Train X leaves New York at 1 am and travels east at a constant speed of x mph. If train Z leaves New York at 2 am and travels east, at what constant rate of speed will train Z have to travel in order to catch train X at exactly 5.30 am ?

Answer: Option D

Let the speed of train Z be z mph
Distance travelled by train X in 1 hr = x miles
Relative speed of train Z w.r.t. train X = (z - x) mph
To catch train X at 5.30 am, train Z will have to cover x miles at relative speed in 3 hr 30 min, i.e., $$\frac{7}{2}$$ hrs
$$\begin{array}{rl}& \therefore \left(z-x\right)\times \frac{7}{2}=x\\ & \Rightarrow \frac{7}{2}z=\frac{9}{2}x\\ & \Rightarrow z=\left(\frac{9}{2}\times \frac{2}{7}\right)x\\ & \Rightarrow z=\frac{9}{7}x\end{array}$$