A rectangular red carpet measures 6 ft by 12 ft. It has a dark red border of uniform width 6 inches all around the edges. What is the area (in square feet) of this dark red border?

Introduction / Context:
This aptitude question involves finding the area of a border around a rectangle. It tests your comfort with unit conversion (inches to feet) and with the idea that the area of a border equals the difference between the outer and inner rectangles.

Given Data / Assumptions:

• Outer rectangle (the entire carpet) has length 12 ft and width 6 ft.
• The border all around is 6 inches wide.
• 1 foot = 12 inches, so 6 inches = 0.5 ft.
• The inner rectangle (central part without border) is smaller by this border width from all sides.
• We seek the area of the border region in square feet.

Concept / Approach:
The border forms a frame around a central rectangle. The area of the border is: Area of border = Area of outer rectangle − Area of inner rectangle We first compute the outer area from the original dimensions. Then we subtract twice the border width from each dimension to find the inner dimensions, compute the inner area and finally subtract.

Step-by-Step Solution:
Step 1: Convert the border width to feet. Given 6 inches, and 12 inches = 1 ft, we get 6 inches = 0.5 ft. Step 2: Outer rectangle dimensions: length L = 12 ft, width W = 6 ft. Step 3: Outer area A_outer = L * W = 12 * 6 = 72 square feet. Step 4: The border is 0.5 ft wide on each side, so total reduction in length is 2 * 0.5 = 1 ft, and total reduction in width is also 1 ft. Step 5: Inner rectangle dimensions: length = 12 − 1 = 11 ft, width = 6 − 1 = 5 ft. Step 6: Inner area A_inner = 11 * 5 = 55 square feet. Step 7: Area of dark red border = A_outer − A_inner = 72 − 55 = 17 square feet.

Verification / Alternative check:
We can think of the border as a union of four rectangles around the central area, but computing their individual areas is more tedious and will still sum to 17 square feet. Since the difference of areas approach is straightforward and produces a positive, reasonable result, it confirms that the answer is correct.

Why Other Options Are Wrong:
9 square feet and 15 square feet: These are too small given that the border runs all around a fairly large carpet. They come from incorrect inner dimensions or missing part of the border area.
18 square feet: This is close but still off by 1 square foot and arises from mishandling one of the dimensions or forgetting that both sides shrink by twice the border width.
12 square feet: This would correspond to shrinking only in one direction or other incomplete reasoning.

Common Pitfalls:
Many students forget to convert inches to feet and attempt to mix units, which leads to wrong areas. Another common issue is subtracting the border width only once instead of twice (left and right, or top and bottom). Always remember that both dimensions are reduced on two sides when there is a uniform border.

Final Answer:
The area of the dark red border is 17 square feet.