The Train Puzzle

Qusetion :

A train travels from city A to city B at a speed of 40 mph. On the return journey from city B to city A, the train travels at a speed of 60 mph. What is the average speed of the train for the entire round trip?

Solution:

To find the average speed for the entire round trip, we use the formula for average speed:

$\text{Average Speed}=\frac{\text{Total Distance}}{\text{Total Time}}$

First, let's find the total distance traveled.

The distance from city A to city B is the same as the distance from city B to city A. Let's denote this distance as $𝐷$.

So, the total distance for the round trip is $2𝐷$.

Now, let's find the total time taken for the round trip.

Time taken for the first leg of the trip (from city A to city B): $\text{Time}=\frac{\text{Distance}}{\text{Speed}}=\frac{𝐷}{40}$

Time taken for the return trip (from city B to city A): $\text{Time}=\frac{\text{Distance}}{\text{Speed}}=\frac{𝐷}{60}$

The total time for the round trip is the sum of these two times.

$\text{Total Time}=\frac{𝐷}{40}+\frac{𝐷}{60}$

Now, let's substitute the expressions for total distance and total time into the formula for average speed.

$\text{Average Speed}=\frac{2𝐷}{\frac{𝐷}{40}+\frac{𝐷}{60}}$

$\text{Average Speed}=\frac{2𝐷}{\frac{3𝐷}{120}+\frac{2𝐷}{120}}$

$\text{Average Speed}=\frac{2𝐷}{\frac{5𝐷}{120}}$

$\text{Average Speed}=\frac{2𝐷\times 120}{5𝐷}$

$\text{Average Speed}=\frac{240𝐷}{5𝐷}$

$\text{Average Speed}=48\text{ mph}$

So, the average speed of the train for the entire round trip is 48 mph.