1988 AHSME Problems/Problem 25

Problem

$X, Y$ and $Z$ are pairwise disjoint sets of people. The average ages of people in the sets $X, Y, Z, X \cup Y, X \cup Z$ and $Y \cup Z$ are $37, 23, 41, 29, 39.5$ and $33$ respectively. Find the average age of the people in set $X \cup Y \cup Z$.

$\textbf{(A)}\ 33\qquad \textbf{(B)}\ 33.5\qquad \textbf{(C)}\ 33.6\overline{6}\qquad \textbf{(D)}\ 33.83\overline{3}\qquad \textbf{(E)}\ 34$

Solution

Let the variables $X$, $Y$, and $Z$ represent the sums of the ages of the people in sets $X$, $Y$, and $Z$ respectively. Let $x$, $y$, and $z$ represent the numbers of people who are in sets $X$, $Y$, and $Z$ respectively. Since the sets are disjoint, we know, for example, that the number of people in the set $X \cup Y$ is $x+y$ and the sum of their ages is $X+Y$, and similar results apply for the other unions of sets. Thus we have $X=37x$, $Y=23y$, $Z=41z$, $X+Y=29(x+y) \implies 37x + 23y = 29x + 29y \implies 8x = 6y \implies y = \frac{4}{3}x$, and $X+Z = 39.5(x+z) \implies 37x + 41z = 39.5x + 39.5z \implies 74x + 82z = 79x + 79z \implies 3z = 5x \implies z = \frac{5}{3}x$. Hence the answer is $\frac{X+Y+Z}{x+y+z} = \frac{37x + 23\times\frac{4}{3}x + 41\times\frac{5}{3}x}{x+\frac{4}{3}x+\frac{5}{3}x} = \frac{136x}{4x} = 34$, which is $\boxed{\text{E}}$. Notice that we did not need all three of the means of the unions of the sets - any two of them would have been sufficient to determine the answer.