Question

Find the fraction which becomes to $\frac{2}{3}$ when the numerator is increasing by $2$ and equal to $\frac{4}{7}$ when the denominator is increased by $4$

Answer 1

Consider that the fraction is $\frac{x}{y}$.

According to the question,

$\frac{x+2}{y}=\frac{2}{3}$

$3(x+2)=2y$

$3x-2y=-6$ .............. (1)

And,

$\frac{x}{y+4}=\frac{4}{7}$

$7x=4(y+4)$

$7x-4y=16$ ............. (2)

From equation (1) and (2),
$2\times \left(1\right)-\left(2\right)⟹$

$6x-4y=-12$

$7x-4y=16$

$..............$

$-x=-28$

$⟹x=28$

On substituting $x=28$ in $\left(1\right)$, we get
$3\left(28\right)-2y=-6$

$2y=90$

$y=45$

Therefore, the fraction becomes $\frac{28}{45}$.