Sum of a Number and Its Reciprocal

Date: 01/28/2002 at 20:21:16
From: Rebeka Lee
Subject: Fractions (solving brain teaser)

The sum of a number and its reciprocal is 5 1/5.  What is the number?

I've tried working the problem out through the guessing technique, 
but I am nowhere near the answer.

Date: 01/28/2002 at 21:23:15
From: Doctor Ian
Subject: Re: Fractions (solving brain teaser)

Hi Rebeka,

Let's begin by giving the number a name, like 'n'. The reciprocal of n 
is 1/n. So the sum of the number and its reciprocal is n + 1/n.

The problem tells us that this sum is equal to 5 1/5, so

  n + 1/n = 5 + 1/5


In this form, it seems pretty obvious that the number must be 5, 
doesn't it?  But suppose we want to take no chances.  Then we can use 
'algebra' to smoke out the number:

     n^2/n + 1/n = 25/5 + 1/5           Get common denominators.
  
     (n^2 + 1)/n = (25 + 1)/5           Add the fractions.

      5(n^2 + 1) = n(25 + 1)            Cross multiply denominators.

        5n^2 + 5 = 26n                  Simplify.

  5n^2 - 26n + 5 = 0                    Subtract 26n from both sides.

 (5n - 1)(n - 5) = 0                    Factor.

Now, in this form we can see that either (n-5) must be zero, in which 
case n must be 5; or (5n - 1) is zero, in which case n must be 1/5.  

Now that's interesting, because what _wasn't_ obvious from looking at

  n + 1/n = 5 + 1/5

is that 

  1/(1/5) = 1 * (5/1)                   To divide by a fraction,
                                        invert and multiply. 

So in fact, 

  (1/5) + 1/(1/5) = 1/5 + 5

                  = 5 1/5
                  
This is actually one of the nice things about the algebraic approach.  
Lots of times, it can show you that there are more solutions than you 
might otherwise have thought to search for!

Does this help? 

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