Question

A train overtakes 2 persons who are walking in the same direction in which the train is going, at the rate of 2 km/hr and 4 km/hr and passes them completely in 9 and 10 seconds respectively. The length of train is
(a) 45m
(b) 50m
(c) 54m
(d) 72m

Answer 1

Hint: Use the formula to calculate speed. Thus time is equal to distance by speed. Form 2 equations for both persons and solve these equations to get the length of the train.

Complete step-by-step answer:
.
It is said that the train overtakes 2 persons who are walking in the same direction.
The speed of \[{{1}^{st}}\] person = 2 km/hr \[=2\times \dfrac{5}{18}\] m/s \[=\dfrac{5}{9}\] m/sec.
We need to convert kmph to m /sec by multiplying speed with \[\dfrac{5}{18}\].
The speed of \[{{2}^{nd}}\] person = 4 km/hr \[=4\times \dfrac{5}{18}\] m/s \[=\dfrac{10}{9}\] m/sec.
Let us consider the length of the train as x meters and speed of train as y m/sec.
The time taken by train to overtake \[{{1}^{st}}\] person = 9 sec.
The time taken by train to overtake \[{{2}^{nd}}\] person = 10sec.
We know the formula that speed = distance / speed.
\[\therefore \] Time = distance / speed.
Let us consider the case of \[{{1}^{st}}\] person.
Time = distance / speed.
\[\dfrac{x}{y-\dfrac{5}{9}}=9-(1)\]
Similarly in the case of \[{{2}^{nd}}\] person,
\[\dfrac{x}{y-\dfrac{10}{9}}=10-(2)\]
Simplify equation (1) and equation (2) and find the value of x and y.
Equation (1) \[\Rightarrow x=9\left( y-\dfrac{5}{9} \right)\].
Cross multiply and simplify,
\[x=9y-5\]
Equation (2), \[\dfrac{x}{y-\dfrac{10}{9}}=10\]
Cross multiply and simplify,
\[\begin{align}
  & x=10\left[ y-\dfrac{10}{9} \right]\Rightarrow 9x=10\left( 9y-10 \right) \\
 & x=10y-\dfrac{100}{9} \\
\end{align}\]
Equate both values and get the value of y.
\[\begin{align}
  & 9y-5=10y-\dfrac{100}{9} \\
 & 10y-9y=\dfrac{100}{9}-5 \\
 & y=\dfrac{100-45}{9}=\dfrac{55}{9} \\
\end{align}\]
Now substitute the value of equation (1).
\[\begin{align}
  & x=9y-5 \\
 & x=9\times \dfrac{55}{9}-5=55-5=50 \\
\end{align}\]
Thus we got the length of the train = x = 50m.
\[\therefore \] Length of train = 50m.
\[\therefore \] Option (b) is the correct answer.

Note: Speed can be denoted by m/s, which is meter per second. It is a gentle walking speed. Speed in km /hr is kilometer every hour. Thus 1 km/hr is like a very slow walking speed. Km/hr is often used for monitoring speed of vehicles. So we convert the speed of people from km/hr to m/s.