Math Problem Statement
A cyclist left Town A at 08:00 and traveled to Town B at a speed of 20 km/h throughout the entire journey. A scooter left Town B at 08:30 and traveled toward Town A. The scooter maintained the same speed until it passed the cyclist at 11:00. The cyclist had covered 4/9 of the journey when they passed each other. The speed of the scooter decreased by 10 km/h after passing the cyclist, and it continued at this new speed for the remainder of the journey.
Solution
Let's analyze the information and solve for the unknowns.
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Determine the Distance Between Towns A and B
Let represent the total distance between Town A and Town B.
The cyclist travels at a constant speed of km/h, and by 11:00, they had traveled for:
During this time, the cyclist covered of the journey, so:
Since speed , we have: Simplifying:
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Calculate the Speed of the Scooter
The scooter left Town B at 08:30 and passed the cyclist at 11:00, meaning it traveled for:
Let be the speed of the scooter before it decreased. The distance covered by the scooter in this time is because the cyclist covered .
Therefore: Simplifying:
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Determine the New Speed of the Scooter After Passing the Cyclist
The speed of the scooter decreased by km/h after passing the cyclist, so its new speed is:
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Time for the Scooter to Complete the Journey
After passing the cyclist, the scooter still had of the journey remaining to reach Town A, a distance of:
With the new speed of km/h, the time taken to cover this distance is:
Therefore, the scooter will reach Town A at:
Summary of Answers:
- Distance between Town A and Town B: km
- Initial speed of the scooter: km/h
- New speed of the scooter after passing the cyclist: km/h
- Time the scooter reaches Town A:
Would you like any part of this broken down further, or do you have any other questions?
Related Questions:
- How would the solution change if the cyclist’s speed increased by 5 km/h?
- What would happen if the scooter reduced its speed by 15 km/h instead of 10 km/h?
- If the scooter encountered a stop midway, how would this affect the arrival time?
- How would the solution change if the cyclist left 30 minutes earlier?
- How can we calculate the point of meeting if speeds and start times are different?
Tip: Drawing a timeline or diagram can often simplify problems involving moving objects and multiple events.
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Math Problem Analysis
Mathematical Concepts
Algebra
Distance and Speed
Proportions
Formulas
Distance = Speed x Time
Proportion of distance traveled = Total Distance x (Fraction traveled)
Theorems
Direct Proportion in Distance Problems
Suitable Grade Level
Grades 9-11