The Method of False Position

Date: 02/06/2004 at 23:37:09
From: Lynne
Subject: Math Riddle

There is a quantity such that 2/3 of it, 1/2 of it, and 1/7 of it
added together becomes 33.  What is the quantity?  Solve the problem
by the method of false position.

I know that using the method of false position, we are looking for 
the root of an equation, and need a way of making a guess that is 
better than our previous guess.

Date: 02/07/2004 at 14:59:35
From: Doctor Douglas
Subject: Re: Math Riddle

Hi Lynne -

Thanks for writing to the Math Forum.

Let's let 'x' be the unknown quantity, and we can write the following
equation:

  x*(2/3) + x*(1/2) + x*(1/7) = 33

Assume x = 42.  This is a convenient first guess because it is
divisible by all of the denominators {3,2,7}, which makes our life a 
bit easier on the first step.

  42*(2/3) + 42*(1/2) + 42*(1/7) = 28 + 21 + 6 = 55.

So our first guess of 42 is approximately too big by a factor of 
55/33 = 5/3.  Our next guess is therefore

  42*(3/5) = 126/5 or 25.2.

We plug this, our second guess, in for x in the equation above 
to obtain 

  (126/5)*(2/3) + (126/5)(1/2) + (126/5)(1/7) = 84/5 + 63/5 + 18/5
        = 165/5 = 33

which is exactly what we want it to be, and we've therefore found
the value of x for which the equation is true.  There is of course a
more direct method to solve the original equation using algebra
techniques, but I think that this problem shows you how the method of
false position works so that you can also apply it to cases where you
cannot simply solve for x.

- Doctor Douglas, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 02/07/2004 at 15:53:54
From: Lynne
Subject: Thank you (Math Riddle)

Thank you!

Date: 02/09/2004 at 09:46:13
From: Doctor Peterson
Subject: Re: Thank you (Math Riddle)

Hi, Lynne.

Just a quick note to add on to Dr. Douglas's reply.  I assume you are
aware that this method was used in ancient Egypt.  See this page:

  The Egyptians' Method of False Position
    http://mathforum.org/library/drmath/view/62036.html 

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/