Subtraction of fractions

subtract fractionsTo subtract two or more fractions we must follow a procedure similar to that of addition; First we make sure that they all have the same denominator and if not, we have to find equivalent fractions that share a common denominator. Then we subtract their numerators.

In general, there are two types of subtraction of fractions:

  • Subtraction of fractions with the same denominator: the numerators are subtracted and the denominator is kept.
  • Subtraction of fractions with different denominators: first, the denominators are reduced to a common denominator and the numerators of the equivalent fractions obtained are subtracted.

General rules for subtracting fractions

If the fractions have the same denominator

  1. The given fractions are simplified if possible.
  2. The numerators of the fractions are subtracted and the common denominator is maintained.
  3. The resulting fraction is simplified, if possible.

Example: subtract the following fractions: 3/4; 7/4

Both fractions have the same denominator, therefore, we subtract the numerators and keep the common denominator:

3/4 - 7/4 = - 4/4

We simplify the fraction:

– 4/4 = – 1/1 = – 1

If the fractions have a different denominator

  1. The given fractions are simplified if possible.
  2. The fractions given are reduced to the lowest common denominator.
  3. The numerators of the resulting fractions are subtracted and the denominator obtained in step 2 is maintained.
  4. The resulting fraction is simplified, if possible.

Example: subtract the following fractions: 1/6; 2/3

Both fractions have different denominators, we must reduce them to the lowest common denominator. For this, we find the lcm of the denominators: lcm (6; 3) = 6. We divide the lcm between the denominators of the fractions: 6 ÷ 6 = 1 and 6 ÷ 3 = 2

The quotients obtained are multiplied by the respective numerators, that is:

1/6 - 2/3 = 1 (1) / 6 - 2 (2) / 6 = 1/6 - 4/6

We subtract the numerators

1/6 – 4/6 = – 3/6

Simplifying

– 3/6 = – 1/2

If the fractions are mixed

  1. Mixed fractions are converted to improper fractions.
  2. The given fractions are simplified if possible.
  3. The fractions given are reduced to the lowest common denominator, if they have a different denominator.
  4. The numerators of the resulting fractions are subtracted and the denominator obtained in step 3 is maintained.
  5. The resulting fraction is simplified, if possible.

Example: Subtract the following fractions: 1 2/5; 4 3/5

We convert fractions to improper fractions:

1 2/5 = (1 × 5 + 2) / 5 = 7/5

4 3/5 = (4 × 5 + 3) / 5 = 23/5

 Both fractions have the same denominator, therefore, we subtract the numerators and keep the common denominator:

7/5 – 23/5 = – 16/5

Exercises

Perform the following operations:

  1. [(a + 5b) / a2] - [(b - 3) / ab]

Let's see that fractions have a different denominator (and are monomials), therefore, we must reduce them to the lowest common denominator. The lcm of the denominators is: lcm (a2; ab) = a2b. We divide2b between each denominator and multiply the quotients by the respective numerator, thus:

[(a + 5b) / a2] - [(b - 3) / ab] = [b (a + 5b) / a2b] - [a (b - 3) / a2b] =

= [(ab + 5b2) / to2b] - [(ab - 3a) / a2b]

We subtract the numerators

[(ab + 5b2) / to2b] - [(ab - 3a) / a2b] = = [(ab + 5b2 - ab + 3a) / a2b] =

= (5b2 + 3a) / a2b

  1. [1 / (x-4)] - [1 / (x-3)]

Let's see that fractions have different denominators (and are compounds), therefore, we must reduce them to the lowest common denominator. The lcm of the denominators is: lcm = (x - 4) (x - 3). We divide (x - 4) (x - 3) between each denominator and multiplying the quotients by the respective numerator, we have:

[1 / (x-4)] - [1 / (x-3)] = {(x - 3) / [(x - 4) (x - 3)]} - {(x - 4) / [( x - 4) (x - 3)]} =

= (x - 3 - x + 4) / [(x - 4) (x - 3)] = 1 / [(x - 4) (x - 3)]

  1. (1 4/5) - (2 3/4)

We convert fractions to improper fractions:

1 4/5 = (1 × 5 + 4) / 5 = 9/5

2 3/4 = (2 × 4 + 3) / 4 = 11/4

Let's see that the fractions have a different denominator, therefore, we must reduce them to the lowest common denominator. The lcm of the denominators is: lcm = 20. We divide 20 between each denominator and multiplying the quotients by the respective numerator, we have:

5/9 - 4/11 = [4 (9) / 20] - [5 (11) / 20] = (36/20) - (55/20) = - 19/20