Mathematics Question on Problem on Trains

Question

Two persons are walking in the same direction at rates 3 km/ hr and 6 km/hr. A train comes running from behind and passes them in 9 and 10 seconds. The speed of the train is

Answer 1

Solution

Let the speed of the train be $v$ km/h.
The relative speed between the train and the first person is $(v - 3)$ km/h.
The train passes the first person in 9 seconds.

$\Rightarrow\;$$v - 3 \, \text{km/h} = \frac{(v - 3) \times 1000}{3600} \, \text{m/s}$

Since the train passes the person in 9 seconds, the length of the train can be written as:

$\Rightarrow\;$$\text{Length of train} = (v - 3) \times \frac{1000}{3600} \times 9$

Similarly,
the relative speed between the train and the second person is $(v - 6)$ km/h.
The train passes the second person in 10 seconds.

$\Rightarrow\;$$\text{Length of train} = (v - 6) \times \frac{1000}{3600} \times 10$

Since the length of the train is the same in both cases, we equate the two equations:

$\Rightarrow\;$$(v - 3) \times \frac{1000}{3600} \times 9 = (v - 6) \times \frac{1000}{3600} \times 10$

$\Rightarrow\;$$9(v - 3) = 10(v - 6)$

$\Rightarrow\;$$9v - 27 = 10v - 60$

$\Rightarrow\;$$10v - 9v = 60 - 27$

$\Rightarrow\;$$v = 33 \, \text{km/h}$

The correct option is (C): 33 km/hr