Beg Algebra: Mixed Numbers

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Mixed Numbers
You know, a mixed number is just a whole number together with some fraction. Let me show you what a mixed number would look like. For example, if you had 4 and , that's an example of a mixed number because you'll notice it has the form "number with a fraction." Now, just like "of" always means multiplication, whenever you hear "and," that always means addition. So when you read this as "4 and ," what that means is you have 4 plus . So this actually equals, camouflaged, 4 + . So this is what in fact is meant by this expression. Now, suppose that you have this, but you want to write it as one fraction. What do you do? Well, the idea is to sort of punch out that 4 and put it in with the fraction. So, you've got to bring out, you've got to start punching: Pow! Pow! Pow! Poo! Poo! Poo! See, and you just punch it, and the idea is to punch that out. And how are we punching it? Well, it's right here, you see? We're going to take this, this addition problem, and write this addition problem out and solve this and add it. And once we do this, we get it. So, we'll put the punching monkey away and we'll see how to actually do this.
Well, whenever you see a whole number like 4, and you have fractions running around, always remember there's an invisible 1 underneath it. So actually, hidden there is an invisible 1. 4 over 1 is just equal to 1. Now, that gives us the following fraction-addition question: What's 4 over 1 plus 3 over 4? Well, we know what to do. When adding fractions, remember, we have to add Porsches to Porsches and peaches to peaches. So, here I've got four 1's--and by the way, notice that is really consistent with what's going on here, because what is 4 than just one, two, three, four, four 1's. So, here I've got four 1's, and here I've got three fourths. I can't combine them. I have to get a common denominator. So what's the common bottom? Well, the common bottom, we can easily see, requires me to multiply the top and the bottom here by a 4. Because if I do that, then 4 times 1 is 4, which now matches with this bottom. But to keep everything absolutely on the up-and-up, I've got to do it on the top, because 4 divided by 4 is 1, 1 multiplied by anything is the anything, so I haven't changed anything yet. But now I've written it in this form.
Again, I want to remind you of a great temptation you might feel. You might say, "hey wait a minute, before we do it, I can actually simplify that." Don't, because then you'll revert back to the original question. The idea here is to keep it, in some sense, unsimplified but having them match on the bottom. Remember, we can only add and subtract fractions with the same bottom. And then what's the rule? We just add or subtract the tops. So, for example, here I see 16 fourths and I'm adding 3 fourths. So how many fourths do I have? Well, it's 16 plus 3 of them, which is 19 fourths. So, if you're given a mixed fraction, you can always convert it to one fraction, which is not mixed in this way. And it turns out that the other side is true. Notice that this fraction here is actually a large fraction. The 19 is actually bigger than the 4. Whenever you have a fraction where the top is larger than the bottom, then you can actually do the long division and, in fact, write out the number in a mixed way.
Let's take a look at an example. Let's take 23 and divide it by 5. Well, if you want to write this as a mixed fraction, so, some whole number part plus some fractional part, the way you do it just long divide. So, take 5, and let's long divide it by 23. Well, 5 times what gives me 23 or just under? Well, the answer would be 4, because 4 times 5 is 20, and when I subtract, I get a number that is indeed less than 5. I get a remainder of 3. So this is called the remainder. This is called the quotient, and this is the remainder. And so what do I see? So...we see that 23 divided by 5 is equal to the whole number 4 plus the remainder 3 divided by 5. You see I still have to divide, I still have to have 3 divided by 5; 5 has to go into 3, so I represent it that way. So that's the answer, but, of course, if you want to write it as a mixed one and you want to be really fancy, you write it 4 and 3/5, and that's the answer. Remember, whenever you see an "and" you add; so, 4 and 3/5.
By the way, it's always good to check your answer whenever you can. How would you check the answer here? You would take 4 + 3/5 and verify that when you add those two fractions together--remember, that's an invisible fraction, 4 over 1--you do indeed get 23 over 5. I'll let you try that, but you can actually check your answer always. But the important thing here is that the mixed fraction 4 and 3/5 means 4 plus the number 3/5. Great! Now, we're all mixed up.
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