Question

7 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: Define proportion

A proportion is a statement that two ratios are equal. For two rectangles to have a proportional relationship between lengths and widths, the ratio of length to width for one rectangle should be equal to the ratio of length to width for the other rectangle.
Let the width of the rectangle be \(w\) and the length be \(l = w + 5\). The ratio of length to width is \(\frac{l}{w}=\frac{w + 5}{w}=1+\frac{5}{w}\).

Step2: Check if ratio is constant

If we have two rectangles with widths \(w_1\) and \(w_2\) (\(w_1
eq w_2\)), the ratios of their lengths to widths are \(\frac{w_1 + 5}{w_1}=1+\frac{5}{w_1}\) and \(\frac{w_2+ 5}{w_2}=1+\frac{5}{w_2}\). Since \(w_1
eq w_2\), \(\frac{5}{w_1}
eq\frac{5}{w_2}\), so \(1+\frac{5}{w_1}
eq1+\frac{5}{w_2}\).

Answer:

No, there is no proportional relationship between the lengths and widths of these rectangles. Because if we assume the width of a rectangle is \(w\) and length is \(l=w + 5\), the ratio of length to width \(\frac{l}{w}=\frac{w + 5}{w}=1+\frac{5}{w}\). For two different rectangles with widths \(w_1\) and \(w_2\) (\(w_1
eq w_2\)), the ratios \(\frac{w_1 + 5}{w_1}\) and \(\frac{w_2+ 5}{w_2}\) are not equal.