Is there a faster way to add/subtract fractions then having to draw a factor tree each time?

Question

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.

Answer 1

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.

Answer 2

Google or Youtube search: The fast way to adding and subtracting fractions. https://www.youtube.com/watch?v=lKiR4BBt6TE . Number 3 has like factors. Cancel them out yields your result.