Weird Fraction Behavior?

Date: 04 Nov 1994 08:41:21 -0000
From: Jeff Neill
Subject: Weird Precalculus Question

If you look at the fractions  (16/64) and (19/95), you may notice that 
if you cancel out the second number in the numerator with the first 
number in the denominator the fraction remaining is equivalent to that 
of the original equation. Ex. in the fraction (16/64) if you cancel out 
the second number in the numerator (6) with the first number in the 
denominator (6), you end up with (1/4), which is equal to (16/64). 
The only restrictions are that the numbers canceling must be the same 
number, as in the above example (a 6 for a 6). Also the numbers for the 
original fraction are restricted to two digits (10-99). How many more 
of these numbers can you find? If you write a program, could you please 
send it to me? I am very interested in finding out how this would be done.

Jeff Neill

From: Dr. Ken
Date: Fri, 4 Nov 1994 12:49:10 -0500 (EST)

Jeff!

Essentially, what you've got here is a very special case of the relationship
between a number and its base ten representation.  It's a neat problem.

You can model the situation with the following equation.

    n    10n + x
    -  = _______ 
    d    10x + d

The right side of the equation is the value of the fraction before your
"cancellation", and the left side of the equation reflects what you get
after you cancel.  If you solve for x in terms of n and d, you'll get a
general formula for the kinds of numbers you can use.  Enjoy, and let us
know what you come up with!

-Ken "Dr." Math