Question

the area of a rectangle is 3,429 square centimeters. the width of the rectangle is \\(\frac{5}{8}\\) times the area. what is the length, in centimeters, of the rectangle?

Explanation:

Step1: Recall the formula for the area of a rectangle

The area \( A \) of a rectangle is given by \( A = \text{length} \times \text{width} \), so \( \text{length} = \frac{A}{\text{width}} \).

Step2: Calculate the width of the rectangle

The width is \( \frac{5}{8} \) times the area. The area \( A = 3429 \) square centimeters. So the width \( w=\frac{5}{8}\times3429=\frac{5\times3429}{8}=\frac{17145}{8} \) centimeters.

Step3: Calculate the length of the rectangle

Using the formula for length, \( \text{length}=\frac{A}{w} \). Substitute \( A = 3429 \) and \( w=\frac{17145}{8} \) into the formula:
\[
\text{length}=3429\div\frac{17145}{8}=3429\times\frac{8}{17145}
\]
Simplify the fraction. Notice that \( 3429\times2 = 6858 \), \( 17145\div5 = 3429 \). So \( \frac{3429\times8}{17145}=\frac{8}{5}=1.6 \)? Wait, no, let's do the division properly. \( 3429\div17145=\frac{3429}{17145}=\frac{1}{5} \) (since \( 17145\div3429 = 5 \)). Then \( \frac{1}{5}\times8=\frac{8}{5} = 1.6 \)? Wait, that can't be right. Wait, no, let's re - check.

Wait, the area is \( A = l\times w \), so \( l=\frac{A}{w} \). \( w=\frac{5}{8}A \), so \( l=\frac{A}{\frac{5}{8}A}=\frac{8}{5}=1.6 \)? Wait, the \( A \) cancels out? Wait, that's a key point. If \( w = \frac{5}{8}A \), then \( l=\frac{A}{w}=\frac{A}{\frac{5}{8}A}=\frac{8}{5}=1.6 \) centimeters. Wait, let's verify with the numbers. \( A = 3429 \), \( w=\frac{5}{8}\times3429 = \frac{17145}{8}=2143.125 \). Then \( l=\frac{3429}{2143.125}=1.6=\frac{8}{5} \). Yes, because \( 3429\div2143.125=\frac{3429\times8}{2143.125\times8}=\frac{27432}{17145}=\frac{8}{5}=1.6 \).

Answer:

The length of the rectangle is \(\frac{8}{5}\) (or \(1.6\)) centimeters.