To solve the problem, we need to find the area of the shaded region, which is the frame between the larger rectangle and the smaller rectangle within it. Here's how we can break it down:

Given:

• The width of the frame is 1 unit.
• The dimensions of the outer rectangle are 9 units (width) and 6 units (height).

Step-by-Step Solution:

1. Outer Rectangle Area: The area of the larger (outer) rectangle is calculated by multiplying its width by its height: $\text{Outer Area} = 9 \times 6 = 54 \text{ square units.}$

2. Dimensions of the Inner Rectangle: The inner rectangle is framed by a 1-unit wide border on all sides. Therefore, the dimensions of the inner rectangle will be:

   • Width: $9 - 2(1) = 7$ units (since 1 unit is subtracted from both sides of the width).
   • Height: $6 - 2(1) = 4$ units (since 1 unit is subtracted from both sides of the height).

3. Inner Rectangle Area: Now, calculate the area of the inner rectangle: $\text{Inner Area} = 7 \times 4 = 28 \text{ square units.}$

4. Shaded Region Area (Frame): The area of the shaded region is the difference between the outer rectangle's area and the inner rectangle's area: $\text{Shaded Area} = 54 - 28 = 26 \text{ square units.}$

Thus, the area of the shaded region is $\boxed{26}$ square units.

Would you like more details on any of the steps, or have any questions?

Related Questions:

1. What happens if the frame width is increased to 2 units? How would the area of the shaded region change?
2. If the outer rectangle’s dimensions were changed to 10 by 8 units, what would be the new shaded area with a 1-unit frame?
3. Can you calculate the perimeter of the inner rectangle based on the dimensions provided?
4. How would you solve this if the frame width were variable instead of fixed at 1 unit?
5. If the dimensions of the outer rectangle are not given, but the frame area is, how can you deduce the dimensions of the outer rectangle?

Tip:

Always ensure that you subtract twice the frame width when calculating the inner dimensions because the frame exists on both sides of the rectangle.