A train passes two bicycles moving in the same direction at a speed of 10 km/h and 15 km/h, respectively in 10 sec and 11 sec respectively. The speed of the train is:

Concept used:

If a train crosses a bicycle/man/pole it covers it own length.

Calculation:

Let's assume the speed is k km/h,

Then, according to the question,

⇒ (k - 10) × 10/3600 = (k - 15) × 11/3600

⇒ (k- 10) × 10 = (k - 15) × 11

⇒ 10k - 100 = 11k - 165

⇒ k = 65

∴ The speed of the train is 65 km/h

Alternate Method  

Let the length of the train be "l" meters.

When the train overtakes the first bicycle, the relative speed between the train and the bicycle is (x - 10) m/s, where x is the speed of the train in m/s.

Similarly, when the train overtakes the second bicycle, the relative speed between the train and the bicycle is (x - 15) m/s.

We know that the time taken to overtake the first bicycle is 10 seconds and the time taken to overtake the second bicycle is 11 seconds.

Therefore, we can set up the following equations:

l/(x - 10) = 10

l/(x - 15) = 11

Simplifying these equations, we get:

l = 10(x - 10)

l = 11(x - 15)

Equating these two expressions for "l", we get:

10(x - 10) = 11(x - 15)

Simplifying this equation, we get:

10x - 100 = 11x - 165

x = 65 km/hr

Therefore, the speed of the train is 65 km/hr.