Regarding +/- fractions: what are some mental tests you can apply to uncommon fraction denominators?

Question

When adding and subtracting fractions: what if there is no uncommon factor (for example 4=2,2 and 6=2,3). Does that always mean to use the LCM? What if the LCM is too big or time consuming to calculate. That means you factor them, yes? Your mind should know this at a glance, right? Last thing you'd want is to waste time writing all the multiples of a number if there are over 6 of them. Also, what if one denominator is prime and the other isn't (ex: 7 and 8). Does that mean you should use the criss-cross "Butterfly Method" described on Youtube? Sorry if I am posting too many questions. Does it sound like I am progressing on this subject and is there anything else I need to know? What are mental tests you can apply to the denominators to decide which path to take?

Answer 1

When adding and subtracting fractions, you can use whatever common denominator you want. And yes, the LCM is the smallest possible such denominator. So for example (using the denominators from your question): \begin{align*} \frac{1}{4} + \frac{5}{6} &= \frac{1\cdot 3}{12} + \frac{5\cdot 2}{12} = \frac{3}{12} + \frac{10}{12} = \frac{13}{12} \\ \frac{2}{7} + \frac{3}{8} &= \frac{2\cdot 8}{7\cdot 8} + \frac{3\cdot 7}{7\cdot 8} = \frac{16}{56} + \frac{21}{56} = \frac{37}{56}. \end{align*} But again, there is nothing magic about using the LCM. In the first example, if you wanted to use a common denominator of $24$ and just cross-multiply to find the numerators, that would work fine: $$\frac{1}{4} + \frac{5}{6} = \frac{1\cdot 6}{4\cdot 6} + \frac{5\cdot 4}{4\cdot 6} = \frac{6}{24} + \frac{20}{24} = \frac{26}{24} = \frac{13}{12}.$$