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'Around and Back' printed from https://nrich.maths.org/
This solution was sent in by Harry from Riccarton High School,
Christchurch, New Zealand.
A cyclist and a runner are pracising on a race track going round at
constant speeds, $V_c$ for the cyclist and $V_r$ for the runner.
Let the fraction of the circuit covered by the runner when they
first meet be $x$ and then the cyclist will have covered $1 + x$
circuits. Equating the time taken gives the first equation:
$${x \over V_r} = {1 + x \over V_c}.$$
Similarly the time taken between the cyclist first passing the
runner and the finish gives the second equation: \begin{equation}{1
- x\over V_r} = {x\over V_c}. \end{equation} The ratio of $V_c$ to
$V_r$ from the two equations gives: \begin{equation*}{V_c\over V_r}
= {1 + x\over x} = {x \over 1 - x} \end{equation*} Hence
\begin{equation*}x^2 = 1 - x^2. \end{equation*} From this we get $x
= \sqrt{1\over 2}$ and this gives the ratio of the speeds as
\begin{equation*}{V_c\over V_r} = {{1 + 1/\sqrt 2}\over {1/\sqrt
2}} = \sqrt 2 + 1. \end{equation*}