A hotel has 80 rooms and charges the same amount per night for each

Question

A hotel has 80 rooms and charges the same amount per night for each occupied room. This is the hotel's only source of revenue. In September, the average (arithmetic mean) number of rooms occupied per night was 60. How many nights in September were all the rooms in the hotel occupied?

(1) On the nights when not all the rooms of the hotel were occupied, the average number of rooms occupied was 40.

(2) In September, the total revenue for the nights when all the rooms of the hotel were occupied was twice the revenue for the nights when not all the rooms were occupied.

chetan2u · Accepted Answer

A hotel has 80 rooms and charges the same amount per night for each occupied room. This is the hotel's only source of revenue. In September, the average (arithmetic mean) number of rooms occupied per night was 60. How many nights in September were all the rooms in the hotel occupied?
 Let the amount per night be x, so total revenue is 60*30*x (average*number of days in September * charge per night)
 
1. On the nights when not all the rooms of the hotel were occupied, the average number of rooms occupied was 40.
That is on night when full booked, 80 rooms are occupied and when not fully booked, average of rooms is 40. 
Hence, the overall average 60, right in Center of the two constituent averages 40 and 80, tells us that the number of days of these averages is same. So 15 days for fully occupied and remaining 15 for certain rooms empty. 
Sufficient

2. In September, the total revenue for the nights when all the rooms of the hotel were occupied was twice the revenue for the nights when not all the rooms were occupied.
Revenue when fully booked = 80x per day. 
If z days were fully occupied, revenue = 80xz
The remaining days it is half of this, so 40xz. 
Total revenue = 80xz+40xz=120xz=1800x
120z=1800…..z=15
Sufficient again

D

egmat · Answer

Here is a  detailed video solution  for this interesting question.
See how this question can be solved in the test environment using " Owning the Dataset " approach.

https://www.youtube.com/watch?v=SBR3UHJtVZM

Let us know if you have any questions.

gmatophobia · Answer

September has 30 days.

Maximum number of bookings possible in the month of September= 80 * 30 = 2400

Actual number of bookings in the month of September= 60 * 30 = 1800

Statement 1

1. On the nights when not all the rooms of the hotel were occupied, the average number of rooms occupied was 40.

Let's assume out of 30 days, all the rooms of the hotel were booked for n days.

Number of bookings made when all the rooms of the hotel were booked = 80*n
Number of bookings made when all the rooms of the hotel were not booked = 40*n

80n + 40n = 1800

120n = 1800

n = 15

The statement is sufficient and we can eliminate B, C, and E.

Statement 2

2. In September, the total revenue for the nights when all the rooms of the hotel were occupied was twice the revenue for the nights when not all the rooms were occupied.

Let's assume out of 30 days, all the rooms of the hotel were booked for n days.
The charge for each night = $x

Revenue when all the rooms of the hotel were booked = 80*n*x
Revenue when all the rooms of the hotel were not booked = 40*n*x
Total Revenue = 120nx

From the question premise, we know that a total of 1800 bookings were made in September. Hence, revenue in September = 1800 * x

120nx = 1800x

As x is not equal to zero, we can divide both sides by x.

120n = 1800

n = 15

This statement is also sufficient.

Option D

parthwagh · Answer

We know that the hotel has 80 rooms and that the average occupancy rate for September was 60 rooms per night. Since September has 30 days, the total number of room-nights occupied over the month was 60 rooms/night * 30 nights = 1800 room-nights.

Statement (1): On the nights when not all the rooms of the hotel were occupied, the average number of rooms occupied was 40.

If we let y be the number of nights when not all rooms were occupied, then the average occupancy on these nights would contribute 40y room-nights.
On the x nights when all rooms were occupied, there would be 80x room-nights.
The total number of room-nights for the month is 1800, so we have:
40y + 80x = 1800

Since we know the average is 60, we can infer that half the nights had to be fully occupied to achieve this average because the average of 40 and 80 is 60. This means x = y, and since there are 30 nights in September, we have x + y = 30. If x = y, then 2x = 30, which means x = 15.

So statement (1) is sufficient alone because it allows us to conclude that 15 nights were fully occupied.

Statement (2): In September, the total revenue for the nights when all the rooms were occupied was twice the revenue for the nights when not all the rooms were occupied.

Let the revenue per room per night be R.
For the x nights when all rooms were occupied, the revenue would be 80xR.
For the y nights when not all rooms were occupied, the revenue would be 40yR (from the previous statement, we know the average occupancy was 40 on these nights).
According to this statement:
80xR = 2 * 40yR
80xR = 80yR
x = y

Again, since x = y and there are 30 nights in September, we have x + y = 30, which means 2x = 30, and therefore x = 15.

So statement (2) is also sufficient alone because it allows us to conclude that 15 nights were fully occupied.

Thus, the answer is D, as each statement alone provides sufficient information to determine the number of nights in September that all the rooms were occupied.

gmat2324 · Answer

I don't understand how statement 2 is sufficient - how is it that 40 is number of rooms booked when not fully occupied

chetan2u · Answer

gmat2324

It is no where taken that for 40 days rooms were not fully occupied. You get 40 after taking out revenue for fully occupied.

Statement 2 says that when fully occupied revenue is twice the revenue when not fully occupied days are combined.

When fully occupied, revenue is 80*x per day where x is the revenue per day. 
Now, when we take that the room was occupied for z days, so revenue generated is (80*x)*z. 
Since revenue for days not fully occupied is HALF, the revenue for those days becomes (80*x)*z/2 or 40*x*z.

When both are combined, the total revenue becomes 80xz+40xz or 120xz.
This should be equal to avg number of rooms * number of days * rent per day = 60*30*x = 1800x

Thus 120xz=1800x or z=15

ISHAAN96 · Answer

Hi,

For S1 shouldnt

Number of bookings made when all the rooms of the hotel were booked = 80*n
Number of bookings made when all the rooms of the hotel were not booked = 40* (30-n)?

In the explanation both have been taken as "n".

Thanks.

tunglamym · Answer

I believe there is an error in your solution. When solving for answer choice B, you assumed that the average number of rooms occupied on the days when the hotel is not fully booked is 40. This assumption was derived from answer choice A

azulonx · Answer

You are correct.

Number of bookings made when all the rooms of the hotel were not booked = 40* (30-n)

TOTAL NO. OF ROOM BOOKED: total No. of rooms occupied WHEN NOT FULLY BOOKED + Total No. of rooms occupied when FULLY BOOKED. 
80n + 1200 - 40n=1800
N= 15

azulonx · Answer

I believe you can find the answer to the problem with weighted averages and without any of the statements (n=15).

Is there an issue with this problem? I saw this problem in the gmat prep test too and was confused because I got the answer without Statement 1 or 2.
@bunnuel please your guidance.  KarishmaB

KarishmaB · Answer

Hotel has 80 rooms and there are 30 nights in Sept. 
Average number of rooms occupied per night was 60. What this means is that if I were to write the number of occupied rooms on each night, I would get data which could look like this:
... 39 rooms, 49 rooms, 50 rooms, 60 rooms, 63 rooms ... 80 rooms, 80 rooms, 80 rooms (there will be 30 values here, one for each night)
The avg here is 60.

Question: How many nights in September were all the rooms in the hotel occupied?
I need additional data for this. The average can be 60 in many many ways. 
e.g.
40, 40, 40 ... (15 times) ... 80, 80, 80 (15 times)
40, 60, 60, 60 ... (28 times), 80
etc. 
 azulonx

(1) On the nights when not all the rooms of the hotel were occupied, the average number of rooms occupied was 40.

There are two types of nights -
when all rooms were occupied i.e. 80 rooms occupied;
when not all rooms were occupied - we have been given that on all these nights exactly 40 rooms were occupied.

Then the entire room night data has only 2 numbers - 40 and 80. For average to be 60, the data has to be exactly this:
40, 40, 40 ... (15 times) ... 80, 80, 80 (15 times)

So all rooms were occupied for exactly 15 nights.

Sufficient alone.

(2) In September, the total revenue for the nights when all the rooms of the hotel were occupied was twice the revenue for the nights when not all the rooms were occupied.

On average 60 rooms were occupied on each night and assuming revenue of $1 per room per night, we get revenue of $1800 for all 30 nights. 
So revenue for nights when all 80 rooms were occupied was $1200 and revenue when not all rooms were occupied was $600.

In how many nights will we get revenue of 1200 when all 80 rooms are occupied? In 1200/80 = 15 nights. 
So all rooms were occupied for exactly 15 nights.

Sufficient alone.

Answer (D)

FGMATS · Answer

Thanks for the explanation! I got this wrong on my mock exam because I made an oversight of the word "for the nights" in statement 2! So I thought, we have 2 variables (# of days that were fully booked, and #of booked rooms on the nights that were not fully booked).

HarshavardhanR · Answer

https://gmatclub.com/forum/download/file.php?id=83857 An algebraic approach to solve this question. --- Harsha GMAT-Club-Forum-4u9g4p15.png

VaibhavKulkarni · Answer

In the video by eGMAT, statement (2) was solved incorrectly. 
It relied on the data from statement (1).

After many tries, this is how I went about the question, in an easier-to-understand manner.

No. of days in Sept. = 30
Let the no. of days with  full (80) occupancy = n 
Therefore, the no. of days with  <80 occupancy = 30-n

Given:   
Avg. = (D1+D2+...+D30)/30 = 60

Splitting the numerator above into;
sum of days with full occupancy ( n(80) ) + sum of days with less than 80 occupancy ( ∑(D<80) ).

Therefore; 
 n(80) + ∑(D<80) = 1800------------------------(i)

Q: The question is asking for  n ?

Statement (1): 
 ∑(D<80)/(30-n) = 40
 Therefore,  ∑(D<80) = 40*(30-n)---------------(ii)
 
Substituting (ii) in (i), we get  n=15 .

Eliminate B, C, & E.

Statement (2):

Assume cost per room per night =  x

Statement (2) Given:
 x*(80)(n)=2*x*∑(D<80)
 =>  40(n)=∑(D<80)----------------------------(iii)
 
Substituting (iii) in (i), we get  n=15 .

Therefore, D is the answer.

VaibhavKulkarni · Answer

In the video by eGMAT, statement (2) was solved incorrectly. 
It relied on the data from statement (1).

After many tries, this is how I went about the question, in an easier-to-understand manner.

No. of days in Sept. = 30
Let the no. of days with  full (80) occupancy = n 
Therefore, the no. of days with  <80 occupancy = 30-n

Given:   
Avg. = (D1+D2+...+D30)/30 = 60

Splitting the numerator above into;
sum of days with full occupancy ( n(80) ) + sum of days with less than 80 occupancy ( ∑(D<80) ).

Therefore; 
 n(80) + ∑(D<80) = 1800------------------------(i)

Q: The question is asking for  n ?

Statement (1): 
 ∑(D<80)/(30-n) = 40
 Therefore,  ∑(D<80) = 40*(30-n)---------------(ii)
 
Substituting (ii) in (i), we get  n=15 .

Eliminate B, C, & E.

Statement (2):

Assume cost per room per night =  x

Statement (2) Given:
 x*(80)(n)=2*x*∑(D<80)
 =>  40(n)=∑(D<80)----------------------------(iii)
 
Substituting (iii) in (i), we get  n=15 .

Therefore, D is the answer.

A hotel has 80 rooms and charges the same amount per night for each

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Data Sufficiency (DS) Questions

­A hotel has 80 rooms and charges the same amount per night for each

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

A hotel has 80 rooms and charges the same amount per night for each