 So what about the addition and subtraction of mixed numbers? For example, let's take 8 and 2 thirds plus 12 and 4 fifths. So the important thing to remember is that a mixed number is the sum of a whole number and a proper fraction. And so 8 and 2 thirds is really the same as 8 plus 2 thirds, and 12 and 4 fifths is the same as 12 plus 4 fifths. Now since we're adding, and the order of an addition doesn't matter, what we can do is we can add the whole number parts. 8 plus 12 is 20. Next, we'll add the fractional parts, 2 thirds plus 4 fifths. Because their denominators are different, we have to get them to have the same denominator, and a common denominator is the product 3 times 5. So we'll transform 2 thirds and 4 fifths into fractions with denominators of 3 times 5. So we get fractions with numerators 10 and 12, so we'll add them, and we get a sum 2 thirds plus 3 fifths is 22 over 3 times 5, which we might try to reduce. But since 22 has no factors in common with 3 or 5, we'll multiply the denominator to get our sum 22 fifteenths. And so our fractional parts add together to get 22 fifteenths, and our answer 20 plus 22 fifteenths. Now there's nothing wrong with the answer 20 plus 22 fifteenths, but here's a useful rule for success in life and in mathematics. Answer questions in the same language they're asked in using the same dialect. So if you're asked a question in a natural language, What price should we charge for a cup of coffee to maximize profit? You should answer the question in the same language with the same dialect. So a good answer, We should charge $200 per cup. But a bad answer, 2 ton for a cup of Gov. And another bad answer, X is equal to 2. So here the thing to note is that we were given mixed numbers, and since we were given mixed numbers, we should present the answer as a mixed number. And so 22 fifteenths, well that's really 15 fifteenths and 7 fifteenths. But 15 fifteenths, well that's a whole cake, plus 7 fifteenths. Equals means replaceable, so 22 fifteenths is 1 plus 7 fifteenths, so we'll replace. We now have a sum of whole numbers, which we can do. We have a whole number plus a fraction, which we can write as a mixed number, 21 and 7 fifteenths. What about 12 minus 5 eighths? Well there's two ways we can do this. We can do this the easy way, or we can do this the hard way. Let's do the hard way first. Because we're subtracting an eighth, we need to express 12 in terms of eighths, so we note two things. First of all, for any number, a once is equal to a. And so that means 12 is 12 once. Now, we want this fraction to have a denominator of 8, so it's missing a factor of 8, so we'll supply it. And we'll multiply that out. And so 12 is 96 eighths. We're subtracting 5 eighths, and so we have 91 eighths. Now, 91 eighths is not a bad answer, but remember we want to answer questions in the same language they're asked in using the same dialect. And so here we're given a problem involving a whole number and a proper fraction, which means that we should write our answer in terms of a whole number and a proper fraction. In other words, we need to write our answer as a mixed number. Now, in order to do that, we have to go back to our mixed number theorem. Let a divided by b equals q with remainder r. Then a b is q and r b. So if I want to convert 91 eighths into a mixed number, I have to divide 91 by 8, and that gives me 11 remainder 3, and so my mixed number is 11 and 3 eighths. Now, that was a hard way. Can we do things the easy way instead? And one way to gain insight into the easy way is to look at what cashiers do when they accept money. And one thing cashiers like to do is like to have people pay with the smallest bill possible. In this case, suppose you have $12 in your wallet and you want to pay for something that costs 5 eighths of a dollar. Well, you could give the cashier all $12 in your wallet, but you're more likely to just hand over a single bill which is more than the cost of the item. Now, the mathematical equivalent of that involves two things. First, we note that 12 is 11 plus 1. Now, we still need to be able to subtract 5 eighths, so we'll rewrite that 1 as 8 eighths. And this relies on the theorem that as long as a is not equal to 0, a eighths is equal to 1. Equals means replaceable, so 12 is 11 plus 8 eighths. We'll subtract 5 eighths. Now, this 11, we kept in our wallet and we still have it. The 8 eighths we handed over, the cashier took 5 eighths of it and returned to us 3 eighths. And so what we have is 11 plus 3 eighths, which is the mixed number 11 and 3 eighths. How about 12 minus 3 and 3 fifths? Since we need to be able to subtract fifths, then we can write 12 as 11 and 5 fifths. Now, I want to subtract 3 and 3 fifths. And the important thing to remember is that when you're subtracting a number, subtract all of the number. So I'm subtracting 3 and 3 fifths. I want to subtract 3 and I also want to subtract 3 fifths. And there's a whole number of subtraction I can do, 11 minus 3. And there's a subtraction of fractions with the same denominator that I can do, 5 fifths minus 3 fifths. And so my difference is 8 plus 2 fifths or 8 and 2 fifths.