A train crosses a pole in 15 seconds, while it crosses a 100m long platform in 25 seconds. The length of the train is
(a) 125m
(b) 135m
(c) 150m
(d) 175m

Hint: Let us assume length of train be L, then speed of train while crossing pole will be $\dfrac{L}{15}$, as $speed=\dfrac{dis\tan ce}{time}$. Again speed of train while passing the platform will be \[\dfrac{\left( 100+L \right)}{25}\], as the corresponding distance will be $\left( 100+L \right)$ that has to be covered in 25 seconds. We will equate these two equations to find the length of train = L, therefore, $\dfrac{L}{15}=\dfrac{\left( 100+L \right)}{25}$.

Complete step-by-step solution -
It is given in the question that the train passes a pole in 15 seconds, and the same train when it passes the platform of 100m long it takes 25 seconds, then we have to find the length of the train. Let us assume that the length of the train is L. We know the formula that $speed=\dfrac{dis\tan ce}{time}$.
When train cross pole in 15 seconds, it means the total distance travelled by train is its actual length which is L, thus speed of train while crossing pole will be = $\dfrac{\text{Distance}\ \text{travelled}}{\text{Time}\ \text{taken}}$, here distance covered = L m, and time taken = 15 seconds. Thus $speed=\dfrac{L}{15}m/s...(i)$.
Now, the train crosses the platform of 100m in 25 seconds which means the train will have to cover a distance of $\left( 100+L \right)$m to completely pass through the platform. So, speed of train while passing through platform will be = $\dfrac{\text{Distance}\ \text{Travelled}}{\text{Time}\ \text{Taken}}$= \[\dfrac{\left( 100+L \right)}{25}.....(ii)\].

Since equation (i) amd (ii) both are giving the speed of train, so, on equating equation 1 and equation 2, we get $\dfrac{L}{15}=\dfrac{\left( 100+L \right)}{25}$, on cross multiplying the obtained equation we get, $25L=15\left( 100+L \right)$ solving further, we get, $25L=1500+15L$, simplifying further, $25L-15L=1500$, therefore, $10L=1500$ and finally $L=150$m. Therefore the length of the train will be L= 150m and option c) is the correct answer.

Note: Generally students misunderstand that the length of train is equal to the length of the platform which train passes in 25 seconds. But actually the length of the platform is less than the length of the train. If we consider length of train equal to length of platform then speed of train comes out 4m/s but we know that the speed of train is 10m/s and if we consider speed of train as 4m/s then length of train will be 60m and we know that length of train is 150m thus it contradict our assumption.