 So we've talked about addition, subtraction, and multiplication of fractions. We'll talk about division of fractions, but let's talk about it as a broader issue of what are called compound fractions. And compound fractions are based on the following useful relationship. The quotient A divided by B is the same as the fraction A B. And what that means is that any time you have a fraction, you can treat it as a division problem, and any time you have a division problem, you can treat it as a fraction. So for example, we want to write as a fraction, 157 divided by 47 by our theorem, 157 divided by 47 is 15747s. Yeah, that isn't too exciting, but it has some important implications because we already know how to do a bunch of things with fractions. So let's talk about division. It's possible that at some point you may have learned a way to divide fractions. To divide a fraction, invert and multiply. Do yourself a favor and forget this method. Instead, we can divide with fractions by using what are called compound fractions. If the dividend or divisor is a fraction, we get a compound fraction. For example, two-thirds divided by five. Well, our theorem says that A divided by B is the fraction A B's. That's a fraction with A as the numerator and B as the denominator. So two-thirds divided by five is two-thirds fifths. Or three-fourths divided by two-fifths. Well, that's a fraction with numerator three-fourths and denominator two-fifths. Eight divided by one-fifth. Well, that's a fraction with numerator eight and denominator one-fifth. Note that sometimes we use a slash to save space. So two-thirds divided by five could be written as a fraction two-slash three over five. Three-fourths divided by two-fifths. That's three-slash four over two-slash five. Eight divided by one-fifth. That's eight over one-slash five. And while we could do this, it's generally not a good idea. And the thing to remember is that paper is cheap. Try to save paper at the cost of understanding. Especially because the amount of space we've saved in using the slash is almost nothing in comparison to the amount of space we would have used if we wrote the fractions out in full. So how do you simplify these compound fractions? When confronted with a compound fraction, it's helpful to keep the following ideas in mind. For any fraction, if we multiply by the denominator, we'll get just the numerator. Second, for any fraction a over b and any non-zero number n, the fraction a over b is the same as the fraction na over nb. We can multiply numerator and denominator by the same thing and not change the fraction. For example, let's say we want to find two-thirds divided by five. So the first thing to remember is that a division gives us a compound fraction. So let's rewrite this division as a fraction. Now our numerator is a fraction with a denominator of three. And we can get rid of that denominator by multiplying by it. So let's multiply the numerator by three. But we can't just alter the numerator. We have to alter the denominator as well. So we also need to multiply the denominator by three. Now we can carry out the multiplication. In the numerator, two-thirds times three is two, and five times three is fifteen, which gives us our final answer to fifteenths. How about three-fourths divided by two-fifths? Well, again, we know that a division can be rewritten as a fraction. To get rid of this denominator four in the numerator, we'll multiply numerator by four. But we also have to multiply denominator by four. In the numerator, three-fourths times four is in the denominator, two-fifths times four is. Now we still have a fraction in the denominator, so let's multiply the denominator by five. We also have to multiply the numerator by five. Multiplying our denominator eight-fifths by five gives us multiplying our numerator three by five gives us and we have our final answer, fifteen-eighths. Eight divided by one-fifth, well, that's the same as the fraction eight over one-fifth. And we can get rid of our fraction in the denominator by multiplying numerator and denominator by five. Multiplying our numerator together gives us forty. Multiplying our denominator together gives us one. And forty-once is just equal to forty. Who says the dividend or divisor have to be single fractions? Compound fractions also include expressions like three over five plus one-seventh or five-thirds plus one-half over two-fifths plus three-sevenths. And these may look a little frightening, but the key to life is you control the step size. What does that mean? Well, let's consider a fraction like three over five plus one-seventh. So the first thing to recognize here is that we have this fraction one-seventh with denominator seven. So if we multiply numerator and denominator of our compound fraction by seven, we have a chance of getting rid of the seven. So I'll multiply my numerator together. Three times seven is twenty-one. And our denominator will use the distributive property. That's five times seven plus one-seventh times seven. Well, five times seven is thirty-five. One-seventh times seven is one. So our denominator, thirty-five plus one, that's thirty-six. And since I have lots of free time in my hands and nothing more important to do, I'll reduce this fraction twenty-one thirty-six. And even though I have so much free time that I'm going to reduce fractions, I'd like to do this at least somewhat efficiently. So let's start off with twenty-one. That factors as three times seven. And so the only question I really care about is whether three or seven is a factor of thirty-six. Now seven is not a factor of thirty-six, but three is. Thirty-six is three times twelve. And now we can remove that common factor and get our final answer, seven-twelfths.