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# On Friday morning at a particular hotel, 5/9 of the rooms were cleaned

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Math Expert
Joined: 02 Sep 2009
Posts: 59587
On Friday morning at a particular hotel, 5/9 of the rooms were cleaned  [#permalink]

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19 Feb 2017, 23:47
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Difficulty:

75% (hard)

Question Stats:

66% (02:48) correct 34% (03:13) wrong based on 89 sessions

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On Friday morning at a particular hotel, 5/9 of the rooms were cleaned in preparation for weekend visitors. Of the rooms that were not cleaned, 1/3 were planned to be vacant for the weekend and therefore did not need to be cleaned. Of the rooms that needed to be cleaned but were not, the staff was able on Friday afternoon to clean 3/5 before check-in time. What fraction of the hotel's rooms were cleaned on Friday before check-in time?

A. 11/15
B. 7/9
C. 37/45
D. 13/15
E. 8/9

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Re: On Friday morning at a particular hotel, 5/9 of the rooms were cleaned  [#permalink]

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20 Feb 2017, 00:45
So if 5/9 were cleaned 4/9 were not.
of this 4/9 fraction that should be cleaned but were not = 1-1/3 = 2/3, so 8/27 should be cleaned but were not
3/5 of there were cleaned before Friday so they cleaned 3/5*8/27 = 8/45
so in total 5/9+4/45 = 33/45 = 11/15

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Re: On Friday morning at a particular hotel, 5/9 of the rooms were cleaned  [#permalink]

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15 Sep 2019, 23:49
Bunuel wrote:
On Friday morning at a particular hotel, 5/9 of the rooms were cleaned in preparation for weekend visitors. Of the rooms that were not cleaned, 1/3 were planned to be vacant for the weekend and therefore did not need to be cleaned. Of the rooms that needed to be cleaned but were not, the staff was able on Friday afternoon to clean 3/5 before check-in time. What fraction of the hotel's rooms were cleaned on Friday before check-in time?

A. 11/15
B. 7/9
C. 37/45
D. 13/15
E. 8/9

Given:
1. On Friday morning at a particular hotel, 5/9 of the rooms were cleaned in preparation for weekend visitors.
2. Of the rooms that were not cleaned, 1/3 were planned to be vacant for the weekend and therefore did not need to be cleaned.
3. Of the rooms that needed to be cleaned but were not, the staff was able on Friday afternoon to clean 3/5 before check-in time.

Asked: What fraction of the hotel's rooms were cleaned on Friday before check-in time?

Let the total number of rooms in the hotel be x

1. On Friday morning at a particular hotel, 5/9 of the rooms were cleaned in preparation for weekend visitors.
Rooms cleaned for weekend visitors = 5x/9

2. Of the rooms that were not cleaned, 1/3 were planned to be vacant for the weekend and therefore did not need to be cleaned.
Rooms not cleaned = 4x/9
Of the rooms that were not cleaned, Rooms did not need to be cleaned = 4x/27
Of the rooms that were not cleaned, Rooms needed to be cleaned = 8x/27

3. Of the rooms that needed to be cleaned but were not, the staff was able on Friday afternoon to clean 3/5 before check-in time.
Rooms cleaned before check-in time = (3/5)*(8x/27) + 5x/9= 8x/45 + 5x/9 = 33x/45 = 11x/15

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Re: On Friday morning at a particular hotel, 5/9 of the rooms were cleaned  [#permalink]

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23 Sep 2019, 18:19
Bunuel wrote:
On Friday morning at a particular hotel, 5/9 of the rooms were cleaned in preparation for weekend visitors. Of the rooms that were not cleaned, 1/3 were planned to be vacant for the weekend and therefore did not need to be cleaned. Of the rooms that needed to be cleaned but were not, the staff was able on Friday afternoon to clean 3/5 before check-in time. What fraction of the hotel's rooms were cleaned on Friday before check-in time?

When dealing with problems like this, I like to organize my thoughts using a "Matrix Box", a common strategy on the GMAT useful when you are dealing with two overlapping groups with dichotomous categories. (Notice, we have rooms that are clean or dirty and rooms that are vacant or used, with subtotals and totals for each combination.)

Here is what the blank table might look like:
$\begin{matrix} & \text{Clean} & \text{Dirty} & \text{Totals} \\ \text{Vacant} & \text{____} & \text{____} & \text{____} \\ \text{Used} & \text{____} & \text{____} & \text{____} \\ \text{Totals} & \text{____} & \text{____} & \text{____} \end{matrix}$
Matrix boxes are simplest if we can use concrete numbers. The problem (or the opportunity!) here is that we are only given ratios. Because we don't have any concrete numbers, we can invent our own total amount, using leverage inside the problem to guess what number would work the best. Because of the fractions in the problem, we want a number divisible by both $$9$$ and $$5$$. Normally, using $$45$$ here would work. However, the problem also states that, once we have taken $$5/9$$ of the hotel rooms away, we have to divide the remainder by $$1/3$$. An additional factor of $$3$$ is need to make the numbers pretty. Thus, if we were to pick $$45*3=135$$ as our total value, we wouldn't have to deal with ugly math.

Watch how easy a Matrix Box allows us to organize our data. If 5/9 of the rooms were cleaned, 4/9 weren't. Let's start there:
$\begin{matrix} & \text{Clean} & \text{Dirty} & \text{Totals} \\ \text{Vacant} & \text{____} & \text{____} & \text{____} \\ \text{Used} & \text{____} & \text{____} & \text{____} \\ \text{Totals} & \text{75} & \text{60} & \text{135} \end{matrix}$
1/3 of the dirty rooms were vacant+dirty. Thus, we can determine:
$\begin{matrix} & \text{Clean} & \text{Dirty} & \text{Totals} \\ \text{Vacant} & \text{____} & \text{20} & \text{____} \\ \text{Used} & \text{____} & \text{40} & \text{____} \\ \text{Totals} & \text{75} & \text{60} & \text{135} \end{matrix}$
The problem then states, "Of the rooms that needed to be cleaned but were not..." -- in other words, the dirty+used rooms -- "the staff was able on Friday afternoon to clean 3/5 before check-in time." Thus, $$3/5$$ of the $$40$$ dirty+used rooms were cleaned by Friday afternoon. $$3/5$$ of $$40$$ is $$24$$. Adding this to the rooms already cleaned gives us the total number of rooms cleaned by Friday afternoon: $$75 + 24 = 99$$. The fraction of the total would be:

$$\frac{99}{135} = \frac{9*11}{9*15} = \frac{11}{15}$$

(By the way, for those of you that are interested in additional help on Matrix Boxes, here is a fun little video I put together that explains the "GMAT Jujitsu" of Matrix Boxes: https://www.youtube.com/watch?v=Qa9CpW-mv1c&t=8s. Enjoy!)
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Re: On Friday morning at a particular hotel, 5/9 of the rooms were cleaned  [#permalink]

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19 Nov 2019, 19:22
1
Bunuel wrote:
On Friday morning at a particular hotel, 5/9 of the rooms were cleaned in preparation for weekend visitors. Of the rooms that were not cleaned, 1/3 were planned to be vacant for the weekend and therefore did not need to be cleaned. Of the rooms that needed to be cleaned but were not, the staff was able on Friday afternoon to clean 3/5 before check-in time. What fraction of the hotel's rooms were cleaned on Friday before check-in time?

A. 11/15
B. 7/9
C. 37/45
D. 13/15
E. 8/9

We can let the total number of rooms in the hotel = 270. Thus, 5/9 x 270 = 150 rooms were cleaned on Friday morning, and 120 rooms were not. Since 1/3 x 120 = 40 rooms did not need to be cleaned, 80 rooms needed to be cleaned. Since 3/5 x 80 = 48 rooms were cleaned on Friday afternoon, a total of 150 + 48 = 198 rooms were cleaned on Friday. Therefore, the fraction of rooms cleaned before check-in time was 198/270 = 22/30 = 11/15.

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Re: On Friday morning at a particular hotel, 5/9 of the rooms were cleaned   [#permalink] 19 Nov 2019, 19:22
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