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 algorithm. 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 \(\frac{1}{3}+\frac{1}{6}\). The LCD = 6, so the sum becomes \(\frac{2}{6}+\frac{1}{6}=\frac{3}{6}\) or \(\frac{1}{2}\). This is reducible.

Now consider \(\frac{1}{3}+\frac{1}{2}\). The LCD = 6, so the sum becomes \(\frac{2}{6}+\frac{3}{6}=\frac{5}{6}\). This is clearly not reducible.

I finally found an article explaining the theory behind when a fraction sum is reducible or irreducible. Please try our exploration tools before checking out number theory details.

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Did you really Explore some denominators first?

It's way more fun to explore first.

But if you insist, here's why.