Question

The sum of the numerator and denominator of a fraction is equal to $7$. Four times the numerator is $8$ less than $5$ times the denominator. Find the fraction

Answer 1

Let the fraction be $\frac{x}{y}$. Then the first condition is $x+y=7$. The second condition translates to $4x=5y-8$. Thus we get two equations
$x+y=7,....\left(1\right)$
$4x-5y=-\mathrm{8....}\left(2\right)$
Multiply $\left(1\right)$ by $4$ and $\left(2\right)$ by $-1$, we get
$4x+4y=28$
$-4x+5y=8$.
Adding these two, we obtain $9y=36$. Hence $y=4$. Now $x+y=7$ gives $x=7-y=7-4=3$.
The required fraction is $\frac{3}{4}$.