Question

The perimeter of a rectangle is 70cm. If its length is decreased by 5Cm and its width is increased by 5cm, its area will increase by 50cm2. find the length and the width of the original rectangle?

Answer 1

`L=25cm and W=10cm`

Explanation:

Let `L and W` be the original length and width respectively:
`P=2L+2W=70 cm`
`2W=70-2L`
`W=35-L`
`A=L*W`
`(L-5)(W+5)= L*W+50`
`(L-5)(35-L+5)=L*(35-L)+50`
`(L-5)(40-L)=35L-L^2+50`
`40L-L^2-200+5L=35L-L^2+50`
`45L-200=35L+50`
`10L=250`
`L=25 cm`
`W=10 cm`

Answer 2

Length: 25 cm
Width:  10 cm

Explanation:

Let me show you how to make a Guess Chart

1) Here is how to limit the number of guesses:

• The length has to be a number greater than 5.
• Probably it is a whole number.
• The most likely length is a multiple of 5.
• L + W must add up to 35.

2) So try these numbers for guessing Length`:` `10, 15, 20, 25, 30`

Length .|.. Width. . |. . Area  || L - 5 |. W + 5 | Area + 50
⚊⚊⚊⚊|⚊⚊⚊⚊⚊|⚊⚊⚊⚊||⚊⚊⚊|⚊⚊⚊⚊|⚊⚊⚊⚊⚊⚊⚊⚊⚊⚊
. . . 10 . .| . . . 25 . . .| . .250 . .||. . .5. .| . . 30. . .| 150 `larr` not 300
⚊⚊⚊⚊|⚊⚊⚊⚊⚊|⚊⚊⚊⚊||⚊⚊⚊|⚊⚊⚊⚊|⚊⚊⚊⚊⚊⚊⚊⚊⚊⚊
. . . 15. . | . . . 20 . . .| . .300. ..|| . 10 . |. . 25 . . | 250 `larr` not 350
⚊⚊⚊⚊|⚊⚊⚊⚊⚊|⚊⚊⚊⚊||⚊⚊⚊|⚊⚊⚊⚊|⚊⚊⚊⚊⚊⚊⚊⚊⚊⚊
. . . 20 . .| . . ..15 . .. | . .300 . .|| . 15 . |. . .20. . | 300 `larr` not 350
⚊⚊⚊⚊|⚊⚊⚊⚊⚊|⚊⚊⚊⚊||⚊⚊⚊|⚊⚊⚊⚊|⚊⚊⚊⚊⚊⚊⚊⚊⚊⚊
`color(red)( . . .25. .| . . . 10. . .| . .250 . .|| . .20 .|.. . 15 . | 300` `larr` `250 + 50!`
⚊⚊⚊⚊|⚊⚊⚊⚊⚊|⚊⚊⚊⚊⚊⚊⚊⚊⚊⚊⚊⚊⚊⚊⚊⚊⚊⚊⚊⚊⚊⚊⚊
. . .30 . . | . . . . . . . . | . . . . . .

• Don't bother to try 30. You have already found the right answer.

• This chart generated the right answer in less than 3 minutes, with no math except for some extremely simple computations.

• There was no need to set up an equation.

• In actual practice, once I saw that 10×25 was way too small, I jumped ahead to 25×10 and just landed on the right answer

• If the right answer doesn't just fall out of a Guess Chart right away, then resort to an equation.
  .........................................

Here is an equation you can use if the Guess Chart isn't working for you:

Set up an equation

(L - 5)(W + 5) = LW + 50

1) FOIL out the binomials
LW + 5L - 5W - 25 = LW + 50

2) Subtract LW from both sides
5L - 5W - 25 = 50

3) Add 25 to both sides
5L - 5W = 75

4) Divide all the terms on both sides by 5
L - W = 15

5) Add this to L + W = 35 `larr` from perimeter

. . L + W = 35
+ L -  W = 15
———————
 2L. . . . . =  50

6) Divide both sides by 2

L = 25 `larr` same answer as the Guess Chart