<s>[INST] If the numerator of a fraction is increased by $2$ and the denominator by $1$, it becomes $5/8$.... If the numerator of a fraction is increased by $2$ and the denominator by $1$, it becomes $\displaystyle \frac{5}{8}$ and if the numerator and the denominator of the same fraction are each increased by $1$, the fraction becomes equal to $\displaystyle \frac{1}{2}$. Find the fraction.
I tried,
Let the numerator of the fraction be $x$,
Let the denominator of the fraction be $y$
Therefore, Original fraction is $\displaystyle \frac{x}{y}$.
$$
\left\{
\begin{eqnarray*}
\frac{x+2}{y+1}=\frac{5}{8} \\[2mm]
\frac{x+1}{y+1}=\frac{1}{2} \\
\end{eqnarray*}
\right.
$$
What should I do now?
 [/INST] $\dfrac{x+2}{y+1}=\dfrac{5}{8}$ and $\dfrac{x+1}{y+1}=\dfrac{1}{2}=\dfrac{4}{8}$
Dividing first equation by second equation you would get
$$\dfrac{x+2}{x+1}=\dfrac{5}{4}=\dfrac{3+2}{3+1}$$
This should be easy now..
</s>