A runs at 1 2/3 (that is, 5/3) times the speed of B. If A gives B a start of 40 m and they finish together, how far from A’s starting line should the winning post be placed?
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A75 m
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B200 m
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C100 m
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D125 m
Answer
Correct Answer: 100 m
Explanation
Introduction / Context:Starts (head-starts) in races shift the effective distance each runner must cover. If two runners finish together, the time taken by both is equal. With a start advantage to the slower runner, we relate distances to speeds using the time equality for the same finishing instant.
Given Data / Assumptions:
- A’s speed : B’s speed = 5 : 3.
- Head-start to B = 40 m.
- Let race length (distance A runs) be D.
Concept / Approach:Time = distance / speed. With equal finish time: D / vA = (D − 40) / vB. Substitute vA / vB = 5/3 to solve for D.
Step-by-Step Solution:
D / (5/3 vB) = (D − 40) / vB ⇒ (3D/5) = D − 40.Multiply by 5: 3D = 5D − 200 ⇒ 2D = 200 ⇒ D = 100 m.Verification / Alternative check:Times: A takes 100 / (5k) = 20/k; B takes 60 / (3k) = 20/k. Equal ⇒ consistent.
Why Other Options Are Wrong:75, 125, or 200 m do not satisfy the equality when substituted.
Common Pitfalls:Interpreting “start” as extra distance for A instead of reduced distance for B, or inverting the given speed ratio.
Final Answer:100 m