Bedbugs and Beds

Date: 09/14/97 at 21:41:57
From: adi mulzer
Subject: Bedbugs and Beds

Please help me. I can't figure out this problem - I don't even know 
where to begin.

The happy Holiday hotel is blessed with cheerful bedbugs. In each 
single bed you can find 7 bedbugs, and in each double bed 13 bedbugs. 
If there are 106 bedbugs in all, how many double beds are there?

Could you please tell me how you got your answer? My teacher always 
wants me to show my work.

Thank you!

Adi Mulzer

Date: 09/15/97 at 09:12:18
From: Doctor Anthony
Subject: Re: Bedbugs and Beds

In this type of problem you should introduce letters to represent 
unknown quantities. This allows you to write down equations using the 
information given in the question. Finally, you solve the equations to 
find actual values for the unknown quantities.

Let x = number of double beds and  y = number of single beds.

Then   13x + 7y = 106

            13x = 106 -7y

                   106 - 7y
              x = ----------
                     13

Now let y range through the values 1, 2, 3, .... until you find a 
value for 106-7y which is exactly divisible by 13. There are not many 
numbers between 13 and 106 that are multiples of 13, so the search 
will be easy.

Multiples of 13 are 13, 26, 39, 52, 65, 78, 91, 104

values of 106-7y  for y = 0, 1, 2, 3, ... are

                 106, 99, 92, 85, 78, ...

The number 78 appears in both lists, so  x = 78/13 = 6   y = 4

So there are 6 double beds and 4 single beds.

(Check.  Number of bugs = 13 x 6 + 7 x 4 = 106)

-Doctor Anthony,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/