SIMPLIFYING COMPLEX FRACTIONS

Simplify the fractions in the numerator and the denominator separately, then divide the two fractions and simplify.

Simplify the following :

Example 1 :

(2 - 1/4) / (2 + 1/4)

Solution :

In the numerator and denominator, we have fractions.

(1) / (2)

= (7/4) / (9/4)

Since we change the division sign as multiplication sign, we can write the reciprocal of the denominator.

= (7/4) ⋅ (4/9)

= 7/9

Example 2 :

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

Solution :

(1) / (2)

= (7/10) / (1/4)

= (7/10) ⋅ (4/1)

Simplifying 10 and 4, we get

= 14/5

= 2  4/5

Example 3 :

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

Solution :

Simplifying the numerator :

= 1 + 1/4 - 1/3

LCM of 4 and 3 is 12.

= (12 + 3 - 4) / 12

= 11/12 ---(1)

Simplifying the denominator :

= (1 - 1/2 + 1/5)

LCM of 2 and 5 is 10.

= (10 - 5 + 2) / 10

= 7/10 ---(2)

(1) / (2)

= (11/12) / (7/10)

= (11/12) ⋅ (10/7)

= (11 ⋅ 5) / (6 ⋅ 7)

=  55/42

= 1  13/42

Example 4 :

(1/3 + 1/4) / (1 - 1/5)

Solution :

(1) / (2)

= (7/10) / (4/5)

= (7/10) x (5/4)

= 7/8

Example 5 :

(22 + 11 ÷ 2) / (23 - 3 x 4)

Solution :

= (22 + 11 ÷ 2) / (23 - 3 x 4)

Using order of operations, let us simplify the numerator and denominators separately.

(1) / (2)

= (55/2) / 11

=  5/2

= 2  1/2

Example 6 :

(15 - 33) / (17 - 7 x 3)

Solution :

Simplifying the numerator :

= 15 - 33

= -18 -----(1)

Simplifying the denominator :

= (17 - 7 x 3)

Performing multiplication, we get

= 17 - 21

= 4 -----(2)

(1) / (2)

= -18 / 4

= -9/2

= -4  1/2

Example 7 :

(15 + 3 x 5²⁾ / (11 - 25 ÷ 2)

Solution :

(15 + 3 x 5²) / (11 - 25 ÷ 2)

(1) / (2)

= 90 / (5/2)

= 90 (2/5)

= 18(2)

= 36

Example 8 :

(-4 - 11) / (12 - 9 ÷ 2)

Solution :

(1) / (2)

= -15 / (15/2)

= -15 (2/15)

= -2

Example 9 :

(1 - 3/4) / (2 + 1/4)

Solution :

Simplifying the numerator :

= 1 - 3/4

= (4 - 3) / 4

= 1/4------(1)

Simplifying the denominator :

= (2 + 1/4)

= (8 + 1) / 4

= 9/4 ------(2)

(1) / (2) 

= (1/4) / (9/4)

= (1/4) x (4/9)

= 1/9

Example 10 :

(1/2 + 1/3 - 1/6) / (1/12 - 1/4)

Solution :

(1) / (2)

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

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

= -4