One night a certain hotel rented 3/4 of its rooms, including

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One night a certain hotel rented 3/4 of its rooms, including 2/3 of their air-conditioned rooms. If 3/5 of its rooms were air-conditioned, what percent of the rooms that were not rented were air-conditioned?

A. 20
B. 33 1/3
C. 35
D. 40
E. 80

The OA is E

Source: GMAT Paper Tests

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by Brent@GMATPrepNow » Sat Jan 19, 2019 8:41 am
swerve wrote:One night a certain hotel rented 3/4 of its rooms, including 2/3 of their air-conditioned rooms. If 3/5 of its rooms were air-conditioned, what percent of the rooms that were not rented were air-conditioned?

A. 20
B. 33 1/3
C. 35
D. 40
E. 80

The OA is E

Source: GMAT Paper Tests
This question is a great candidate for the Double Matrix method, whichcan be used for most questions featuring a population in which each member has two characteristics associated with it.

Here, we have a population of motel rooms, and the two characteristics are:
- air conditioning or no air conditioning
- rented or not rented

So, we can set up our matrix like this:
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Now since we're asked to find a certain PERCENTAGE, let's choose a nice value for the total number of motel rooms.
Notice that the question includes the fractions 3/4, 2/3 and 3/5. So, let's choose a number that works well with all of these fractions.
Since 60 is the least common denominator of 3/4, 2/3 and 3/5, let's say that there are 60 motel rooms altogether.
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Now that we're set up, we can use the given information to determine the number of rooms to place in each of the four boxes.

If 3/5 of its rooms were air-conditioned
So, 3/5 of the 60 rooms are air conditioned. 3/5 of 60 = 36, which means 36 rooms have AC and the remaining 24 rooms do not have AC...
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...motel rented 3/4 of its rooms
3/4 of 60 = 45, so 45 rooms are rented and the remaining 15 rooms are not rented.
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...a certain motel rented ....2/3 of its air-conditioned rooms
So of the 36 rooms with AC, 2/3 were rented.
2/3 of 36 = 24. So, 24 rooms had AC AND were rented.
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Now that we know the number of rooms in 1 box, and we know the sums of the rows and columns, we can fill in the remaining boxes.
Image

Question: what percent of the rooms that were not rented were air-conditioned?
There were 15 unrented rooms and 12 of them were air conditioned.
12/15 = 4/5 = [spoiler]80% = E[/spoiler]

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To learn more about the Double Matrix method, watch our free video: https://www.gmatprepnow.com/module/gmat- ... ems?id=919

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- https://www.beatthegmat.com/mba/2011/05/ ... question-1
- https://www.beatthegmat.com/mba/2011/05/ ... question-2
- https://www.beatthegmat.com/mba/2011/05/ ... question-3
- https://www.beatthegmat.com/ds-quest-t187706.html
- https://www.beatthegmat.com/overlapping- ... 83320.html
- https://www.beatthegmat.com/finance-majo ... 67425.html
- https://www.beatthegmat.com/ds-french-ja ... 22297.html
- https://www.beatthegmat.com/sets-t269449.html#692540
- https://www.beatthegmat.com/in-costume-f ... tml#692116

Cheers,
Brent
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by Scott@TargetTestPrep » Sun Jan 27, 2019 9:51 am
swerve wrote:One night a certain hotel rented 3/4 of its rooms, including 2/3 of their air-conditioned rooms. If 3/5 of its rooms were air-conditioned, what percent of the rooms that were not rented were air-conditioned?

A. 20
B. 33 1/3
C. 35
D. 40
E. 80

The OA is E

Source: GMAT Paper Tests
We can let the number of rooms = 60. (We choose this convenient number 60 because it is the least common multiple (LCM) of the denominators 3, 4, and 5.)

We see that 3/4 x 60 = 45 rooms were rented, and 60 - 45 = 15 rooms were not rented.

We also see that 3/5 x 60 = 36 rooms are air conditioned, and 60 - 36 = 24 rooms are not air conditioned.

Thus, 2/3 x 36 = 24 air conditioned rooms were rented, so 36 - 24 = 12 air conditioned rooms were not rented.

Thus, the percent of rooms that were not rented that were air conditioned is 12/15 = 4/5 = 80%.

Answer: E

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