How many small squares with sides 1/2 inch long are needed to completely cover a square region that measures 4 ft by 4 ft?

Introduction / Context:
This question connects unit conversion with area concepts. We have a large square region measured in feet, and we want to know how many smaller squares measured in inches are needed to cover it without overlap or gaps. This is a classic tiling problem that reinforces the idea that the number of tiles equals the ratio of total area to the area of one tile when there is perfect coverage.

Given Data / Assumptions:

• Large square side length = 4 ft.
• Small square side length = 1/2 inch.
• Squares cover the region exactly with no overlap and no uncovered space.

Concept / Approach:
We first convert the large square side from feet to inches because the tile side is given in inches. Then we can either compute the areas and divide, or more simply count how many small squares fit along each side and then square that count, since the region is also a square. The key conversion is:
1 ft = 12 inches.

Step-by-Step Solution:
Step 1: Convert the side of the large square. Side length = 4 ft = 4 * 12 = 48 inches. Step 2: Side of each small square = 1/2 inch. Step 3: Number of small squares along one side of the large square. Count along a side = 48 / (1/2) = 48 * 2 = 96. Step 4: Since the region is square, there are also 96 small squares along the perpendicular side. Total small squares = 96 * 96 = 9216.

Verification / Alternative check:
Using areas: area of large square = 4 ft * 4 ft = 16 ft^2. In square inches, 1 ft^2 = 144 in^2, so area = 16 * 144 = 2304 in^2. Area of one small square = (1/2 in)^2 = 1/4 in^2. The number of tiles is 2304 / (1/4) = 2304 * 4 = 9216. This matches the earlier computation, confirming the result.

Why Other Options Are Wrong:
10246, 12345 and 7527 are arbitrary large numbers that do not correspond to any natural square of an integer number of tiles along each side.
4096 equals 64^2 and would correspond to only 64 tiles along each side, which would happen if the side were 32 inches instead of 48 inches, so it is inconsistent with the given dimensions.

Common Pitfalls:
Students sometimes convert 4 ft incorrectly or forget that 1/2 inch tiles give twice as many along each side compared to 1 inch tiles. Others mistakenly divide area in feet squared by area in inches squared without proper conversion, leading to incorrect results. Working consistently in one unit system and deciding whether to use side lengths or areas at the start helps avoid confusion.

Final Answer:
A total of 9216 squares with side 1/2 inch are needed to cover the 4 ft by 4 ft square region.