Similarly, if you are asked to add m(1/3)+n(1/2), you are being asked to add different things. You have m "thirds" of something and are adding n "halves," so you cannot immediately add them together. Fortunately, the same factor can have many different ways of being written.
The Many Forms of 1:
Remember that n(1/n) = 1, this means that n/n = 1. This should make a certain amount of sense, because if I split something into 5 pieces, and I have 5 of those pieces, I should have the whole thing. However, since multiplying by 1 does not change a number m*1 = m*(n/n) = (mn)/n = m.
Although we only stated that an integer times one remains the same, you should feel free to verify that (m/n)*1 = m/n. Thus (m/n)*(a/a) = (ma)/(na) = m/n. Sometimes this is called cancellation, as you can think of the a in the numerator and the a in the denominator cancelling each other out. More practically, this gives us a way to add together fractions which, at first, seem to be counting different things.
Adding Fractions:
To add fractions together you MUST have the same denominator, that way the fractions are counting the same sort of thing. So, how do we arrange for 1/3 and 1/2 to have the same denominator? The key is the commutativity of multiplication, we already know that 2*3 = 3*2. In order to make sure not to change the value of 1/3 we can multiply it by 1, but we can write 1 as 2/2, so 1/3 = (1/3)*(2/2) = (1*2)/(3*2) = 2/6. We can do the same thing to 1/2, multiplying it by 3/3 instead, so 1/2 = (1/2)*(3/3) = (1*3)/(2*3) = 3/6. Notice how we made use of the fact that 2*3 = 3*2?
So, if we return to the original problem of adding m(1/3)+n(1/2), we now know that m(1/3)+n(1/2) = m(2/6)+n(3/6) = (m*2)(1/6) + (n*3)(1/6). Since both of our fractions are counting the same thing, in this case sixths, we can successfully add them together to get (2m+3n)(1/6) = (2m+3n)/6. As long as you remember to only add fractions whose denominators match, so you are adding "similar things," and that you can make denominators match by multiplying by a clever form of 1, you will have a long and successful career adding together fractions.
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