Question

A train takes 18 seconds to pass completely a station 162 m long and 15 seconds through another station 120 m long. The length of the train is:

Answer 1

Hint: Let us assume that the length of the train is “x”. The total distance that the train travelled in 18 seconds is the addition of 162 m with x. Similarly the total distance covered for 120 m long station is the addition of 120 m and x. Now, the speed of the train is the same in both the platform sizes. Equate the speed of the train in platform of 162 m length with the platform of 120 m length. Calculate speed by using the formula $\text{Speed}=\dfrac{\text{Distance}}{\text{Time}}$.

Complete step-by-step answer:
Let us assume that the length of the train is “x”. Now, it is given that the time required by the train to pass completely through the platform of distance 162 m is 18 seconds. It is also given that the time taken to pass through the platform of length 120 m is 15 seconds. Now, the speed of the train is the same in both the cases of different platform lengths.
To find the speed of the train we are going to use the following formula:
$\text{Speed}=\dfrac{\text{Distance}}{\text{Time}}$…………….. Eq. (1)
In the above formula, distance is equal to the addition of length of the platform with the length of the train.
Now, calculating the speed of the train for platform having length of 162 m is by substituting distance as 162 + x and time as 18 seconds in eq. (1) we get,
$Speed=\dfrac{162+x}{18}$……..Eq. (2)
Calculating the speed of the train for platform having length of 120 m is by substituting distance as 120 + x and time as 15 seconds in eq. (1) we get,
$Speed=\dfrac{120+x}{15}$…………Eq. (3)
Equating eq. (2) and eq. (3) we get,
$\dfrac{162+x}{18}=\dfrac{120+x}{15}$……..Eq. (4)
On cross multiplying the above equation we get,
$\begin{align}
  & 15\left( 162+x \right)=18\left( 120+x \right) \\
 & \Rightarrow 2430+15x=2160+18x \\
\end{align}$
Rearranging the above equation by writing the coefficient of x on one side of the equation and the constant terms on the other side of the equation we get,
$\begin{align}
  & 2430-2160=18x-15x \\
 & \Rightarrow 270=3x \\
\end{align}$
Dividing 3 on both the sides of the equation we get,
$\begin{align}
  & \dfrac{270}{3}=x \\
 & \Rightarrow x=90 \\
\end{align}$
Hence, the length of the train is equal to 90 m.

Note: You can minimize the lengthy calculations in the above solution by simplifying the eq. (4) which is given as:
$\dfrac{162+x}{18}=\dfrac{120+x}{15}$
In the above equation, as you can see that denominators are divisible by 3 so multiplying the numerator by 3 on both the sides of the above equation we get,
$\left( \dfrac{162+x}{18} \right)3=\left( \dfrac{120+x}{15} \right)3$
In the above equation, 3 is dividing 18 by 6 times and 3 is dividing 15 by 5 times so using these relations in the above equation we get,
$\Rightarrow \dfrac{162+x}{6}=\dfrac{120+x}{5}$
On cross multiplying the above equation we get,
$\begin{align}
  & 5\left( 162+x \right)=6\left( 120+x \right) \\
 & \Rightarrow 810+5x=720+6x \\
\end{align}$
$\begin{align}
  & \Rightarrow 810-720=x \\
 & \Rightarrow 90=x \\
\end{align}$
You can see from the above that we are getting the same value of x as we have got in the above solution. But the difference is that here the calculations are a bit easier than the above one.