Equal Sum, Product, Quotient of Two Numbers

Date: 10/13/2002 at 20:20:38
From: Liz Ureste
Subject: Math problem

I need to know the sum, product, and quotient of two different numbers 
that are all equal. What are the two numbers?

I don't know how to start the problem.

Date: 10/13/2002 at 22:20:54
From: Doctor Ian
Subject: Re: Math problem

Hi Liz,

Well, if we call the numbers x and y, then we know that the following 
must be true:

  x + y = x * y

  x + y = x / y

  x * y = x / y

What we really want to do is get one of these variables (x or y) by 
itself on one side of an equation, because then we can substitute for 
it in another equation. Let's look at the third equation:

      x * y = x / y

  x * y * y = x
  
      y * y = x / x

        y^2 = 1

          y = 1 or -1

That narrows things down considerably. Now let's look at the first 
equation. Since y must be 1 or -1, one of the following must be true:

  1)  x + 1 = x * 1              

  2)  x - 1 = x * -1

The first one can't be true for any number, since one operation 
changes the value of x and the other doesn't. But what about the 
second equation? Can you find a number that makes it true? 

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