Question

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.
What is the length of the original rectangle?
4 8 16 12

Answer 1

Answer:

8

Step-by-step explanation:

  M = Original perimeter = 40

   M = 2 w + 2 l = 2 ( w + l )

          40 = 2 ( w + l )

Divide both sides by 2 (simplify)

               20 = w + l

               w + l = 20

Subtract w from both sides

       w + l - w = 20 - w

            l = 20 - w

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

          New length 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 length:

       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 2