A rectangle has a length 6 more than its width. If the width is decreased by 2 and the length decreased by 4, the resulting rectangle has an area of 21 square units.

Question

A rectangle has a length 6 more than its width. If the width is decreased by 2 and the length decreased by 4, the resulting rectangle has an area of 21 square units. What is the length of the original rectangle? What is ratio of the original rectangle’s area to the new rectangle’s area? What is the perimeter of the new rectangle?

Answer 1

If we take the original width to be W then the original length is W+6 (6 more than the width).

In the new rectangle the width becomes W-2 and the new length becomes (W+6-4) which simplifies to W+2.

We know that the area of the new rectangle is 21 units^2 so we can substitute the values of the new width and length into the equation for area of a rectangle:

area = width*length

21=(W-2)*(W+2)

Expanding this gives:

21 = W^2 -2W + 2W -4

Simplified this becomes:

21 = W^2 - 4

25 = W^2

W = +/- 5

Given that this is a real rectangle we can take +5 as the value of W which is the width of the original rectangle.

Going back to our initial expression for the original length (W+6) we can substitute the value of W (5+6) to give an original length of 11.

Using the same equation for area and substituting the value for the initial rectangle gives us the area of the original rectangle.

area = width*length

area_original = 5*11

area_original = 55

Using the area of the original rectangle which we just calculated and the area of the new rectangle which we were given in the problem we can set up a ratio:

area_original / area_new = 55/21

Finally, we can use the values we found for the original rectangle to calculate the value of the width and length of the new rectangle which we can plug into the equation for perimeter:

Width_new = W-2

Width_new = 5-2

Width_new = 3

Length_new = W+2

Length_new = 5+2

Length_new = 7

Perimeter = 2*width + 2*length

Perimeter_new = 2*Width_new + 2*Length_new

Perimeter_new = 2*3 + 2*7

Perimeter_new = 6 + 14

Perimeter_new = 20