Video: Writing and Solving a System of Linear Equations in Two Unknowns in Word Problems Involving Ratios

Given that the perimeter of a rectangle equals 72 cm and the ratio between the lengths of two of its sides is 5 : 4, determine its area.

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Video Transcript

Given that the perimeter of a rectangle equals 72 centimeters and the ratio between the lengths of two of its sides is five to four, determine its area.

Let’s consider the rectangle as shown with length π‘₯ centimeters and width 𝑦 centimeters. We are told that its perimeter is equal to 72 centimeters. And the perimeter of a rectangle is the distance around the outside. As the opposite sides of a rectangle are parallel and equal in length, we have the equation π‘₯ plus 𝑦 plus π‘₯ plus 𝑦 is equal to 72. Grouping or collecting like terms, this equation simplifies to two π‘₯ plus two 𝑦 is equal to 72.

We can divide both sides of the equation by two, giving us π‘₯ plus 𝑦 is equal to 36. We are also told in the question that the ratio between the lengths of the two sides is five to four. Therefore, π‘₯ and 𝑦 are in the ratio five to four. We now know the sum of the two quantities and the ratio between them.

We can calculate one part of the ratio by dividing the sum of the quantities by the number of parts. In this case, one part or one share is equal to 36 divided by nine. This is equal to four. Our next step is to multiply each of the parts by four. Five multiplied by four is equal to 20. And four multiplied by four is equal to 16. This means that the lengths π‘₯ and 𝑦 are 20 centimeters and 16 centimeters, respectively.

We were asked to determine the area of the rectangle. We do this by multiplying the length by the width. Two multiplied by 16 is equal to 32. Therefore, 20 multiplied by 16 is 320. The area of the rectangle is 320 square centimeters.

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