Question

A square has 6 1/2 cm width and 4 1/2 cm length. Byron calculated the area of the rectangle as 22 square centimeters. What is the error in his reasoning and what is the correct answer?
A. Bryon found the perimeter of the rectangle. The correct area is 24 1/4 square centimeters.
B. Bryon found the perimeter of the rectangle. The correct area is 22 square centimeters.
C. Bryon found the perimeter of the rectangle. The correct area is 29 1/4 square centimeters.
D. Byron found the perimeter of the rectangle. The correct area is 11 square centimeters

Answer 1

To find the area of a rectangle, you multiply the width by the length. In this case, the width is \(6 \frac{1}{2}\) cm and the length is \(4 \frac{1}{2}\) cm.

First, let's convert the mixed numbers to improper fractions:
- Width: \(6 \frac{1}{2} = \frac{13}{2}\) cm
- Length: \(4 \frac{1}{2} = \frac{9}{2}\) cm

Now, calculate the area:

\[
\text{Area} = \text{Width} \times \text{Length} = \left(\frac{13}{2}\right) \times \left(\frac{9}{2}\right)
\]

Calculating this gives:

\[
\text{Area} = \frac{13 \times 9}{2 \times 2} = \frac{117}{4} \text{ square centimeters}
\]

Now, convert \(\frac{117}{4}\) to a mixed number by performing the division:

\[
117 \div 4 = 29 \text{ remainder } 1 \Rightarrow 29 \frac{1}{4}
\]

So, the correct area of the rectangle is \(29 \frac{1}{4}\) square centimeters.

Now to analyze Byron's calculation:

- By saying he calculated the area as 22 square centimeters, he must have made an error in his calculation.
- It is likely that he confused the calculation of area with another calculation (possibly the perimeter).

To check all the answer choices:

A. Byron found the perimeter of the rectangle. The correct area is \(24 \frac{1}{4}\) square centimeters. (Incorrect)

B. Byron found the perimeter of the rectangle. The correct area is \(22\) square centimeters. (Incorrect)

C. Byron found the perimeter of the rectangle. The correct area is \(29 \frac{1}{4}\) square centimeters. (Correct)

D. Byron found the perimeter of the rectangle. The correct area is \(11\) square centimeters. (Incorrect)

The correct answer is **C**. Byron found the perimeter of the rectangle. The correct area is \(29 \frac{1}{4}\) square centimeters.