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A Simple Formula?


Date: 02/21/97 at 19:19:53
From: Ross Mannell
Subject: Why a simple formula works

When looking at a maths problem of the form, "If one person takes two 
hours to paint a wall and another only one hour, how long will it take 
the two to paint the wall together?" I have been told that this simple 
formula gives the result:

   a + b
   -----
   a * b

I have wondered why this is so. Would the following apply for three 
workers?

   a + b + c
   ---------
   a * b * c

Forgive my ignorance, but I am plagued by the need to know.

Ross Mannell


Date: 02/22/97 at 03:09:39
From: Doctor Mike
Subject: Re: Why a simple formula works

Hi Ross,   

If the a and b of your formula mean how many hours it takes each 
person, like a = 2 and b = 1, then the formula gives 3/2 or 1 hour and 
30 minutes.  This is not reasonable, since one of the painters alone 
could do it in 1 hour. 

Actually, your formula is UPSIDE DOWN. It should be (a*b)/(a+b). 
Here's why.
  
Let h be the number of hours it takes for the two of them to do it
together. The first person could do it all in a hours, but can paint 
only a fraction of the wall working h hours. Assuming a constant rate 
of painting, that fraction is h/a. That is, the first painter will 
paint h/a of the wall in h hours.  

Similarly, the second painter will paint h/b of the wall in that time. 
So, when will they be done? When the whole wall is painted, which 
happens whenever those 2 fractions add up to 1.  

You need to solve the following equation:

  (h/a) + (h/b) = 1  
      h*b + h*a = a*b           Multiplying through by a*b
        h*(b+a) = a*b           Factoring out h
              h = (a*b)/(a+b)   Dividing both sides by b+a

Okay, now say 3 painters could do it in a hours, b hours or c hours, 
respectively. Let h be the number of hours they all must work to get 
it all painted. The first painter finishes h/a of the wall in that 
amount of time. The other 2 painters manage to paint h/b of the wall 
and h/c of the wall in that time.  Since h is the time to finish the 
job, all 3 of these fractions must add up to the whole wall, or 1.  
So you have the equation:
     
        h     h     h         
       --- + --- + ---  =  1 
        a     b     c    
  
Solve this for h (start by multiplying through by a*b*c) and you will 
get the generalized formula for 3 painters. It's not the formula you 
wrote above but I think you can finish it off.
  
I hope this helps.      

-Doctor Mike,  The Math Forum
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