A ball is thrown vertically upward with an initial velocity of 150m...

The formula to calculate the velocity of the ball after $3s$ is given by

${v}_{3} = u - 3g . . . \left(1\right)$

The formula to calculate the velocity of the ball after $5s$ is given by

${v}_{5} = u - 5g . . . \left(2\right)$

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

$\dfrac{{v}_{3}}{{v}_{5}} = \dfrac{u - 3g}{u - 5g} . . . \left(3\right)$

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

$\begin{cases}\dfrac{{v}_{3}}{{v}_{5}} & = & \dfrac{150 - 30}{150 - 50} \\ & = & \dfrac{120}{100} \\ & = & \dfrac{6}{5} \\ & = & \dfrac{5 + 1}{5}\end{cases}$

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