Henry Ernest Dudeney/Puzzles and Curious Problems/71 - The Three Motor-Cars/Solution

by : $71$

 * The Three Motor-Cars

Solution

 * $6 \tfrac 2 3$ minutes.

Proof
The trick of this puzzle is to consider the relative speeds instead of absolute speeds.

Let $A$, $B$ and $C$ denote the current positions of Andrews, Brooks and Carter respectively.

Let $d$ be the distance between where $A$ and $B$ are now.

Let $v_A$, $v_B$ and $v_C$ units per minute be the speeds of $A$, $B$ and $C$ respectively.

Let $t$ be the number of minutes since the start for Brooks to pass Carter.

We have:

Write $x = v_A - v_B$ and $y = v_A - v_C$.

Then $v_B - v_C = y - x$.

So we have:

therefore Brooks passes Carter $6 \tfrac 2 3$ minutes after Andrew does.