Question

Find a fraction which reduces to $\frac{2}{3}$ if the numerator and the denominator are each increased by 1, and reduces to $\frac{3}{5}$ if the numerator and the denominator are each decreased by 2.

• A. $\frac{13}{12}$
• B. $\frac{11}{17}$
• C. $\frac{9}{4}$
• D. $\frac{13}{3}$

Answer 1

The correct option is B $\frac{11}{17}$

Let the fraction be $\frac{x}{y}.$
Given, if the numerator and the denominator are each increased by $1$, then fraction $=\frac{x+1}{y+1}=\frac{2}{3}$
$⟹3x+3=2y+2$
$⟹3x-2y=-1$ ....................... $\left(1\right)$
Also, if the numerator and the denominator are each decreased by $2$, then fraction $=\frac{x-2}{y-2}=\frac{3}{5}$
$⟹5x-10=3y-6$
$⟹5x-3y=4$ ....................... $\left(2\right)$
Multiplying equation $\left(1\right)$ with $5$ we get, $15x-10y=-5$ ...$\left(3\right)$

Multiplying equation $\left(2\right)$ with $3$ we get, $15x-9y=12$ ....$\left(4\right)$

Subtracting equations $\left(3\right)$ from $\left(4\right)$, we get $y=17$

Substituting $y=17$ in the equation $\left(2\right)$, we get $5x-3\left(17\right)=4$

$⟹x=11$
Hence, the fraction is $\frac{11}{17}.$