Operations and calculations

3. Subtraction

3.5. Subtracting fractions

There are three steps to follow when subtracting fractions:

  1. Ensure the fractions have a common denominator (the numbers below the fraction bars must be the same).
  2. Subtract 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.


3/8 - 1/8 (three eighths minus one eighth)
  1. The denominators are the same (8), so we can go ahead and subtract the numerators.
  2. 3 - 1 = 2, which placed over the denominator gives 2/8.
  3. This can be simplified by dividing both the numerator and the denominator by 2, giving the answer 1/4.
3/8 - 1/8 is the same as 3-1/8 which equals 2/8 and simplifies to 1/4

1/3 - 1/6 (one third minus one sixth)

  1. The denominators are not the same, so we need to use the least common denominator (the smallest multiple they have in common).

    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.

    1 x 2 = 2

    3 x 2 = 6

    This gives us an adjusted fraction of 2/6

  2. 2/6 - 1/6 = 1/6
  3. This fraction cannot be simplified so the answer is 1/6.
2/6 - 1/6 is the same as 2-1/6 which is equal to 1/6

3/4 - 2/5 (three quarters minus two fifths): Method 1

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

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

    4 x 5 = 20

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

    3 x 5 = 15

    2 x 4 = 8

    Placing these over the common denominator gives us adjusted fractions of 15/20 and 8/20.

  2. 15/20 - 8/20 = 7/20
  3. This fraction cannot be simplified, so the answer is 7/20.
15/20 - 8/20 is the same as 15 - 8/20 which equals 7/20

3/4 - 2/5 (three quarters minus two fifths): Method 2

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

    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 4 and 5.

    Multiples of 4: 1 x 4 = 4, 2 x 4 = 8, 3 x 4 = 12, 4 x 4 = 16, 5 x 4 = 20

    Multiples of 5: 1 x 5 = 5, 2 x 5 = 10, 3 x 5 = 15, 4 x 5 = 20

    The least common denominator is 20.

    Multiply the numerator and denominator of the first fraction (3/4) by 5:

    3 x 5 = 15

    4 x 5 = 20

    This gives us an adjusted fraction of 15/20.

    Multiply the numerator and denominator of the second fraction (2/5) by 4:

    2 x 4 = 8

    5 x 4 = 20

    This gives us an adjusted fraction of 8/20.

  2. 15/20 - 8/20 = 7/20
  3. This fraction cannot be simplified, so the answer is 7/20.
15/20 - 8/20 is the same as 15 - 8/20 which equals 7/20