SIMPLIFYING COMPLEX FRACTIONS WITH VARIABLES

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

Simplify the following :

Example 1 :

(1 + 1/x) /(1 - 1/x²)

Solution :

(1) / (2)

= [(x + 1)/x] / [(x + 1) (x - 1)/x²]

= [(x + 1)/x] ⋅ [x²/(x + 1) (x - 1)]

= x/(x - 1)

Example 2 :

[4 + 12/(2x - 3)] / [5 + 15/(2x - 3)]

Solution :

Simplifying the numerator :

= [4 + 12/(2x - 3)]

= [4(2x - 3) + 12] / (2x - 3)

= (8x - 12 + 12) / (2x - 3)

= 8x / (2x - 3)  -------(1)

Simplifying the denominator :

= [5 + 15/(2x - 3)]

= [5(2x - 3) + 15] / (2x - 3)

= (10x - 15 + 15) / (2x - 3)

= 10x / (2x - 3)  -------(2)

(1) / (2)

= 8x / (2x - 3) / 10x / (2x - 3)

= [8x / (2x - 3)]  ⋅ [(2x - 3)/10x]

= 4/5

Example 3 :

[(2/x) - 5/(x + 3)] / [(3/x) + 3/(x+3)]

Solution :

Simplifying the numerator :

= [(2/x) - 5/(x + 3)]

Least common multiple of x and (x + 3) is x (x + 3)

= [2(x + 3) - 5x] / x(x + 3)

= (2x + 6 - 5x) / x(x + 3)

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

Simplifying the denominator :

= [(3/x) + 3/(x+3)]

= [3(x + 3) + 3x] / x(x + 3)

= (3x + 9 + 3x) / x(x + 3)

= (6x + 9) / x(x + 3)

= 3(2x + 3) / x(x + 3) ------(2)

(1) / (2)

= [(-3x + 6) / x(x + 3)] / [3(2x + 3) / x(x + 3)]

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

= 3(2 - x) / 3(2x + 3)

= (2 - x) / (2x + 3)

Example 4 :

[x/(x + 2) - x/(x - 2)] / [x/(x + 2) + x/(x - 2)]

Solution :

Simplifying the numerator :

= [x/(x + 2) - x/(x - 2)]

Taking the least common multiple, we get

= [x(x - 2) - x(x + 2)] / (x + 2) (x - 2)

= (x² - 2x - x² - 2x) / (x + 2) (x - 2)

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

Simplifying the denominator :

=  [x/(x + 2) + x/(x - 2)]

= [x (x - 2) + x (x + 2)] / (x + 2) (x - 2)

= (x² - 2x + x² + 2x) / (x + 2) (x - 2)

= 2x² / (x + 2) (x - 2) -----(2)

(1) / (2)

= -4x / 2x²

= -2/x

Example 5 :

[2a - (1/8a)] / [4 + (1/a)]

Solution :

= [2a - (1/8a)] / [4 + (1/a)]

Simplifying the numerator :

= 2a - (1/8a)

= (16a² - 1)/8a

Expanding the numerator, we get

= ((4a)² - 1)/8a

= (4a + 1)(4a - 1) / 8a

Simplifying the denominator :

= 4 + (1/a)

= (4a + 1)

Dividing the numerator and denominator, we get

= [(4a + 1)(4a - 1) / 8a] / (4a + 1)

= (4a + 1)(4a - 1) / 8a(4a + 1)

= (4a - 1) / 8a

Example 6 :

[4/x² - 3/x] / [1/x² + 2/3x]

Solution :

= [4/x² - 3/x] / [1/x² + 2/3x]

Simplifying the numerator :

= [4/x² - 3/x]

= (4 - 3x)/x²

Simplifying the denominator :

= [1/x² + 2/3x]

= (3 + 2x)/3x²

Dividing the numerator and denominator, we get

= [(4 - 3x)/x² ] / [(3 + 2x)/3x²]

= [(4 - 3x)/x² ] ⋅ 3x²/ (3 + 2x)

= 3(4 - 3x) / (3 + 2x)

Example 7 :

[5 - 1/a] / [1/a² - 25]

Solution :

= [5 - 1/a] / [(1/a²) - 25]

Simplifying the numerator :

= [5 - 1/a]

= (5a - 1) / a

Simplifying the denominator :

=  [(1/a²) - 25]

= (1 - 25a²) / a²

= (1 - a²a²) / a²

= 1² - (5a)² / a²

= (1 + 5a)(1 - 5a) / a²

Dividing the numerator by the denominator, we get

= [(5a - 1) / a] / [(1 + 5a)(1 - 5a) / a²]

= [(5a - 1) / a]  ⋅ [a² / (1 + 5a) (1 - 5a)]

= [-(5a - 1) / a]  ⋅ [a² / (5a + 1)(5a - 1)

= - a / (5a + 1)

Example 8 :

x - [(x + 2) / 3]

Solution :

= x - [(x + 2) / 3]

= [3x - (x + 2)]/3

= (3x - x - 2)/3

= (2x - 2)/3

Example 9 :

2x/3 - [(2x + 1) / 6]

Solution :

= 2x/3 - [(2x + 1) / 6]

= (2x/3) ⋅ (2/2) - [(2x + 1) / 6]

= (4x/6) - [(2x + 1) / 6]

= [4x - (2x +1)]/6

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

= (2x - 1)/6

Example 10 :

(x + 6)/2 - [(1 - 3x) / 7]

Solution :

= (x + 6)/2 - [(1 - 3x) / 7]

= [7(x + 6) - 2(1 - 3x)] / 14

= (7x + 42 - 2 - 6x) / 14

= (x + 40) / 14