Question

zahra has paper rectangles of different sizes. every rectangle is 5 centimeters longer than it is wide. is there a proportional relationship between the lengths and widths of these rectangles? explain.

Explanation:

Step1: Let width be \( w \)

Let the width of the rectangle be \( w \) centimeters. Then the length \( l=w + 5 \) centimeters.

Step2: Check proportionality

For two rectangles with widths \( w_1,w_2 \) and lengths \( l_1=l_1+5,l_2=l_2 + 5 \). If they were proportional, \(\frac{l_1}{w_1}=\frac{l_2}{w_2}\). Substitute \( l_1=w_1 + 5 \) and \( l_2=w_2 + 5 \). \(\frac{w_1 + 5}{w_1}=1+\frac{5}{w_1}\) and \(\frac{w_2 + 5}{w_2}=1+\frac{5}{w_2}\). Since \(\frac{5}{w_1}
eq\frac{5}{w_2}\) (unless \( w_1 = w_2 \), but rectangles are of different sizes), \(\frac{l_1}{w_1}
eq\frac{l_2}{w_2}\)

Answer:

No, there is no proportional relationship between the lengths and widths of these rectangles. Because if \(l = w+5\), for two different widths \(w_1\) and \(w_2\) (\(w_1
eq w_2\)), \(\frac{l_1}{w_1}=1 + \frac{5}{w_1}\) and \(\frac{l_2}{w_2}=1+\frac{5}{w_2}\) and \(\frac{5}{w_1}
eq\frac{5}{w_2}\) so \(\frac{l_1}{w_1}
eq\frac{l_2}{w_2}\)