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1. The sum of the numerator and denominator of a fraction is 11. If 1 is added to the numerator and 2 is subtracted from the denominator, it becomes $\frac{2}{3}$ . The fraction is :

A. $\frac{5}{6}$

B. $\frac{6}{5}$

C. $\frac{3}{8}$

D. $\frac{8}{3}$

Solution:

Let the fraction be

\frac{x}{y}

Then,

\begin{aligned} \Leftrightarrow x + y = 11..... (i) \\ \Leftrightarrow \frac{x + 1}{y - 2} = \frac{2}{3} \\ \Leftrightarrow 3 (x + 1) = 2 (y - 2) \\ \Leftrightarrow 3 x - 2 y = - 7..... (ii) \end{aligned}

Solving (i) and (ii), we get:
x = 3 and y = 8
So, the fraction is

\frac{3}{8}

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