The length of a rectangle is five inches less than twice its width. If the rectangle has a perimeter of 80 inches, find the length and width

Question

I don't understand how to turn this word problem into an equation. I really need help. Thank you very much!

Answer 1

Thanks for your question!  Let's step through it by breaking down the sentence.
 
But first we need to know the formula to find the perimeter of a rectangle.  It is:
 
                  2*length            +           2*width                  =         perimeter
 
         two times the length   plus   two times the width    equals   the perimeter
 
--- Based on that formula, let's use "2x" to represent the width

--- Next, in the sentence it says "the length... is five inches less than twice the width"; we can show that as:
 
                                              twice its   five inches
                                                width          less
                                         2*    (2x      -       5)
 
     ---- to simplify this, we have to use the algebraic concept of distribution which says to multiply everything in the parenthesis by "2".  When we do, we get:
 
                                                    (4x - 10) 
 
--- Now let's write a math "sentence" to match the word sentence above:
 
                             the length         the width        the perimeter
                              (4x - 10)     +       2x        =        80
 
--- from here we will solve for "x" (aka the width) first:
 
     --- we do so by combining the "x"s in the equation
 
                       (4x - 10)  +  2x  =  80
 
     --- let's rearrange things to see it easier
 
                          4x  + 2x - 10  =  80
 
     ----- we can now sum (add) the "x" numbers to get

                                   6x - 10  =  80
 
     ----- we now reduce the equation by adding "10" to each side (as the left side has "- 10" showing)
 
                        6x - 10  +  10  =  80 + 10
 
      ----- this leaves us with
                                        6x  =  90
    
      ----- to further solve for "x" we will need to divide both sides by "6" (as currently we have 6*"x", aka 6x)
 
                                        6x  =  90
                                         6   =  6
 
                                        x   = 15    - width of the rectangle
 
We now know that the width of the rectangle is 15.  Now let's find the length!
 
To do that we will need to use our little handy equation we built above:
 
              the length          the width        the perimeter                                        
              2*(2x - 5)     +      2*x        =        80
 
----- from that formula we need "2*(2x - 5) and can use our new width number of 15 to substitute in for "x":
 
               the length                                                             
               2*(2x - 5)      =  ?    
               2*(2(15) - 5)  = ?
               2*(30 - 5)      = ?    
               2*(25)           = ?
                               50  = ?          = the length
 
We can check our numbers by putting them all together:
 
         the length        the width        the perimeter
               50       +        30         =         80
 
Voila! The numbers add up!

Answer 2

Read the quesiton carefully.  The word "is" can be translated as equal ( = ).  Therefore "length is" means length equals five less than twice the width (or 2 X width minus 5).

 

Set your variables:

L = length

W = Width

 

Equation 1:  L = (2*W) - 5

Equation 2:  (2*L)+(2*W) = 80

 

move 2W to the other side in equation 1 and add equation 1 and 2 together (you can also graph this and look for the intersect or use substition).

 

L - 2W = -5

2L + 2W = 80

 

3L = 75

L = 25

Plug into either equation and solve for W:

2(25) +2(W) = 80;  W = 15.

 

Plug values into equation 1 and check your work:

25 - 2*15 = -5

 

Tim

 

Answer 3

Let w = width

and length = l = 2w - 5

then perimeter = 2 * (l+w) = 2 * (2w-5 + w) = 6w - 10 = 80

6w = 90

w = 15 in

l = 2w -5 = 25 in

Answer 4

Hi Cassidy--

 

Let's call the width w, and let's call the length "2w - 5" for "five inches less than twice its width."

 

You know that the perimeter is the sum of all 4 sides, so in this case

 

w + w, plus "2w-5" plus the other "2w - 5" = 80 inches.

 

If you add up all the w's, there are 6 of them (w + w + 2w + 2w), and subtract the two 5's (because each one is "minus 5"),

 

you get 6w - 10 = 80, which means...

 

6w = 90, which means

 

w (the width) is 15 inches.

 

to get the length, you do 2w - 5, which is 2(15) - 5, which is 30 - 5, which is

 

length = 25.

 

To check it, it's 2 widths plus 2 lengths, and they should equal 80

 

15 + 15 + 25 + 25 = 80.

 

WOO HOO!

 

Hope this helps.

 

Matt