Prime Factorization and Reducing Fractions

Continuing our discussion about the prime factorization, we focus on using it to reduce fractions containing large numbers.  In high school math, this application – happily – has resurfaced.

You can read about prime factorization in my Oct 14 post.  Let’s assume you’re old hat at it now.

We are familiar with reducing fractions.  For instance,

40/35=8/7

We know that the reason for the reduction is that 5 divides into both 40 and 35, so can be cancelled out top and bottom:

40/35=(8*5)/(7*5)=8/7

Of course, it’s easy to tell that 40/35 reduces to 8/7.  However, what about reducing a fraction like 98/154?  Well, you can cut both down by 2, then realize 7 goes into both, and so on.  However, let’s try the prime factorization method.  First, we’ll rewrite the fraction with each number’s prime factorization:

98/154=(2*7*7)/2*7*11)

Note that 2*7 occurs in both the top and the bottom.  It’s called the greatest common factor (the GCF), since it’s the largest combination that occurs in both prime factorizations.  Cancel the GCF top and bottom and you get the reduced fraction:

(2*7*7)/(2*7*11)=7/11

Like with finding the LCM (see my Oct 17 post), the prime factorization method for reducing fractions becomes more useful as the numbers get bigger and less familiar.  Try it and see :)

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

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