Two trains start at the same time, P from A to B and Q from B to A. If they arrive at B and A, respectively, $2\frac{1}{2}$ hours and $10$ hours after they passed each other, and the speed of P is $90\text{km/hr}$, then the speed of Q in $\text{km/hr}$ is?

Let the speed of trains P and Q be $S_1$ and $S_2$ respectively, and the times taken by them to reach their destinations after meeting each other be $t_1$ and $t_2$.We have the standard formula for objects moving towards each other and continuing to destinations after crossing:
$$\frac{S_1}{S_2}=\sqrt{\frac{t_2}{t_1}}$$
Given data:$S_1=90\text{km/hr}$$t_1=2\frac{1}{2}\text{ hours}=\frac{5}{2}\text{ hours}$$t_2=10\text{ hours}$ Substituting these values into the formula:
$$\frac{90}{S_2}=\sqrt{\frac{10}{\frac{5}{2}}}$$
$$\frac{90}{S_2}=\sqrt{\frac{10\times 2}{5}}=\sqrt{\frac{20}{5}}=\sqrt{4}=2$$
$$2\times S_2=90⟹S_2=\frac{90}{2}=45\text{km/hr}$$