when 2 is added to the denominator of a fraction it changed to 1/4 and

1.	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» ★ 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 $\sf \dfrac{3}{10}$ .