the first one they have in common.

    The denominators are 6 and 7.

    Multiples of 6: 1 x 6 = 6, 2 x 6 = 12, 3 x 6 = 18, 4 x 6 = 24, 5 x 6 = 30, 6 x 6 = 36, 7 x 6 = 42

    Multiples of 7: 1 x 7 = 7, 2 x 7 = 14, 3 x 7 = 21, 4 x 7 = 28, 5 x 7 = 35, 6 x 7 = 42

    The least common denominator is 42.

    Multiply the numerator and denominator of the first fraction (2/6) by 7:

    2 x 7 = 14

    6 x 7 = 42

    This produces the adjusted fraction 14/42.

    Multiply the numerator and denominator of the second fraction (4/7) by 6:

    4 x 6 = 24

    7 x 6 = 42

    This produces the adjusted fraction 24/42.

  2. 14/42 + 24/42 = 38/42
  3. This fraction can simplified to 19/21 by dividing both the numerator (38) and the denominator (42) by 2.
14/42 plus 24/42 is the same as 14+24/42 which equals 38/42 and simplified to 19/21