Question

The sum of the numerator and denominator of a fraction is $3$ less than twice the denominator. If the numerator and denominator are decreased by $1$, the numerator becomes half the denominator. Determine the fraction.

Answer 1

Let the fraction $=\frac{x}{y}$

First condition :

$\Rightarrow x+y=2y-3$

$\Rightarrow x-y=-3$

Second condition :

$\Rightarrow \frac{x-1}{y-1}=\frac{1}{2}$

$\Rightarrow 2x-2=y-1$

$\Rightarrow 2x-y=1$

Subtract two equations ;

$\Rightarrow 2x-x=1+3\Rightarrow x=4$

substitute in above equation

$\Rightarrow y=1+8\Rightarrow y=7$

$\therefore$ the fraction $=\frac{4}{7}$