The formula to calculate the velocity of the ball after $3 \mathrm{s}$ is given by

$v_3=u-3g ...\left(1\right)$

The formula to calculate the velocity of the ball after $5 \mathrm{s}$ is given by

$v_5=u-5g ...\left(2\right)$

Divide equation (1) by equation (2) to obtain the required ratio.

$\frac{v_3}{v_5}=\frac{u-3g}{u-5g} ...\left(3\right)$

Substitute the values of the known parameters into equation (3) to calculate the required ratio.

$\begin{array}{rcl}\frac{v_3}{v_5}& =& \frac{150-30}{150-50}\\ & =& \frac{120}{100}\\ & =& \frac{6}{5}\\ & =& \frac{5+1}{5}\end{array}$

Comparing the above equation with the given expression, it can be written that $x=5$.