To determine the distance between Town A and Town B, let's analyze the information step by step.

Given Information:

1. Cyclist leaves Town A at 08:00 and travels at a constant speed of 20 km/h.
2. Scooter leaves Town B at 08:30 and travels toward Town A at a constant speed until it meets the cyclist at 11:00.
3. At the meeting time (11:00), the cyclist has covered $\frac{4}{9}$ of the journey.
4. After passing the cyclist, the scooter's speed decreases by 10 km/h.

Steps to Solve:

Step 1: Determine the Total Time the Cyclist Has Traveled Until They Meet

• The cyclist started at 08:00 and met the scooter at 11:00.
• This means the cyclist traveled for 3 hours.

Step 2: Calculate the Distance Covered by the Cyclist When They Met

• The cyclist's speed is 20 km/h, so the distance they covered in 3 hours is: $\text{Distance} = \text{Speed} \times \text{Time} = 20 \, \text{km/h} \times 3 \, \text{hours} = 60 \, \text{km}$
• Since this distance represents $\frac{4}{9}$ of the total journey, we can set up an equation to find the total distance $D$: $\frac{4}{9} D = 60$ $D = \frac{60 \times 9}{4} = 135 \, \text{km}$

Answer:

The distance between Town A and Town B is 135 km.

Would you like further details, or do you have any questions?

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1. What would be the scooter's speed before it passed the cyclist?
2. How far did the scooter travel before meeting the cyclist?
3. What was the scooter’s speed after it passed the cyclist?
4. How much time did the scooter take to meet the cyclist after it left Town B?
5. What is the remaining distance for the scooter after it passes the cyclist?

Tip: When dealing with fractions of a journey in relative speed problems, you can often find total distances by setting up proportions based on the partial distances given.