Reducing Fractions with Large Numbers

Date: 01/30/98 at 09:37:30
From: Pat
Subject: Reducing fractions

Dear Dr. Math,

A neighborhood child asked for help with a math problem involving 
reducing fractions. The numbers involved were relatively high (i.e. 
1742/4395). Is there a "trick" that would allow me to go through the 
least amount of possibilities before declaring that 4,5,6,7,8,9, digit 
number fractions may or may not be reduced?

I would also like to know if you might suggest a book that shows some 
math tricks that would help to make math a "cool" and enjoyable 
exercise for children and parents alike. I feared and disliked math 
until I took statistics and then I regretted not realizing how much 
fun it could be. I would like to pass this concept on to my eight-
year-old while he is still young.   

Thank you,
P.M.

Date: 01/30/98 at 12:10:17
From: Doctor Rob
Subject: Re: Reducing fractions

Indeed, there is a trick, and it is never taught at the elementary 
school level. It is called "Euclid's Algorithm". The idea is to find 
the largest common factor of the numerator and denominator. It goes 
like this.

Take the larger of the two numbers, and divide it by the smaller, 
getting a quotient and remainder. If the remainder is zero, the answer 
is the smaller of the two numbers. If the remainder is not zero, 
replace the larger number by this remainder, and repeat the above.

Example:  52740 and 20904.

   52740 = 2*20904 + 10932, replace 52740 by 10932.
   20904 = 1*10932 + 9972, replace 20904 by 9972.
   10932 = 1*9972 + 960,
    9972 = 10*960 + 372,
     960 = 2*372 + 216,
     372 = 1*216 + 156,
     216 = 1*156 + 60,
     156 = 2*60 + 36,
      60 = 1*36 + 24,
      36 = 1*24 + 12,
      24 = 2*12 + 0,

so the answer is 12, and 52740/12 = 4395, 20904/12 = 1742, so, reduced 
to lowest terms, 20904/52740 = 1742/4395.

This always works, and always gives the largest common factor.  
Furthermore, as you can see from the example, it doesn't take very 
many steps. In the worst case it takes five times the number of digits 
in the smaller number, but usually less than half that many.

For a book, try _Mathematics and the Imagination_.  There are lots of 
books about recreational mathematics which might be appropriate.

-Doctor Rob,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/