On Friday morning at a particular hotel, 5/9 of the rooms...

We know that on Friday morning, 5/9 of the rooms were cleaned. So the number of rooms that were clean at check in equals 5/9 + whatever fraction of rooms were cleaned on Friday afternoon.

We know that on Friday morning 4/9 of the rooms were not cleaned. However of those rooms, 1/3 did not need to be cleaned. This means that 2/3 of the 4/9 of the rooms that were not cleaned DID still need to be cleaned. 2/3 * 4/9 = 8/27. So 8/27 of the rooms DID still need to be cleaned.

Then we learn that of the rooms that DID still need to be cleaned, 3/5 were cleaned on Friday afternoon. This means that 3/5 of the 8/27 rooms that DID still need to be cleaned were cleaned on Friday afternoon. 3/5 * 8/27 = 1/5 * 8/9 = 8/45. So 8/45 of the rooms were cleaned on Friday afternoon.

So we add the 5/9 that were cleaned Friday morning to the 8/45 of the rooms that were cleaned on Friday afternoon = 5/9 + 8/45 = 25/45 + 8/45 = 33/45 = 11/15.

AAPL 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

The OA is A.

I don't have clear this PS question. Please, can any expert explain it for me? Thanks.

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.

Answer: A