 What about the subtraction of fractions? For example, how about this subtraction? To begin with, remember that the subtraction A minus B can be read as from A, remove B, leaving A minus B. So how you speak influences how you think, so we can read this as from 812, remove 312. Well, if I have 8 of something and I remove 3 of the same thing, I should have 5 left, and so I should end up with 512. Arithmetic is bookkeeping, so what we did is we did the subtraction 812 minus 312, and we ended up with 512, which we'll write in fraction form. Well, that worked out pretty well. Let's try this subtraction. So from two-thirds, we'll remove one-fifth, but we can't remove what isn't there. And so as with addition, we need a common denominator. So again, we can find a common denominator by multiplying the denominators together. So a common denominator will be 3 times 5, and so our fraction two-thirds, our denominator is missing the factor of 5, so we'll put it in as long as we pay for it. And again, it's convenient to leave our denominator in factored form and multiply out the numerator. And in our fraction one-fifth, the denominator is missing a factor of 3. So we'll multiply the denominator by 3 and multiply the numerator by 3 as well, giving us the equivalent fraction. So now from 10 of something, I'm going to remove 3 of the same type of thing. Well, if I do that, I'm going to have 7 left, and they're still the same type of things, either 5 times 3's or 3 times 5's. I'll use 5 times 3's. If it's not written down, it didn't happen. This was the subtraction two-thirds minus one-fifth, and since 7 has no factors in common with 5 or with 3, it's safe to multiply those denominator factors together to get 15. As with addition, the lowest common denominator is the least common multiple of the denominators. It's sometimes useful to find it, but it's never necessary. For example, let's evaluate 5-12's minus 5-18's. So again, we could find the lowest common denominator, but why? A common denominator is 12 times 18. So our fraction 5-12's, the denominator is missing the factor of 18, so we'll supply it. We'll multiply our numerator together, but leave our denominator in factored form. The other fraction, 5-18's, our denominator is missing the factor of 12, so we'll supply it. Again, we'll multiply our numerator factors together, but leave our denominator in factored form. Now we have the same denominator, 18 times 12, which is the same as 12 times 18, so we can subtract our numerators, 90 minus 60, and we get our answer 30 over 18 times 12. Again, if it's not written down, it didn't happen. This is what we got when we subtracted 5-12's minus 7-18's. And it's a kindness to reduce to lowest terms, but again, that's not even really necessary. But if we want to, because we've left our denominator in factored form, that becomes much easier. And we might stare at this and say, well, the numerator looks pretty easy to factor. That's 3 times 10, and 3 is a factor of 18. So I can factor 18 as 3 times 6, leave the 12 alone, remove the common factor, say, hey, 10 is pretty easy to factor 2. That's 2 times 5. And since 2 is a factor of 6, I'll factor 6 as 2 times 3. And I haven't done anything with the 12, so they're still there, and now I can remove a common factor. My numerator is 5, which doesn't factor any further, and since it has no factors in common with the denominator, I can multiply those remaining factors together and get my final answer 536.