Let's break down and solve the problem together.

Problem Translation:

A person is cycling with a speed of 16 km/h in the first hour. Every subsequent hour, the speed decreases to $\frac{3}{4}$ of the speed from the previous hour. What is the total distance traveled after cycling for 5 hours?

Step-by-Step Solution:

1. Initial Speed and Decay Factor:

   • In the first hour, the speed is 16 km/h.
   • The speed decreases to $\frac{3}{4}$ of the previous hour's speed each hour. This is a geometric progression.

2. Speed for Each Hour:

   • First hour: $16$ km/h.
   • Second hour: $16 \times \frac{3}{4} = 12$ km/h.
   • Third hour: $12 \times \frac{3}{4} = 9$ km/h.
   • Fourth hour: $9 \times \frac{3}{4} = 6.75$ km/h.
   • Fifth hour: $6.75 \times \frac{3}{4} = 5.0625$ km/h.

3. Total Distance Traveled: The total distance is the sum of the distances traveled each hour, which is speed multiplied by time (in this case, 1 hour per interval): $\text{Total distance} = 16 + 12 + 9 + 6.75 + 5.0625 = 48.8125 \text{ km}.$

Final Answer:

The total distance traveled after 5 hours is 48.8125 km.

Let me know if you want further details or if anything needs clarification. Here are some related questions:

1. What is the formula for a geometric progression, and how is it used in this context?
2. If the person cycled for 6 hours, what would be the total distance?
3. How does the decrease in speed affect the total distance covered in each hour?
4. How could you calculate the average speed for the entire 5-hour journey?
5. What if the speed decreased by half instead of $\frac{3}{4}$ per hour—what would the total distance be?

Tip: For problems involving decreasing rates or repeated percentages, identifying a geometric progression helps simplify the solution!