emmak 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?
81
88
160
550
710
Let the number of republicans be 100x and that of the democrats be 100y. Now, we are given that 80x like apple pies. Again, 16x like both apple pie and eclairs amongst the republicans.
Also, 55y like eclairs amongst the democrats.
Also given that 16x = 55y
Thus, the total number of people in the group =\(100x+100y = 100(x+y) = 100*y*(\frac{55}{16}+1) = 100*y*(\frac{71}{16})\)
Now, the number of people has to be an integer.Also, as 71 and 16 are co-primes, thus 100y has to be a multiple of 16. Thus, 100y = 16k, where k is a non negative integer.
Thus, the total number of people = 100(x+y) = \(16k*\frac{71}{16}\) = 71k. From the given options, for k=10, option E matches.
E.
The question stated 20% of the Republicans who liked apple pie also liked eclairs, not 16% as you did. I think the question has a wrong number because the way you did is right.