Problem:
X,Y, and Z are pairwise disjoint sets of people. The average ages of people in the sets X,Y,Z,X∪Y,X∪Z and Y∪Z are given in the table below.
SetAverage age ofpeople in the setX37Y23Z41X∪Y29X∪Z39.5Y∪Z33
Find the average age of the people in the set X∪Y∪Z.
Answer Choices:
A. 33
B. 33.5
C. 33.66
D. 33.833
E. 34
Solution:
Let the numbers of elements in the sets be given by ∣X∣=x,∣Y∣=y and ∣Z∣=z. Then ∣X∪Y∣=x+y,∣X∪Z∣=x+z and ∣Y∪Z∣=y+z. The given information can be summarized in the following 3 equations in 3 unknowns:
x+y37x+23y=29;x+z37x+41z=39.5;y+z23y+41z=33.
Simplifying these equations, we obtain 4x=3y;5x=3z;5y=4z. We want the value of the fraction
x+y+z37x+23y+41z
Making the substitutions y=4x/3 and z=5x/3, we obtain
x+34x+35x37x+23(34x)+41(35x)=3x+4x+5x111x+92x+205x=34
Queries. Where did we use the given that X,Y and Z are disjoint? Also, did we need all the information which was given?