In a three-floor building, floor 2 is above floor 1, and floor 3 is above floor 2. Each floor has two flats: Flat A is to the west of Flat B. Each flat has an area of 576 ft² and a distinct number of rooms. The area of each room in one of the flats on the even-numbered floor is 64 ft². The total number of rooms on the third floor is 7. The flat with rooms of area 72 ft² is on an odd-numbered floor. The flat directly above the flat with 9 rooms has rooms of area 288 ft². There is exactly one floor between the flat with room area 192 ft² and the flat with room area 72 ft². The flat with 6 rooms has no flat to its west.

What is the area of a room in Flat B on the 3rd floor?

Question

In a three-floor building, floor 2 is above floor 1, and floor 3 is above floor 2. Each floor has two flats: Flat A is to the west of Flat B. Each flat has an area of 576 ft² and a distinct number of rooms. The area of each room in one of the flats on the even-numbered floor is 64 ft². The total number of rooms on the third floor is 7. The flat with rooms of area 72 ft² is on an odd-numbered floor. The flat directly above the flat with 9 rooms has rooms of area 288 ft². There is exactly one floor between the flat with room area 192 ft² and the flat with room area 72 ft². The flat with 6 rooms has no flat to its west.

What is the area of a room in Flat B on the 3rd floor?

Accepted Answer

**Solution:**

1. Each flat has an area of 576 ft².

2. The flats are arranged such that Flat A is always to the west of Flat B on each floor.

3. The total number of rooms on the 3rd floor is 7.

4. The flat with room area 72 ft² is on an odd-numbered floor (either 1st or 3rd floor).

5. The flat directly below the flat with 9 rooms has rooms of area 288 ft².

6. There is exactly one floor between the flat with room area 192 ft² and the flat with room area 72 ft².

7. The flat with 6 rooms has no flat to its west, meaning it must be Flat B on its floor.

Let's analyze step-by-step:

- Since the flat with 9 rooms is mentioned, and the flat below it has rooms of area 288 ft², the flat with 9 rooms must be on the 2nd or 3rd floor, and the flat below it on the floor beneath.

- The flat with room area 64 ft² is on an even-numbered floor (floor 2).

- The flat with room area 72 ft² is on an odd-numbered floor (floor 1 or 3).

- The flat with room area 192 ft² is separated by exactly one floor from the flat with room area 72 ft².

- Since floor 2 is even, and floor 3 is odd, the flat with room area 72 ft² could be on floor 3, and the flat with room area 192 ft² on floor 1 (or vice versa).

- The flat with 6 rooms has no flat to its west, so it must be Flat B.

- Total rooms on floor 3 are 7, so Flat A and Flat B together have 7 rooms.

- Since no two flats have the same number of rooms, possible room distributions on floor 3 are (3,4) or (4,3).

- The flat with room area 72 ft² is on floor 3 (odd floor), so Flat A or Flat B on floor 3 has room area 72 ft².

- The flat with 9 rooms is on floor 2 or 3, so the flat below it (floor 1 or 2) has room area 288 ft².

- Using these clues, the area of each room in Flat B on floor 3 is:

$$	ext{Area of each room} = \frac{	ext{Total area of flat}}{	ext{Number of rooms}} = \frac{576}{4} = 144 	ext{ ft}^2$$

**Answer:** 144 ft²

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