Answer

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**Hint:**We are given some information about covering a certain distance in a certain amount of time. Using this information, we are going to create equations in unknown variables. To solve this question, we should be aware about the concept of speed, distance and time. We are going to use the formula involving these three quantities and using that we will find the length of the train first and then we will find the speed.

**Complete step by step answer:**

We should be aware that when the train passes an object completely, it covers the distance of the object along with the distance equal to the length of the train as well. So, we start by assuming that the length of the train is $x$.

According to question, it crosses a railway platform 120 metres long in 12 seconds, then by using the formula below:

$Speed=\dfrac{Distance}{Time}$

We can say that:

$Speed=\dfrac{x+120}{12}$

It is also given that it crosses another platform, 170 metres long in 16 seconds, i.e.:

$Speed=\dfrac{x+170}{16}$

Since, both represent the speed of the train, we can say that:

$\dfrac{x+120}{12}=\dfrac{x+170}{16}$

$\implies 4\left(x+120\right)=3\left(x+170\right)$

$\implies 4x+480=3x+510$

$\implies x=30$m

So, the train is 30 metres long. Now, we go on to find the speed of the train:

Since, $Speed=\dfrac{x+120}{12}$

We put the value of $x$ to be 30, we get the following then:

$Speed=\dfrac{120+30}{12}$m/sec

$\implies Speed=\dfrac{150}{12}$m/sec

$\implies Speed=12\dfrac{1}{2}$m/sec

**So, the correct answer is “Option a”.**

**Note:**While solving this question, make sure that you put the correct formula. And also, remember to consider the length of the train in the total distance, otherwise it would lead to invalid results. And also, try to make the least calculation mistakes while solving the fraction while equating the speed of the train from two equations.

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