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Mixed numbers and improper fractions

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We're now going to learn how to go from mixed numbers to improper fractions and vice versa. So first a little bit of terminology. What is a mixed number? Well, you've probably seen someone write, let's say, 2 and 1/2. This is a mixed number. You're saying why is it a mixed number? Well, because we're including a whole number and a fraction. So that's why it's a mixed number. It's a whole number mixed with a fraction. So 2 and 1/2. And I think you have a sense of what 2 and 1/2 is. It's some place halfway between 2 and 3. And what's an improper fractional? Well an Improper fraction is a fraction where the numerator is larger than the denominator. So let's give an example of an improper fraction. I'm just going to pick some random numbers. Let's say I had 23 over 5. This is an improper fraction. Why? Because 23 is larger than 5. It's that simple. It turns out that you can convert an improper fraction into a mixed number or a mixed number into an improper fraction. So let's start with the latter. Let's learn how to do a mixed number into an improper fraction. So first I'll just show you kind of just the basic systematic way of doing it. It'll always give you the right answer, and then hopefully I'll give you a little intuition for why it works. So if I wanted to convert 2 and 1/2 into an improper fraction or I want to unmix it you could say, all I do is I take the denominator in the fraction part, multiply it by the whole number, and add the numerator. So let's do that. I think if we do enough examples you'll get the pattern. So 2 times 2 is 4 plus 1 is 5. So let's write that. It's 2 times 2 plus 1, and that's going to be the new numerator. And it's going to be all of that over the old denominator. So that equals 5/2. So 2 and 1/2 is equal to 5/2. Let's do another one. Let's say I had 4 and 2/3. This is equal to -- so this is going to be all over 3. We keep the denominator the same. And then new numerator is going to be 3 times 4 plus 2. So it's going to be 3 times 4, and then you're going to add 2. Well that equals 3 times 4 -- order of operations, you always do multiplication first, and that's actually the way I taught it how to convert this anyway. 3 times 4 is 12 plus 2 is 14. So that equals 14 over 3. Let's do another one. Let's say I had 6 and 17/18. I gave myself a hard problem. Well, we just keep the denominator the same. And then new numerator is going to be 18 times 6 or 6 times 18 plus 17. Well 6 times 18. Let's see, that's 60 plus 48 it's 108, so that equals 108 plus 17. All that over 18. 108 plus 17 is equal to 125 over 18. So, 6 and 17/18 is equal to 125 over 18. Let's do a couple more. And in a couple minutes I'm going to teach you how to go the other way, how to go from an improper fraction to a mixed number. And this one I'm going to try to give you a little bit of intuition for, why what I'm teaching you actually works. So let's say 2 and 1/4. If we use the -- I guess you'd call it a system that I just showed you -- that equals 4 times 2 plus 1 over 4. Well that equals, 4 times 2 is 8 plus 1 is 9, 9 over 4. I want to give you an intuition for why this actually works. So 2 and 1/4, let's actually draw that, see what it looks like. So let's put this back into kind of the pie analogy. So that's equal to one pie. Two pies. And then let's say a 1/4 of a pie. A 1/4 is like this. 2 and 1/4, and ignore this, this is nothing. It's not a decimal point -- actually, let me erase it so it doesn't confuse you even more. So go back to the pieces of the pie. So there's 2 and 1/4 pieces of pie. And we want to re-write this as just how many 1/4s of pie are there total. Well if we take each of these pieces -- I need to change the color -- if we take each of these pieces and we divide it into 1/4s, we can now say how many total 1/4s of pie do we have? Well we have 1, 2, 3, 4, 5, 6, 7, 8, 9 fourths. Makes sense, right? 2 and 1/4 is the same thing as 9/4. And this will work with any fraction. So let's go the other way. Let's figure out how to go from an improper fraction to a mixed number. Let's say I had 23 over 5. So here we go in the opposite direction. We actually take the denominator, we say how many times does it go into the numerator. And then we figure out the remainder. So let's say 5 goes into 23 -- well, 5 goes into 23 four times. 4 times 5 is 20. And the remainder is 3. So 23 over 5, we can say that's equal to 4 and in the remainder 3 over 5. So it's 4 and 3/5. Let's review what we just did. We just took the denominator and divided it into the numerator. So 5 goes into 23 four times. And what's left over is 3. So, 5 goes into 23, 4 and 3/5 times. Or another way of saying that is 23 over 5 is 4 and 3/5. Let's do another example like that. Let's say 17 over 8. What does that equal as a mixed number? You can actually do this in your head, but I'll write it out just so you don't get confused. 8 goes into 17 two times. 2 times 8 is 16. 17 minus 16 is 1. Remainder 1. So, 17 over 8 is equal to 2 -- that's this 2 -- and 1/8. Because we have one 8 left over. Let me show you kind of a visual way of representing this too, so it actually makes sense how this conversion is working. Let's say I had 5/2, right? So that literally means I have 5 halves, or if we go back to the pizza or the pie analogy, let's draw my five halves of pizza. So let's say I have one half of pizza here, and let's say I have another half of pizza here. I just flipped it over. So that's 2. So it's 1 half, 2 halves. So that's three halves. And then I have a fourth half here. These are halves of pizza, and then I have a fifth half here, right? So that's 5/2. Well, if we look at this, if we combine these two halves, this is equal to 1 piece, I have another piece, and then I have half of a piece, right? So that is equal to 2 and 1/2 pieces of pie. Hopefully that doesn't confuse you too much. And if we wanted to do this the systematic way, we could have said 2 goes into 5 -- well, 2 goes into 5 two times, and that 2 is right here. And then 2 times 2 is 4. 5 minus 4 is 1, so the remainder is 1, and that's what we use here. And of course, we keep the denominator the same. So 5/2 equals 2 and 1/2. Hopefully that gives you a sense of how to go from one mixed number to an improper fraction, and vice versa, from an improper fraction to a mixed number. If you're still confused let me know and I might make some more modules. Have fun with the exercises.