Is there a faster way to add/subtract fractions then having to draw a factor tree each time? Do you really have to draw a factor tree and work with primes every time you encounter adding or subtracting fractions? 
Not this way - LCM(8,15)...
15: 15, 30, 45, 50, 65, *80* --
8: 8, 16, 24, 32, 40, 48, 56, 64, 72, *80* -- 

This makes adding and subtracting fractions quite a lot of work. 
What is the most efficient and effective practice in regards to dealing with adding or subtracting fractions? Is there a faster way to add or subtract fractions? I heard of the "Butterfly Method" but it involves a lot of rules. The factor tree seemed easier. I came here to see if determining the least common denominator of two fractions can be done even more efficiently.

We don't need the least common multiple to add fractions.
But if you want the least common multiple (lcm) of $x$ and $y$, where $x$ and $y$ are BIG, first use the Euclidean Algorithm to find the greatest common divisor $\gcd(x,y)$ efficiently. Then use the fact that $\operatorname{lcm}(x,y)=\frac{xy}{\gcd(x,y)}$.
For very large numbers, this is far more efficient than factoring using the best currently known algorithms. But for smallish familiar numbers, factoring works well.
