Question

The ratio of the length to the width of a rectangle is 5 to 2. Express the width of the rectangle as a fraction of the perimeter of the rectangle. I am not sure how to help my son do the problem.

Answer 1

The perimeter of a triangle is given by the following expression:

$P=2\cdot(w+l)_{}$

Where P is the perimeter, w is the width and l is the length.

We know that the ratio of the length to the width is 5 to 2. This means that for every 5 units of the length there are 2 units of the width, therefore:

$5\cdot l=2\cdot w$

If we isolate the "l" variable on the left, we have:

$l=\frac{2\cdot w}{5}$

We can replace the expression above on the formula for the perimeter.

$P=2\cdot(\frac{2\cdot w}{5}+w)_{}$

We now need to simplify the right side of the equation and isolate the "w" variable.

$\begin{gathered} P=\frac{4w}{5}+2w \\ P=\frac{4w+10w}{5} \\ P=\frac{14w}{5} \\ w=\frac{5P}{14} \end{gathered}$

The width is equal to 5/14 of the perimeter.