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/
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