To find the average speed for the whole trip, we first need to determine the total distance traveled and the total time taken.
Step 1: Calculate the distance for the first part of the trip.
The man travels for 6 hours at a speed of 50 miles per hour.
Distance = Speed × Time = 50 miles/hour × 6 hours = 300 miles.
Step 2: Calculate the time taken for the return trip.
The return trip takes him \( \frac{15}{2} \) hours, which is equal to 7.5 hours.
Step 3: Calculate the distance for the return trip.
Since the return trip is straight back, the distance is the same: 300 miles.
Step 4: Calculate the total distance.
Total distance = Distance going + Distance returning = 300 miles + 300 miles = 600 miles.
Step 5: Calculate the total time.
Total time = Time going + Time returning = 6 hours + 7.5 hours = 13.5 hours.
Step 6: Calculate the average speed for the whole trip.
Average Speed = Total Distance / Total Time = 600 miles / 13.5 hours.
Now, calculating \( \frac{600}{13.5} \):
\[ \frac{600}{13.5} = 44.4444... \text{ miles per hour} \]
To express this as a fraction:
600 ÷ 13.5 = 600 ÷ \frac{27}{2} = 600 × \frac{2}{27} = \frac{1200}{27} = \frac{400}{9} \text{ miles per hour}
Finally, rounding it, we find that the average speed for the whole trip is:
\[ \text{Average Speed} \approx 44.44 \text{ miles/hour} \] Thus,
\[ \text{Final Result: } \frac{400}{9} \text{ miles/hour} \approx 44.44 \text{ miles/hour} \]