Question

Numerator of a fraction is 3less than the denominator.if 4 is added to both the numerator and denominator, the value of fraction increases by 1/8. find the fraction​

Answer 1

Answer:-

Let the fraction be x/y.

Given:

Numerator is 3 less than the denominator.

⟹ Numerator = Denominator - 3

⟹ x = y - 3 -- equation (1)

And,

The fraction increases by 1/8 if 4 is added to both numerator and denominator.

According to the above condition,

$\implies \sf \: \frac{x + 4}{y + 4} = \frac{x}{y} + \frac{1}{8}$

Taking LCM in RHS we get,

$\implies \sf \: \frac{ y - 3 + 4}{y + 4} = \frac{8x + y}{8y} \\ \\ \implies \sf \: \frac{y + 1}{y + 4} = \frac{8(y - 3) + y}{8y} \\ \\ \implies \sf \: 8y(y + 1) = (y + 4)(8(y - 3) + y) \\ \\ \implies \sf \: 8 {y}^{2} + 8y = (y + 4)(8y - 24 + y) \\ \\ \implies \sf \: 8 {y}^{2} + 8y = y(9y - 24) + 4(9y - 24) \\ \\ \implies \sf \: 8 {y}^{2} + 8y = 9 {y}^{2} - 24y + 36y - 96 \\ \\ \implies \sf \: 0 = 9 {y}^{2} - 24y + 36y - 96 - 8 {y}^{2} - 8y \\ \\ \implies \sf \: 0 = {y}^{2} + 4y - 96 \\ \\ \implies \sf \: {y}^{2} + 12y - 8y - 96 = 0$

⟹ y (y + 12) - 8(y + 12) = 0

⟹ (y - 8) (y + 12) = 0

★ y - 8 = 0

y = 8

★ y + 12 = 0

y = - 12

Denominator of a fraction cannot be negative . So - 12 is neglected.

Substitute the value of y in equation (1)

⟹ x = 8 - 3

⟹ x = 5

Therefore, the required fraction x/y is 5/8.