Question

when 2 is added to the denominator of a fraction it changed to 1/4 and if l is added to the numerator it changed to 2/5. then find the fraction..?​

Answer 1

★ Solution :-

Let the value of the numerator be x.

Let the value of the denominator be y.

So, the fraction is

$\sf \leadsto \dfrac{x}{y}$

According to the first case,

$\sf \leadsto \dfrac{x}{y + 2} = \dfrac{1}{4}$

According to the second case,

$\sf \leadsto \dfrac{x + 1}{y} = \dfrac{2}{5}$

By first case,

$\sf \leadsto \dfrac{x}{y + 2} = \dfrac{1}{4}$

$\sf \leadsto 4(x) = 1(y + 2)$

$\sf \leadsto 4x = 1y + 2$

$\sf \leadsto 4x - 1y = 2 \: \: --- (i)$

By second case,

$\sf \leadsto \dfrac{x + 1}{y} = \dfrac{2}{5}$

$\sf \leadsto 5(x + 1) = 2(y)$

$\sf \leadsto 5x + 5 = 2y$

$\sf \leadsto 5x - 2y = -5 \: \: --- (ii)$

Now, by first equation

$\sf \leadsto 4x - 1y = 2$

$\sf \leadsto 4x = 2 + 1y$

$\sf \leadsto x = \dfrac{2 + 1y}{4}$

Now, let's find the value of y by second equation.

$\sf \leadsto 5x - 2y = -5$

$\sf \leadsto 5 \bigg( \dfrac{2 + 1y}{4} \bigg) - 2y = -5$

$\sf \leadsto \dfrac{10 + 5y}{4} - 2y = -5$

$\sf \leadsto \dfrac{10 + 5y - 8y}{4} = -5$

$\sf \leadsto \dfrac{10 - 3y}{4} = -5$

$\sf \leadsto 10 - 3y = -5 \times 4$

$\sf \leadsto 10 - 3y = -20$

$\sf \leadsto -3y = -20 - 10$

$\sf \leadsto -3y = -30$

$\sf \leadsto y = \dfrac{-30}{-3}$

$\sf \leadsto y = 10$

Now, let's find the value of y by first equation.

$\sf \leadsto 4x - 1y = 2$

$\sf \leadsto 4x - 1(10) = 2$

$\sf \leadsto 4x - 10 = 2$

$\sf \leadsto 4x = 2 + 10$

$\sf \leadsto 4x = 12$

$\sf \leadsto x = \dfrac{12}{4}$

$\sf \leadsto x = 3$

Now, we can observe that,

Value of x :- 3

Value of y :- 10

Therefore, the fraction is $\sf \dfrac{3}{10}$.