# need help w/ 'The length of a rectangular piece of cardboard

#### frenzy_252003

##### New member
The length of a rectangular piece of cardboard is 2 cm greater than its width. If the length and the width were each decreased by 1 cm, the area of the cardboard would be decreased by 27 square cm. What are the dimensions of the original piece of cardboard?

thanks a lot..

#### soroban

##### Elite Member
Re: need help with word problem..

Hello, frenzy_252003!

Make sketches and baby-talk your way through it . . .

The length of a rectangular piece of cardboard is 2 cm greater than its width.
If the length and the width were each decreased by 1 cm,
the area of the cardboard would be decreased by 27 cm².
What are the dimensions of the original piece of cardboard?

In the original rectangle, the length is 2 more than the width.
Let $$\displaystyle W$$ = width, then $$\displaystyle W+2$$ = length.
Code:
      * - - - - - - - - - - - *
|                       |
|                       |
W |                       |
|                       |
|                       |
* - - - - - - - - - - - *
W + 2

The area is: $$\displaystyle \:W(W\,+\,2)\:=\:W^2\,+\,2W$$ cm².

The new rectangle has its length and width decreased by 1.
Its length will be: $$\displaystyle \,(W\,+\,2)\,-\,1\:=\:W\,+\,1$$
Its width will be: $$\displaystyle \,W\,-\,1$$
Code:
      * - - - - - - - - *
|                 |
W - 1 |                 |
|                 |
* - - - - - - - - *
W + 1

Its area will be: $$\displaystyle \,(W\,+\,1)(W\,-\,1)\:=\:W^2\,-\,1$$ cm².

The area of the smaller rectangle is 27 cm² less than the original rectangle.

. . $$\displaystyle W^2\,-\,1\;=\;(W^2\,+\,2W)\,-\,27$$

And there is our equation!