Question

In a given fraction, if 1 is subtracted from the numerator and 2 is added to the denominator, it becomes $\frac{1}{2}$. If 7 is subtracted from the numerator and 2 is subtracted from the denominator, it becomes $\frac{1}{3}$. The fraction is
(a) $\frac{13}{24}$

(b) $\frac{15}{26}$

(c) $\frac{16}{27}$

(d) $\frac{16}{21}$

Answer 1

Let the fraction be $\frac{x}{y}$
It is given that $\frac{x-1}{y+2}=\frac{1}{2}$
$$\begin{gathered} \Rightarrow 2x-2=y+2 \\ \Rightarrow 2x-y=4.....\left(\mathrm{i}\right) \end{gathered}$$
Also, $\frac{x-7}{y-2}=\frac{1}{3}$
$$\begin{gathered} \Rightarrow 3x-21=y-2 \\ \Rightarrow 3x-y=19.....\left(\mathrm{ii}\right) \end{gathered}$$
Subtract (ii) from (i), we get
$x=15$
Put the value of x in equation (i), we get;
$$\begin{gathered} 2\left(15\right)-y=4 \\ 30-y=4 \\ \Rightarrow y=30-4=26 \end{gathered}$$
Therefore, the fraction is $\frac{x}{y}=\frac{15}{26}$
Hence, the correct answer is option (b)