Henry Ernest Dudeney/Puzzles and Curious Problems/209 - A Running Puzzle/Solution

Puzzles and Curious Problems by Henry Ernest Dudeney: $209$

$ABCD$ is a square field of $40$ acres.

The line $BE$ is a straight path, and $E$ is $110$ yards from $D$.

In a race Adams runs direct from $A$ to $D$,

but Brown has to start from $B$, go from $B$ to $E$, and thence to $D$.

Each keeps to a uniform speed throughout, and when Brown reaches $E$, Adams is $30$ yards ahead of him.

Who wins the race, and by how much?

Brown wins by $8$ yards.

First let us establish the geometry of the situation.

We have that the field is square, and of $40$ acres.

Hence the field is $40 \times 4840 = 440^2$ square yards, and so $440$ yards on a side.

So $AE = 440 - 110 = 330$ yards, and $\triangle ABE$ is a $3$-$4$-$5$ triangle.

Hence $BE = 550$ yards, and the total distance Brown is to run is $660$ yards.

Adams, however, is to run only $440$ yards, the length of the side of the field.

We have that Brown gets to $E$ after Adams has run $330 + 30 = 360$ yards.

Now let $v$ be the speed that Adams runs.

Then Brown runs $\dfrac {550 v} {360}$ times as fast as Adams.

So while Adams reaches $D$ after $\dfrac {440} v$, Brown reaches $D$ after $\dfrac {660 \times 360} {550 v} = \dfrac {432} v$, in appropriate units.

So Brown gets there first.

So Adams has only managed to get $432$ yards by the time Brown reaches the end.

That is, he has $8$ yards still to go.

$\blacksquare$