A hotel has 80 rooms and charges the same amount per night for each occupied room. This is the hotel's...

Understanding the Question

We need to find: How many nights in September were all 80 rooms occupied?

Given Information

• Hotel has 80 rooms total
• Same price charged per room per night
• September has 30 nights
• Average rooms occupied per night in September = 60

Key Insight

If the average occupancy is 60 rooms per night over 30 nights, then:

• Total room-nights used = \(60 \times 30 = 1{,}800\)
• Total room-nights possible = \(80 \times 30 = 2{,}400\)
• We're 600 room-nights short of full capacity

This means some nights MUST have had fewer than 80 rooms occupied. The question asks us to find exactly how many nights had all 80 rooms occupied.

For a statement to be sufficient, it must narrow us down to exactly ONE value for the number of full nights.

Analyzing Statement 1

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

What This Tells Us

• Full nights: 80 rooms occupied
• Partial nights: 40 rooms occupied on average
• Overall average: 60 rooms per night

Strategic Thinking

This is a weighted average situation. We're mixing:

• Some nights with 80 rooms (full nights)
• Some nights with 40 rooms average (partial nights)
• To get an overall average of 60 rooms

Notice that 60 is exactly halfway between 40 and 80. This suggests an equal split between full and partial nights.

But more importantly: With a fixed total of 1,800 room-nights, and knowing that partial nights average exactly 40 rooms, there's only ONE way to distribute the 30 nights to achieve our constraints.

Think of it this way: If we have x full nights and (30-x) partial nights:

• Full nights contribute: \(80x\) room-nights
• Partial nights contribute: \(40(30-x)\) room-nights
• Total must equal 1,800

This creates a unique constraint with only one solution.

[STOP - Statement 1 is Sufficient!]

This eliminates choices B, C, and E.

Analyzing Statement 2

Now we forget Statement 1 completely and analyze Statement 2 independently.

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.

What This Tells Us

Since the price per room is constant:

• Revenue is directly proportional to the number of room-nights
• Revenue from full nights = 2 × Revenue from partial nights
• Therefore: Room-nights from full nights = 2 × Room-nights from partial nights

Strategic Thinking

We know the total is 1,800 room-nights. If full nights account for twice as many room-nights as partial nights, we're essentially dividing 1,800 into three equal parts:

• Full nights get 2 parts = 1,200 room-nights
• Partial nights get 1 part = 600 room-nights

Since each full night uses exactly 80 rooms:

• Number of full nights = \(1{,}200 \div 80 = 15\) nights

The 2:1 revenue ratio creates a unique constraint. There's only ONE way to split 1,800 room-nights such that one portion is exactly twice the other.

[STOP - Statement 2 is Sufficient!]

This eliminates choices A, C, and E.

The Answer: D

Both statements independently provide enough information to determine exactly how many nights had full occupancy.

Answer Choice D: "Each statement alone is sufficient."