 So what about the multiplication of fractions? So if you want to multiply two fractions, the product of the fraction Ab by Cd is going to be the product of the numerators over the product of the denominators. So for example, if we want to multiply three-fifths by twelve-sevenths, paper is cheap, so let's write down our steps. The product of these two fractions, we will have the product of the numerators over the product of the denominators, and we'll multiply those factors together. Now a common thing we'll have to do is to multiply a whole number by a fraction, so let's multiply fifteen by seven-ninths. And while we're at it, let's go ahead and reduce to lowest terms. So we can multiply two fractions together, but what about a fraction and a whole number? And so for this, we'll want to remember that for any number n, n is the same as n once. So equals means replaceable, so I can replace fifteen with fifteen once. So now I have two fractions, so I can multiply them together, and multiplying them out fifteen by seven is a hundred and five, one by nine is nine, but I do want to reduce. So the thing to remember here is that a divisor is only relevant if it's a common divisor, and since we only care about common divisors, we'll factor the easier number. Nine is three times three. If three is not a factor of a hundred and five, it doesn't matter what other factors it has, none will be in common with a factor of nine. So either a hundred and five is three times something, or it doesn't matter how a hundred and five factors. So let's try it out, and we see that a hundred and five is three times thirty-five. So equals means replaceable, and we'll remove that common factor to get our final answer, thirty-five-thirds. Now, because we're going to try and remove common factors anyway, it might be more efficient to look for common factors before we multiply. So let's go through that same process. Fifteen times seven-ninths. Well, we'll set it up the same way, but this time we won't multiply out fifteen by seven or one by nine. Instead, we'll think about trying to remove common factors before we multiply things out. So again, nine is three times three, and unless three is a factor of seven or fifteen, we don't care what other factors these numbers have, and we find that three is a factor of fifteen. We'll remove those common factors, and now we have a simpler multiplication, five by seven over one times three. So multiplying those numbers out gives us our final answer. This idea of removing common factors before you multiply is very useful once we look at larger products. For example, five-twelfths by sixteen-twenty-fifths by nine. Again, any whole number n is n once, so nine is nine once. We'll set up the multiplication of the numerators and the denominators, but we won't multiply them. Instead, we'll look for common factors. Sixteen is four times four, nine is three times three. So our goal is to find factors of three, four, or five in the denominators. Well, twelve is three times four, and twenty-five is five times five, so we can remove our common factors. And finally, we do have some multiplications remaining, so we get our final answer, twelve-fifths. One particularly important product is twenty-seven-fifty-thirds times fifty-three. So let's set that up. Twenty-seven-fifty-thirds times fifty-three. Well, fifty-three is the same as fifty-three once. We'll set up our product, but we won't multiply it out, and now we'll try to remove some common factors. Now, the first thing to recognize here is that we have a common factor already. Fifty-three is in both numerator and denominator, so we can remove it and get our much simpler fraction, twenty-seven-once, which reduces to twenty-seven. And the thing that's worth noting here is we multiplied this fraction by its denominator and got just the numerator. And this suggests that, in general, for any fraction, a over b times b is going to give us a. In other words, if we multiply a fraction by its denominator, we'll get the numerator by itself.