A runs 500 m in 30 s; B runs 500 m in 35 s. In a 1 km race, how much head-start (in meters) should A give B so that they finish in a dead heat?

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Solve this multiple-choice question and choose the correct option.

Accepted Answer

Introduction / Context:For a dead-heat with a head-start, the slower runner’s race distance is shorter, and the finish times must be equal. Compute each runner’s speed from the given 500 m times and then set equal times for the 1 km event to solve the required start. Given Data / Assumptions: vA = 500/30 = 50/3 m/s. vB = 500/35 = 100/7 m/s. Let head-start to B be x meters. A runs 1,000 m; B runs (1,000 − x) m. Concept / Approach:Equal finish time: 1,000 / vA = (1,000 − x) / vB. Solve for x. Convert to a mixed fraction as needed. Step-by-Step Solution: 1,000 / (50/3) = (1,000 − x) / (100/7).Left: 60 s. Right: (1,000 − x)*7/100.Equate: 60 = (7,000 − 7x)/100 ⇒ 6,000 = 7,000 − 7x ⇒ 7x = 1,000 ⇒ x = 1,000/7 = 142 6/7 m. Verification / Alternative check:Plugging x back equates both times to 60 seconds. Why Other Options Are Wrong:Nearby fractions give unequal finish times; only 1,000/7 m equalizes them. Common Pitfalls:Using average speeds or subtracting times directly without adjusting distances. Final Answer:142 6/7 m

A runs 500 m in 30 s; B runs 500 m in 35 s. In a 1 km race, how much head-start (in meters) should A give B so that they finish in a dead heat?

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