To find the correct area of the rectangle, we can use the formula for the area of a rectangle, which is:
\[ \text{Area} = \text{Length} \times \text{Width} \]
The length of the rectangle is \( 6 \frac{1}{2} \) cm, which can be converted to an improper fraction:
\[ 6 \frac{1}{2} = 6 + \frac{1}{2} = \frac{12}{2} + \frac{1}{2} = \frac{13}{2} \text{ cm} \]
The width of the rectangle is \( 4 \frac{1}{2} \) cm, which can also be converted to an improper fraction:
\[ 4 \frac{1}{2} = 4 + \frac{1}{2} = \frac{8}{2} + \frac{1}{2} = \frac{9}{2} \text{ cm} \]
Now we can calculate the area:
\[ \text{Area} = \frac{13}{2} \times \frac{9}{2} = \frac{13 \times 9}{2 \times 2} = \frac{117}{4} \text{ square centimeters} \]
Now converting \(\frac{117}{4}\) to a mixed number gives:
\[ \frac{117}{4} = 29 \frac{1}{4} \]
Byron calculated the area as 22 square centimeters, which is incorrect. The error in his reasoning is that he did not correctly calculate the multiplication of the dimensions of the rectangle.
Thus, the correct answer to the problem is:
Byron found the perimeter of the rectangle. The correct area is \( 29 \frac{1}{4} \) square centimeters.