 To understand the arithmetic of fractions, it's very important to understand the concept of equivalent fractions. So let's take a look at these in more detail. So remember that given any fraction a over b, and any non-zero number n, a over b is equal to na over nb. While we can pick any n we want, sometimes we want to find n, so we have a specific denominator. For example, write 5 over 8, I mean 5 8's, so it has a denominator of 40. In many cases, mathematics is like building a bridge. We can start at the ends and work towards the middle. So we have 5 8's, and what we'd like to end with is a fraction with a denominator of 40. So our equivalent fractions theorem tells us that we can produce an equivalent fraction by multiplying numerator and denominator by the same non-zero number. So 5 8's will be equal to a fraction that's 5 times something over 8 times the same thing. And so the question you've got to ask yourself is, self, what do I multiply 8 by to get 40? And the answer is 5. If we multiply the denominator by 5, we get our denominator 40, but we also have to multiply the numerator by 5 as well. And now we can calculate. Our numerator is 5 times 5, which is 25, and our denominator is 8 times 5, which is 40, which is what we wanted. With this idea in mind, we can talk about cancelling. No, wait, we should actually talk about reducing fractions. We'll explain why in a moment. This is all based around the idea that equals means replaceable, and our equivalent fraction theorem. Since a fraction like 5 12's is the same as 2 times 5 over 2 times 12, or 10 24, then anywhere we see 5 12's, we can replace it with 10 24. But that also works in the other direction. Anywhere we see 10 24's, we can replace it with 5 12's. Now you might wonder why we want to do that, and the reason we might want to replace 5 12's with 10 24's, we'll show up later when we start to talk about addition and subtraction of fractions. On the other hand, the reason that we might want to replace 10 24's with 5 12's is because we think size does matter, and there's something better about 5 12's. And so we define reduced fractions. If A and B have no common divisors, then A over B is said to be in reduced form. Now here's why it's important not to use the word cancel. What we've done is we've gone from this form, N A over N B, to a simpler form, A over B, by removing a common factor. But you can only remove a common factor when you have factors. And so you can only reduce a fraction when both numerator and denominator are products. The important thing is that cancel only requires that we have the same thing. Removing a common factor requires that we actually have a factor. So how you speak influences how you think. If you say that you've removed a common factor, this is a reminder that you have to have a factor. For example, let's try to put in reduced form the fraction 36 40 seconds. Now there's two ways we can do this. We can do this the hard way, or we can do this the easy way. Let's do it the hard way first. So our theorem says that any time we find a common factor in numerator and denominator, we can remove that common factor. And so that suggests that what we should do is factor and remove common factors. So first we'll factor 36, and 42 will factor as equals means replaceable. So we can replace 36 and 42 with their factored forms. And now we'll remove common factors. So the two is in numerator and denominator, so we'll remove that factor. The three is a common factor, it's in numerator and denominator, so we'll remove that common factor. And there are no more common factors, so we have a reduced form, 2 times 3 over 7. Now we could leave it in this form because factored form is one of the more useful forms to have a fraction in. But if we have nothing better to do, we can also multiply 2 times 3 in the numerator and get our final answer, 6 7s. What's the easy way? In our equivalent fraction theorem, notice that a factor only matters if it's common to both numerator and denominator. It's got to appear in both. And so this means that a factor is only relevant if it's a common factor. So rather than doing the complete factorization of 36 and 42, we'll just look for factors that are present in both of them. So we stare at our numbers, and the first thing we might notice is that both of these are even, so both have a factor of 2. So we can rewrite both of them as 2 times something. Now we do have to do some factorization, and we might look at our two remaining factors, 18 and 21. And the thing that's worth noting is 21 is easy to factor because we only have to do one more thing to it. It's 3 times 7. So 3 and 7 are only relevant if they are factors of 18, so we'll check. 7 is not a factor, so we don't worry about it. 3, on the other hand, is a factor. 18 is 3 times 6. And since 7 and 6 have no common factors, we can stop our factorization here. 6 may have additional factors, but none of them are going to be useful. So equals means replaceable, so we'll replace 36 and 42 with their factored forms. We'll remove our common factors of 2 and 3, and what's left over is 6-7s. As an example of how this is more efficient, let's try to reduce, if possible, 105-6-40ths. So we'll try things the easy way and factor one of them. 105 is 5 times 21, and again, we don't really care about the complete factorization of 640. What we care about is whether 5 is a factor of 640, and it is. So we'll rewrite it as 5 times. 21 is 3 times 7, and again, we don't care what the factorization of 128 is. What we care about is whether it has factors of 3 or 7. And since neither 3 nor 7 is a factor of 128, it doesn't matter what the factors of 128 are, we can just leave it. And so equals means replaceable, so 105-6-40ths is the same as 5 times 21 over 5 times 128. We'll remove those common factors and get our final answer, 21-128.