Question

What is the length of the original rectangle?

The perimeter of a rectangle is equal to 40. If the length is halved and the width is divided by 3, the new perimeter is decreased by 24.

Select one:
a. 12
b. 8
c. 4
d. 16

Answer 1

P = Original perimeter = 40

P = 2 w + 2 l = 2 ( w + l )

40 = 2 ( w + l )

Divide both sides by 2

20 = w + l

w + l = 20

Subtract w to both sides

w + l - w = 20 - w

l = 20 - w


If the length is halved and the width is divided by 3 mean:

New lenth l1 = l / 2 = ( 20 - w ) / 2 = 10 - w / 2

New width w1 = w / 3

The new perimeter is decreased by 24 mean:

P1 = New perimeter = 40 - 24 = 16

P1 = 2 w1 + 2 l1 = 2 ( w1 + l1 )

16 = 2 ( w1 + l1 )

Divide both sides by 2

8 = w1 + l1

w1 + l1 = 8

w / 3 + 10 - w / 2 = 8

Subtract 10 to both sides

w / 3 + 10 - w / 2 - 10 = 8 - 10

w / 3 - w / 2 = - 2

2 w / 6 - 3 w / 6 = - 2

- w / 6 = - 2

Multiply both sides by - 6

( - 6 ) ∙ ( - w / 6 ) = ( - 2 ) ∙ ( - 6 )

w = 12

l = 20 - w = 20 - 12 = 8


Proof:

Original perimeter:

P = 2 w + 2 l = 2 ( w + l ) = 2 ∙ ( 12 + 8 ) = 2 ∙ 20 = 40

New lenth:

l1 = l / 2 = 8 / 2 = 4

New width:

w1 = w / 3 = 12 / 3 = 4

New rectangle will be the square. ( the square is a special case of the rectangle )

New perimeter:

P1 = 2 w1 + 2 l1 = 2 ( 4 + 4 ) = 2 ∙ 8 = 16


The length of the original rectangle:

l = 8

Answer b