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?

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:

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