### Video Transcript

A square and a rectangle have the same length perimeters, and the length of the rectangle is five times its width. What is the ratio between the area of the rectangle and the area of the square in its simplest form?

Letβs firstly consider a square of side length π₯ and a rectangle of width π€ and length π. The perimeter of any shape is the distance around the outside. And, each side of a square is the same length. This means that the perimeter of the square is equal to π₯ plus π₯ plus π₯ plus π₯. This is equal to four π₯. We are told that the length of the rectangle is five times its width. Therefore, π is equal to five π€. The perimeter of the rectangle is therefore π€ plus five π€ plus π€ plus five π€. This is equal to 12π€.

We are also told that the perimeter of both shapes are equal. This means that four π₯ is equal to 12π€. Dividing both sides of this equation by four gives us π₯ is equal to three π€, as four π₯ divided by four is equal to π₯, and 12π€ divided by four is equal to three π€. This means that the side length of the square is equal to three times the width of the rectangle. We can replace each of the lengths π₯ on the square with three π€.

We were asked to work out the ratio between the area of the rectangle and the area of the square. The area of the square is equal to three π€ multiplied by three π€. This is equal to nine π€ squared. Three multiplied by three is equal to nine. And, π€ multiplied by π€ is π€ squared. The area of the rectangle is equal to π€ multiplied by five π€. This equals five π€ squared.

It is important that we write the ratio in the correct order. We are asked for the ratio of the rectangle to the area of the square. Therefore, the rectangle must come first. The ratio is five π€ squared to nine π€ squared. We need to give our answer in its simplest form. We do this by dividing both sides by π€ squared. Five π€ squared divided by π€ squared is equal to five. And, nine π€ squared divided by π€ squared is equal to nine. The ratio of the area of the rectangle to the area of the square in its simplest form is five to nine.