Fraction Addition and Reducible Sums

Fraction Sum Exploration

A common method for adding fractions is to find the lowest common multiple of the denominators. This is called the LCD algorithm, Lowest Common Denominator. Even though we have the smallest possible denominator that allows us to combine the fractions, sometimes the sum will still reduce. I have always wondered why.

Consider `1/3+1/6`. The LCD = 6, so the sum becomes `2/6+1/6=3/6` or `1/2`. This is reducible.

Now consider `1/3+1/2`. The LCD = 6, so the sum becomes `2/6+3/6=5/6`. This is clearly not reducible.

I finally found an article explaining the theory behind when a fraction sum is reducible or irreducible.

Biscuits of Number Theory by Arthur T. Benjamin, Ezra Brown, 2006, MAA

Originally published as:
Reducing the Sum of Two Fractions
Shultz, Harris S. and Ray C. Shiflett. Mathematics Teacher. vol. 98, no. 7 (March 2005): pp.486-490.

If the denominators have a commom prime power in the prime factorizations, then some sums are reducible, otherwise, the sums are always irreducible. For denominators `12=2^2*3` and `15=3*5` there exist numerators for which the sum will reduce since they have a common prime power of `3^1`, and, for example, `1/12+1/15=9/60 = 3/20`. For denominators `8=2^3` and `12=2^2*3` there are no numerators for which the sum reduces, since they do not have a common prime power. Here I am assuming that the numerators are relatively prime to their denominators, i.e., they have a gcd of 1.

Even better, if the common prime power is based on 2, then the sums are always reducible. Otherwise, there exist numerators for which the sum will reduce. The problem then becomes finding those numerators.

10 vs 5 1 2 3 4
1 3/10 1/2 7/10 9/10
3 1/2 7/10 9/10 11/10
7 9/10 11/10 13/10 3/2
9 11/10 13/10 3/2 17/10

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