Dead-heat start in 1 km race: L runs 1 km in 10 min and M in 12 min. What start (in metres) can L give M in a 1 km race so that the finish is a dead heat?

Introduction / Context:
A “start” means the slower runner begins closer to the finish. For a dead heat, the slower runner’s actual running distance must match how far they would get in the faster runner’s finish time.

Given Data / Assumptions:

• L = 1000 m in 10 min
• M = 1000 m in 12 min
• Race length = 1000 m

Concept / Approach:
In L’s finish time (10 min), M would cover (1000 * 10/12) m. The start equals 1000 minus this distance.

Step-by-Step Solution:

Verification / Alternative check:
With this start, both finish together in 10 min (L’s time).

Why Other Options Are Wrong:
Other values would not synchronize finishes.

Common Pitfalls:
Using speed differences instead of proportionate distances in the common finish time.

Final Answer:
1662/3 m