Ben and Bill

Date: 12/21/97 at 10:21:53
From: SHAMALA
Subject: Algebra

Bill + Ben's Age = 91
Bill is twice as old as Ben was when Bill was as old as Ben is now.

I've tried to form an equation but I get stuck in the first phrase. 
Please help.

Thank you,
Shamala

Date: 01/06/98 at 13:32:06
From: Doctor Dawn
Subject: Re: Algebra

Dear Shamala,

Wow! This was a tough question and had me stumped for a while.  At 
first I was trying to assign Ben a variable and Bill a different 
variable, but I couldn't figure out what to put down for the second 
sentence. Since the difference in their ages is constant, I finally 
decided to assign the difference in their ages a variable. I also know 
that Bill is older. Here goes:

First, define the variables

     x = difference in Bill's and Ben's ages
     b = Ben's age

This means

     b + x = Bill's age

and 

     Bill - x = Ben's age.  (This will be important later)   

Next write the English sentences as math equations using the 
variables.

        Bill + Ben = 91
       (b + x) + b = 91
   so       2b + x = 91.      (Equation 1)

Now for the second sentence -

  Bill is twice as old as Ben was, when Bill was as old as Ben is now.

The first Bill is now in the present and is represented by b + x

The second Bill is in the past when he was Ben's age now which is b.
Ben in the past was x years younger than Bill in the past, so Ben in 
the past is represented by: 

       Bill (past) - x = b - x.

The second equation then goes like this.

   Bill  =  twice as old as Ben was 
 (b + x) =  2   *   (b - x)   (Equation 2)

Now take both equations and solve. 

  2b + x = 91
   b + x = 2*(b - x)

Working with equation 2:

     b+x = 2b-2x    (distribute the 2)
      3x = b        (add 2x  and subtract b on each side)

Substituting 3x for b into equation 1:   
 
    2*3x + x = 91
      6x + x = 91
          7x = 91
           x = 13

The difference in their ages is 13.

Using equation 1 again, and knowing x = 13:        

    2b + 13 = 91
                                                                                    
         2b = 78
                                                                                      
          b = 39 (Ben's age)

       Bill = b + x = 39 + 13 = 52.

Bill is 52 years old; Ben is 39 years old. To check,  when Bill was 39 
(Ben's age now), Ben was 26. 52 is twice 26.

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