A 10% head start is not always offset by running 10% faster. The trap is to notice how much distance each runner actually has to cover.
The Race with a Hidden Head Start
Riddle statement
Two runners move at constant speed toward a finish line located 100 meters from the start.
One begins 10 meters ahead, so he only has to run 90 meters. The other starts from the starting line.
By what percentage must the second runner exceed the first runner's speed for both to arrive at exactly the same time?
Show solution
Solution
No, it is not enough.
The runner with the head start begins at meter 10. Therefore, he only needs to run:
meters.
The other runner needs to run the full 100 meters.
To arrive at the same time, their speeds must be in the same ratio as their distances:
That means the runner without the head start must run:
times as fast as the other runner.
And:
In other words, he needs to run approximately 11.11% faster.
Answer: no. Running 10% faster is not enough; he would have to run 11.11% faster.