Problem Solving

swerve wrote:On a Saturday night, each of the rooms at a certain motel was rented for either $40 or $60. If 10 of the rooms that were rented for $60 had instead been rented for $40, then the total rent the motel charged for that night would have been reduced by 25 percent. What was the total rent the motel actually charged for that night?

A. $600
B. $800
C. $1,000
D. $1,600
E. $2,400

The OA is B

Source: GMAT Paper Tests

We can let a = the number of 40-dollar rooms rented and b = the number of 60-dollar rooms rented. Thus, the original rental income for the night was (40a + 60b). If this amount were reduced by 25%, the motel's rental income would be 0.75(40a + 60b). This would result from renting 10 more 40-dollar rooms (a + 10), resulting in a decrease in the number of 60-dollar rooms to (b - 10). Thus, we have:

0.75(40a + 60b) = 40(a + 10) + 60(b - 10)

30a + 45b = 40a + 400 + 60b - 600

200 = 10a + 15b

40 = 2a + 3b

Since the total rental income of the motel, in terms of a and b, is (40a + 60b), which is 20 times (2a + 3b), the total rent is 20(2a + 3b) = 20(40) = $800.

Alternate Solution:

We can let a = the number of 40-dollar rooms rented and b = the number of 60-dollar rooms rented. Thus, the motel charges (40a + 60b) in total for the night. If 10 of the rooms that were rented for $60 had instead been rented for $40; then the motel would have charged 40(a + 10) + 60(b - 10) = 40a + 400 + 60b - 600 = 40a + 60b - 200 instead. We notice that this is a $200 reduction compared to the actual price. Since we are given that the reduction in price would be 25% of the actual figure, the motel charged 200/(0.25) = $800 for the night.

Answer: B