What is the least number of square tiles of side 41 cm required to pave the ceiling of a rectangular room that is 15 m 17 cm long and 9 m 2 cm broad?

Introduction / Context:
This question is about tiling a rectangular surface with square tiles of a fixed size. You are asked to find how many such tiles are needed to cover the ceiling exactly, without cutting any tile. It tests conversion of units, understanding of area and the idea that the number of tiles equals total area divided by area of one tile when dimensions are compatible.

Given Data / Assumptions:

- Length of the room = 15 m 17 cm.
- Breadth of the room = 9 m 2 cm.
- Each tile is a square with side 41 cm.
- Tiles are laid without gaps and without cutting, so the room dimensions are assumed to be exact multiples of 41 cm.
- We need the least number of tiles required to cover the ceiling completely.

Concept / Approach:
First, convert all dimensions into the same unit, preferably centimetres. Then compute the area of the ceiling (which has the same dimensions as the floor). Next, compute the area of a single square tile. If the room dimensions are exact multiples of the tile side, the number of tiles will be the product of the number of tiles along the length and along the breadth, which is also equal to the total area divided by the tile area.

Step-by-Step Solution:
Step 1: Convert room dimensions to centimetres.Step 2: Length = 15 m 17 cm = 15 * 100 + 17 = 1500 + 17 = 1517 cm.Step 3: Breadth = 9 m 2 cm = 9 * 100 + 2 = 900 + 2 = 902 cm.Step 4: Side of one square tile = 41 cm, so its area = 41 * 41 sq. cm.Step 5: Check how many tiles fit along the length: 1517 / 41 = 37, because 41 * 37 = 1517.Step 6: Check how many tiles fit along the breadth: 902 / 41 = 22, because 41 * 22 = 902.Step 7: Therefore, total number of tiles = 37 * 22 = 814.

Verification / Alternative check:
You can compute total ceiling area directly: Area_ceiling = 1517 * 902 sq. cm. Number of tiles should equal Area_ceiling divided by 41^2. Since both 1517 and 902 are exact multiples of 41, the result is an integer, which we already found to be 37 * 22. Performing this multiplication again confirms that the total number of tiles is 814 and no partial tiles are needed.

Why Other Options Are Wrong:
Values like 656, 738, 902 and 960 do not match the product of integer counts along each dimension. They can appear if someone divides the area roughly or rounds the room dimensions or miscomputes 1517 / 41 or 902 / 41. Since the dimensions divide exactly, any answer other than 814 is inconsistent with the exact tiling requirement.

Common Pitfalls:
Common mistakes include failing to convert metres to centimetres correctly, leading to wrong multiples, or approximating division instead of checking for exact divisibility. Some students directly divide total area by tile area without verifying that the result is an integer, which could allow a non-whole number of tiles. Careful unit conversion and exact arithmetic are crucial in tiling problems.

Final Answer:
The least number of square tiles required is 814.