Train from Nagpur to Raipur

A train departs from Nagpur rail station every 1 hour. It takes exactly 5 hours to arrive to Raipur rail station. Another train departs from Raipur 30 minutes after the departure of the first Nagpur train and also every 1 hour. The Raipur trains are older and slower and it takes them 7 hours to arrive at Nagpur. Both train lines operate 24x7. How many trains will the Nagpur train encounter during its journey to Raipur?

Let’s call the distance between the two cities S.

Nagpur train speed is then $\frac{S}{5}$ and Raipur’s is $\frac{S}{7}$.

Assuming a x-axis with its origin O at Nagpur, in 30 minutes, the first train has arrived at $\frac{S}{5}*0.5 = \frac{S}{10}$ from O. Then we have an equivalent problem of calculating the (first) meeting point of two trains moving at opposite directions, with speeds $\frac{S}{5}$ and $\frac{S}{7}$ and distance between them $S_1=0.9*S$.

Suppose they meet at t hours after the departure of the Raipur train. We have: $\frac{S}{5}*t+\frac{S}{7}*t=0.9*S$.

By solving, we get $t=2.625$ hours.

This means that the Nagpur train meets the first Raipur train $2.625+0.5 = 3.125 $ hours after its departure.

From this point forward, I don't know if I am right:

The remaining distance to be traveled by the Nagpur train is $0.375*S$

In the remaining $5-3.125 = 1.875$ hours, the Nagpur train meets 2 more trains (1st one at 0.232143*S and second at 0.089286*S).

Is it correct?

Solution 1:

This is an old trick question.

Without loss of generality we can imagine that we are sitting on the train leaving from Nagpur at 7 a.m., before the train that left Raipur at 12.30 a.m. arrives, but after the arrival of 11.30 p.m. departure. That train will arrive at Raipur at noon, after the 11.30 a.m. departure but before the 12.30 p.m. departure.

So during the journey we will encounter the trains that departed from Raipur any half hour between midnight and noon. That is 12 trains.

No need to calculate the times of the encounters.