Henry Ernest Dudeney/Modern Puzzles/42 - The Puzzle of the Runners/Solution

by : $42$

 * The Puzzle of the Runners

Solution
$20 \tfrac 1 4$ times faster.

Probably not technically achievable.

Proof
While Brown has run $\dfrac 1 6$ of the course, or $\dfrac 4 {24}$ of it, Tomkins has already run $\dfrac 5 6 - \dfrac 1 8$, that is $\dfrac {17} {24}$ of it.

Thus Tomkins's pace has up till now been $\dfrac {11} 4$ of Brown's.

Brown now has $\dfrac 5 6$ of the course to run, while Tomkins has $\dfrac 1 6$ left.

Thus Brown must speed up to $5$ times as fast as Tomkins in order to pass the finish line at the same time as Tomkins.

Thus he must go at $5 \times \dfrac {17} 4$, which is $\dfrac {85} 4$ times as fast as he went at first.

Then we note that $\dfrac {85} 4$ times as fast is the same thing as $\dfrac {85} 4 - 1$ times faster.

Hence the correct answer is $\dfrac {81} 4 = 20 \tfrac 1 4$ times faster.