Question

The numerator of a fraction is 3 less than the denominator. If 2 is added to both the numerator and
thedenominator, then the sum of the new fraction and the original fraction is find the original
fraction.​

Answer 1

Correct Question:

The numerator of a fraction is 3 less than its denominator. If 2 is added to both the numerator and denominator, then the sum of new fraction and the original fraction is 29/20 . Find the original fraction.

Answer:-

Let the fraction be x/y.

Given:

Numerator is 3 less than the denominator.

⟶ Numerator = denominator - 3

⟶ x = y - 3 -- equation (1)

And,

If 2 is added to both numerator and denominator the sum of the original fraction and new fraction is 29/20.

$\longrightarrow \sf \: \frac{x + 2}{y + 2} + \frac{x}{y} = \frac{29}{20} \\ \\ \longrightarrow \sf \: \frac{y(x + 2) + x(y + 2)}{y(y + 2)} = \frac{29}{20} \\ \\ \longrightarrow \sf \: \frac{xy + 2y + xy + 2x}{ {y}^{2} + 2y} = \frac{29}{20}$

Substitute the value of x from equation (1).

$\: \longrightarrow \sf \: \frac{(y - 3)y + 2y + (y - 3)y + 2(y - 3)}{ {y}^{2} + 2y} = \frac{29}{20} \\ \\ \longrightarrow \sf \: \frac{ {y}^{2} - 3y + 2y + {y}^{2} - 3y + 2y - 6 }{ {y}^{2} + 2y } = \frac{29}{20} \\ \\ \longrightarrow \sf \: \frac{2 {y}^{2} - 2y - 6}{ {y}^{2} + 2y} = \frac{29}{20} \\ \\ \longrightarrow \sf \:20(2y ^{2} - 2y - 6) = 29( {y}^{2} + 2y) \\ \\ \longrightarrow \sf \:40 {y}^{2} - 40y - 120 - 29 {y}^{2} - 58y = 0 \\ \\ \longrightarrow \sf \:11 {y}^{2} - 90y -120= 0 \\ \\ \longrightarrow \sf \:11y ^{2} - 110y + 12y - 120 = 0 \\ \\ \longrightarrow \sf \:11y(y - 10) + 12(y - 10) = 0 \\ \\ \longrightarrow \sf \:(11y + 12)(y - 10) = 0$

✯ 11y + 12 = 0

⟶ 11y = - 12

⟶ y = - 12/11

✯ y - 10 = 0

⟶ y = 10

Denominator of a fraction cannot be negative So positive value is taken.

Substitute y = 10 in equation (1)

⟶ x = 10 - 3

⟶ x = 7

∴ The required fraction x/y = 7/10.

Answer 2

Correct Question

The numerator of a fraction is 3 less than its denominator. If 2 is added to both the numerator and denominator, then the sum of new fraction and the original fraction is 29/20 . Find the original fraction.