Rock750 wrote:
75% of the guestrooms at the Stagecoach Inn have a queen-sized bed, and each of the remaining rooms has a king-sized bed. Of the non-smoking rooms, 60% have a queen-sized bed. If 10% of the rooms at the Stagecoach Inn are non-smoking rooms with king-sized beds, what percentage of the rooms permit smoking ?
A. 25%
B. 30%
C. 50%
D. 55%
E. 75%
OFFICIAL SOLUTION
This is an overlapping sets problem, which can be solved most efficiently by using a double set matrix. Our first step in using the double set matrix is to fill in the information given in the question. Because there are no real values given in the question, the problem can be solved more easily using 'smart numbers'; in this case, we can assume the total number of rooms to be 100 since we are dealing with percentages. With this assumption, we can fill the following information into our matrix:
There are 100 rooms total at the Stagecoach Inn.
Of those 100 rooms, 75 have a queen-sized bed, while 25 have a king-sized bed.
Of the non-smoking rooms (let's call this unknown n), 60% or .6n have queen-sized beds.
10 rooms are non-smoking with king-sized beds.
Let's fill this information into the double set matrix, including the variable n for the value we need to solve the problem:
In a double-set matrix, the first two rows sum to the third, and the first two columns sum to the third. We can therefore solve for n using basic algebra:
10 + .6n = n
10 = .4n
n = 25
We could solve for the remaining empty fields, but this is unnecessary work. Observe that the total number of smoking rooms equals 100 – n = 100 – 25 = 75. Recall that we are working with smart numbers that represent percentages, so 75% of the rooms at the Stagecoach Inn permit smoking.
The correct answer is E.
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