Average Speed Calculation: A to B and Back Trip

This problem requires calculating the average speed for a journey covering two different segments with different speeds. The key is to remember that average speed isn't just the average of the two speeds because the time spent on each segment might differ.

Defining Average Speed

The fundamental formula for average speed is:

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

Let's apply this to the scenario.

Step 1: Define Variables and Distances

Let the distance from point A to point B be '$d$' km.

The journey consists of two parts:

• Trip 1: A to B, Distance = $d$ km, Speed = 15 km/h
• Trip 2: B to A, Distance = $d$ km, Speed = 25 km/h

The total distance covered is the sum of the distances for both trips:

Total Distance = $d + d = 2d$ km

Step 2: Calculate Time for Each Part

We need to calculate the time taken for each part of the journey using the formula Time = Distance / Speed.

• Time for Trip 1 (A to B): $t_1 = \frac{d}{15}$ hours
• Time for Trip 2 (B to A): $t_2 = \frac{d}{25}$ hours

Step 3: Calculate Total Time

The total time for the entire journey is the sum of $t_1$ and $t_2$.

Total Time ($T$) = $t_1 + t_2 = \frac{d}{15} + \frac{d}{25}$

To add the fractions, find a common denominator (which is 75):

Total Time ($T$) = $\frac{5d}{75} + \frac{3d}{75} = \frac{5d + 3d}{75} = \frac{8d}{75}$ hours

Step 4: Calculate Average Speed

Now, substitute the Total Distance and Total Time into the average speed formula:

Average Speed = $\frac{\text{Total Distance}}{\text{Total Time}} = \frac{2d}{\frac{8d}{75}}$

To simplify the division by a fraction, multiply by its reciprocal:

Average Speed = $2d \times \frac{75}{8d}$

The '$d$' cancels out:

Average Speed = $\frac{2 \times 75}{8} = \frac{150}{8}$

Simplifying the fraction:

Average Speed = $\frac{75}{4}$ km/h

Converting the fraction to a decimal:

Average Speed = 18.75 km/h

Final Result Summary

The average speed of the man for the entire journey from A to B and back to A is 18.75 km/h.