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1. What is the speed of a train if it overtakes two persons who are walking in the same direction at the rate of a m/s and (a + 1) m/s and passes them completely in b seconds and (b + 1) seconds respectively?

A. (a + b) m/s

B. (a + b + 1) m/s

C. (2a + 1) m/s

D. $\frac{2 a + 1}{2}$ m/s

Solution:

\begin{aligned} Let the length of the train be x metres \\ and its speed be y m/s \\ Then, \\ \frac{x}{y - a} =  b and \\ \frac{x}{y - (a + 1)} = (b + 1) \\ \Leftrightarrow x = b (y - a) and \\ x = (b + 1) (y - a - 1) \\ \Leftrightarrow b (y - a) = (b + 1) (y - a - 1) \\ \Leftrightarrow b y - b a = b y - b a - b + y - a - 1 \\ \Leftrightarrow y = (a + b + 1) \end{aligned}

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