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

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?

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: $$\frac{1}{4} + \frac{5}{6} = \frac{1\cdot 6}{4\cdot 6} + \frac{5\cdot 4}{4\cdot 6} = \frac{6}{24} + \frac{20}{24} = \frac{26}{24} = \frac{13}{12}.$$

• So I can always use just the common denominator? That is really contrary to what all these Youtube tutorials and online guides teach. The Youtube video by MrMcGlover taught this cross multiplication method, but I always see other guides stressing the lowest common denominator. This makes it confusing. – kinesis Oct 16 '14 at 21:42
• You can use any denominator that is a multiple of both of the denominators in question. The LCM is the least of these, but as you point out, it may be simpler just to cross-multiply to avoid having to find the LCM. As you can see above, they both give the same answer. And the reason is that, for example, $\frac{6}{24} = \frac{3}{12}$ in the above --- using a larger common denominator simply means you are multiplying numerator and denominator by a larger number, which of course does not change its value. – rogerl Oct 16 '14 at 21:51
• SO ask yourself "Can I easily determine the LCM within 5-6 multiples?", if not, then cross multiply. Do I ever have to worry about the 'factor tree' or factoring them down to primes (ex: 24 = 2*3*2*2 or 150 = 2*3*5*5) This is another thing I see video tutorials talking about. Khan Academy says it is used to get a LCD. Is it ever needed or helpful? In algebra I know they factor fractions down too. Is the concept more related to algebra than fractions in general? I ran into a situation where cross multiplying yielded an improper fraction, does this mean convert it to proper? – kinesis Oct 17 '14 at 2:03
• It is used to get an LCM, yes. But again, the LCM is just one possible common denominator. You really should work lots of examples (just make them up; pick some interesting-looking denominators) to really understand how this works, and to convince yourself that the LCM is not the only choice. – rogerl Oct 17 '14 at 2:05
• And whether a fraction is proper or improper is irrelevant. – rogerl Oct 17 '14 at 2:05