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

by : $71$

 * The Three Motor-Cars
 * Three motor-cars travelling along a road in the same direction are, at a certain moment, in the following positions in relation to one another.
 * Andrews is a certain distance behind Brooks,
 * and Carter is twice that distance in front of Brooks.
 * Each car travels at its own uniform rate of speed,
 * with the result that Andrews passes Brooks in seven minutes,
 * and passes Carter five minutes later.


 * Now, in how many minutes after Andrews would Brooks pass Carter?

Proof
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 $d_1$ be the distance from $A$ at which Andrews passes Brooks.

Let $d_2$ be the distance from $A$ at which Andrews passes Carter.

Let $d_3$ be the distance from $A$ at which Brooks passes Carter.

Let $t$ be the number of minutes after Andrews that Brooks passes Carter.

We have: