QUESTION

# A train can travel 50% faster than a car. Both start from point A at the same time and reach point B 75kms away from point A at the same time. On the way however, the train lost about 12.5 minutes while stopping at the stations. The speed of the car isa)100kmphb)110kmphc)120kmphd)130kmph

Hint: We need to use the speed distance formula to solve this question. We can take the speed of the car as a variable and use the information given in the question to obtain the corresponding equations and then solve for this variable.

Let the speed of the car be ${{v}_{c}}$. As it is given in the question that the train can travel 50% faster than the car, the speed of the train ${{v}_{t}}$ would be given by
${{v}_{t}}={{v}_{c}}+\dfrac{50}{100}{{v}_{c}}=\dfrac{3}{2}{{v}_{c}}..............(1.1)$
In the question, it is given that the distance travelled by both the train and the car from point A to point B = 75km.
We use the time-speed relation which states that
$\text{time taken=}\dfrac{\text{distance}}{\text{speed}}$
Thus, time taken by the car to reach point B is
${{t}_{c}}=\dfrac{75}{{{v}_{c}}}...................(1.2)$
However, the train loses 12.5 minutes while stopping at the stations, we know that
$\text{time in hours=}\dfrac{\text{time in minutes}}{60}$
Therefore, time lost by the train is
$\text{time lost by train=}\dfrac{12.5}{60}hours............(1.3)$
Thus, time taken by the train to reach point B (by using equation 1.1 and 1.3) is
${{t}_{t}}=\text{time lost by train+}\dfrac{75}{{{v}_{t}}}=\dfrac{12.5}{60}\text{+}\dfrac{75}{\dfrac{3}{2}{{v}_{c}}}...................(1.4)$
As, it is given that the time taken by the train and the car are same, we can equate (1.2) and (1.4) to obtain
\begin{align} & \dfrac{75}{{{v}_{c}}}=\dfrac{12.5}{60}\text{+}\dfrac{75}{\dfrac{3}{2}{{v}_{c}}} \\ & \Rightarrow \dfrac{75}{{{v}_{c}}}-\dfrac{2\times 75}{3{{v}_{c}}}=\dfrac{12.5}{60}\Rightarrow \dfrac{75}{3{{v}_{c}}}=\dfrac{12.5}{60} \\ & \Rightarrow {{v}_{c}}=\dfrac{75\times 60}{3\times 12.5}=120km/hr \\ \end{align}
Thus, the velocity of the car is 120kmph which matches option (c). Hence (c) is the correct option.

Note: We should be careful to convert the time lost by the train into hours before solving the question as all the terms in a valid equation should be in the same units.