guerrero wrote:A group of Republicans and Democrats was surveyed and each member was asked whether they liked apple pies or éclairs. 80% of the Republicans liked apple pie, 55% of the Democrats liked eclairs, and 20% of the Republicans who liked apple pie also liked eclairs. If the number of Republicans who liked both desserts is equal to the number of Democrats who liked éclairs, what is one possible value for the number of members of the group?

A. 81
B. 88
C. 160
D. 550
E. 710

Since this is really a problem about RATIOS, I wouldn't use a matrix here.
To determine the least possible value for the total number of members, plug in the least possible value for the total number of Democrats.

55% of the Democrats liked eclairs.
Since 55/100 = 11/20, the total number of Democrats must be a multiple of 20.

Case 1: Total democrats = 20.
Since 55% liked eclairs, the number of Democrats who liked eclairs = 11/20 * 20 = 11.

The number of Republicans who liked both desserts is equal to the number of Democrats who liked éclairs.
Thus, Republicans who liked both = 11.

20% of the Republicans who liked apple pie also liked eclairs.
Thus, the 11 Republicans who liked both are equal to 20% -- or 1/5 - of the total number of Republicans who liked apple pie:
11 = (1/5)x
x=55.

80% of the Republicans liked apple pie.
Thus, the 55 Republicans who liked apple pie are equal to 80% -- or 4/5 -- of the total number of Republicans:
55 = (4/5)y
y = 275/4.

Resulting sums:
Total Democrats = 20 and total Republicans = 275/4.

The total number of Republicans must be an INTEGER.
Thus, ALL of the values in Case 1 -- including the total number of Democrats -- must be multiplied AT LEAST BY A FACTOR OF 4:
Least total Democrats = 4*20 = 80.
Least total Republicans = 4 * 275/4 = 275.
Least total members = 80+275 = 355.

Thus, the total number of members must be a MULTIPLE OF 355.

The correct answer is E.