Question

The denominator of a fraction is $4$ more than twice the numerator. When both the numerator and denominator are decreased by $6$, then the denominator becomes $12$ times the numerator. Determine the fraction.

Answer 1

Let the numerator and denominator of the fraction be x and y respectively. Then,
Fraction $=\frac{x}{y}$

It is given that
Denominator $=2$ (Numerator) $+4$
$\Rightarrow y=2x+4$
$\Rightarrow 2x-y+4=0$

According to the given condition, we have
$y-6=12(x-6)$
$\Rightarrow y-6=12x-72$
$\Rightarrow 12x-y-66=0$

Thus, we have the following system of equations
$2x-y+4=0$ .(i)
$12x-y-66=0$ ..(ii)

Subtracting equation (i) from equation (ii), we get
$10x-70=0$
$\Rightarrow x=7$

Putting $x=7$ in equation (i), we get
$14-y+4=0$
$\Rightarrow y=18$
Hence, required fraction $=\frac{7}{18}$.