Henry Ernest Dudeney/Modern Puzzles/163 - The Russian Motor-Cyclists/Solution

by : $163$

 * The Russian Motor-Cyclists

Solution

 * $3 \tfrac 1 3$ miles.

The "pretty little rule" as so patronisingly puts it is applied thus:


 * ... divide $15$ by $6$ and add $2$, which gives us $4 \tfrac 1 2$.
 * Now divide $15$ by $4 \tfrac 1 2$, and the result $3 \tfrac 1 3$ miles) is the required distance between the two points.

Thus if $a + x$ and $b$ are the legs of the right triangle:


 * $x = \dfrac b {\dfrac b a + 2}$

which is actually a pretty unwieldy little rule considering how rarely one would end up using it.

Proof
Let $x$ miles be the distance from $A$ to $C$.

Let $y$ miles be the distance from $C$ to $B$.

We have:

The general case can be explored as:

as we were to show.