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.

• $45+15=60\neq50$. $8$ and $15$ clearly have no common factors, so their LCM is $8\cdot15=120$. More often than not, when adding fractions, it is easier to just use product instead of LCM, and cancel common factors in the end. – Jyrki Lahtonen Oct 15 '14 at 8:50
• I caught that mistake. What do you mean by product? What do you mean by cancelling common factors? Are there any tutorials or videos that demonstrate this in action? Is this video on target? youtube.com/watch?v=kZUImLklXHk It says an easy way is to multiply the two denominators to determin a "common denominator" but not "least common denominator". I'm left not really understanding it thus far (only at +5:49 into the video) – kinesis Oct 15 '14 at 8:59

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 $\operatorname{lcm}(x,y)=\frac{xy}{\gcd(x,y)}$.