Lesson Eleven: In Which We Do Not Add Apples to Oranges

Suppose you have one bushel of apples, then someone gives you another two apples, how many apples do you have. Although 1+2 = 3, you certainly wouldn't answer that you have three apples. This is because "bushels of apples" and "apples" are different things. However, if you know that every bushel of apples contains twenty apples, then you know that having a bushel of apples and having twenty apples are the same thing, so if you have a bushel of apples, and you get two more apples, you have twenty apples and two more, or 20+2 = 22 apples.

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.