Q&A #7499

Curious about fractions

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From: Pat Ballew (for Teacher2Teacher Service)
Date: Dec 12, 2001 at 01:49:43
Subject: Re: Curious about fractions

  I'm not sure it has an "official" name, but I call it the "three arrow 
method".  When I do it I draw an arrow upward from the right denominator to 
the left numerator  (this way \) and another from the left denominator to 
the right numerator (this way/) and one across through the two denominators 
(this way  --).  It also works for subtraction by the way... 

15              8
   3         2     15-8     7
  --   -   ---- =  ----  = ----
   4         5      20      20  

The earliest record I can find of this (with the three arrows as straight 
lines drawn between the values to be multiplied) is by Nicholas Chuquet in 
the fifteenth century.  It was used quite regularly in the early arithmetics 
such as that by Recorde.  

WHY it works??? If the two denominators are relatively prime, then their 
product is the least common denominator and the "other denominator" is the 
multiplier to make an equivalent fraction.  [20/4 = 5 and 20/5 = 4 ]

If they are not, you just have to reduce as a final step.  In the U.S. we 
often emphasize finding the Least Common Multiple to be the common 
denominator, but ANY common multiple will work, it just forces you to simplify 
the final answer to get it in simplest terms.  The easy way to find a common 
multiple of two numbers is to multiply them against each other, thus making 
each the co-factor of the others numerator to express the fractions in a 
common denominant unit.  

The nice thing for students is that it also works when you get to algebra 
and beyond and have to add stuff like

   2       +    3x
 -----         -----     =  
  (x-1)         (x+1)  

Hope that helps... good luck

 -Pat Ballew, for the T2T service

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