2452.12 – Doctors and Lawyers Play Dodgeball

A group of doctors and lawyers gathered together for their annual doctors-versus-lawyers dodgeball match. (An actuary officiates.)

The average age of all of the players is forty, the average age of the doctors is thirty-five, and the average age of the lawyers is fifty. Knowing this, what is the ratio of $d$ , the number of doctors, to $l$ , the number of lawyers?

$3:2$
$3:1$
$2:3$
$2:1$
$1:2$

Solution

Because the average age of the doctors is thirty-five, we can undo the formula for calculating averages to find that the combined age of the doctors is $35d$ . We can do the same for the lawyers to get a combined age of $50l$ . Next, since we know the average age for the group, we can equate that to the combined age of all the doctors and lawyers divided by the number of players:

$\dfrac{35d+50l}{d+l}=40$

We now need only solve the equation for $\dfrac{d}{l}$ to have our answer:

$\begin{aligned}35d+50l&=40(d+l) \\5d+50l&=40d+40l \\10l&=5d \\\dfrac{10}{5}&=\dfrac{d}{l}=2:1\end{aligned}$

Therefore, the correct answer is d. This of course gives the doctors a wildly unfair advantage in the upcoming dodgeball game, assuming, of course, that the actuary will first notice this and will then either proceed to ignore the imbalance or let someone else deal with it.