Convert between improper fractions and mixed numbers

Video transcript

Write 7/4 as a mixed number. So right now it's an improper fraction. 7 is larger than 4. Let's write it is a mixed number. So first I'm just going to show you a fairly straightforward way of doing it and then we're going to think a little bit about what it actually means. So to figure out what 7/4 represents as a mixed number, let me write it in different colors. So this is going to be equal to-- the easiest way I do it is you say, well, you divide 4 it 7. If we're dealing with fourths, 4 goes into 7 a total of one time. Let me do this in another color. 1 times 4 is 4. And then what is our remainder? 7 minus 4 is 3. So if we wanted to write this in plain-- well, let me just do the problem, and then we'll think about what it means in a second. So you see that 4 goes into 7 one time, so you have one whole here, and then how much do you have left over? Well, you have 3 left over, and that comes from right over there. That is the remainder when you divide 4 into 7. 3 left over, but it's 3 of your 4, or 3/4 left over. So that's the way we just converted it from an improper fraction to a mixed number. Now, it might seem a little bit like voodoo what I just did. I divided 4 into 7, it goes one time, and then the remainder is 3, so I got 1 and 3/4. But why does that make sense? Why does that actually makes sense? So let's draw fourths. Let's draw literally 7 fourths and maybe it'll become clear. So let's do a little square as a fourth. So let's say I have a square like that, and that is 1/4. Now, let's think about what seven of those mean, so let me copy and paste that. Copy and then paste it. So here I have 2 one-fourths, or you could see I have 2/4. Now I have 3 one-fourths. Now, I have 4 one-fourths. Now this is a whole, right? I have 4 one-fourths. This is a whole. So let me start on another whole. So now I have 5. Now I have 6 one-fourths, and now I have 7 one-fourths. Now, what does this look like? So all I did is I rewrote 7/4, or 7 one-fourths. I just kind of drew it for you. Now, what does this represent? Well, I have 4 fourths here, so this is 4/4. This right here is 3/4. Notice, 7/4 is 4/4 with 3/4 left over. So let me write it this way. 7/4 is 4/4 with 3/4 left over. Now what is 4/4? 4/4 is one whole. So you have one whole with 3/4 left over, so you end up with 1 and 3/4. So that is the 3/4 part and that is your one whole. Hopefully that makes sense and hopefully you understand why it connects. Because you say, well, how many wholes do you have? When you're dividing the 4 into the 7 and getting the one, you're essentially saying how many wholes? So the number of wholes, or you can imagine, the number of whole pies. And then how many pieces do we have left over? Well, we have 3 pieces and each piece is 1/4, so we have 3/4 left over. So we have one whole pie and three pieces, which are each a fourth left over.

Video transcript

Write 5 and 1/4 as an improper fraction. An improper fraction is just a pure fraction where the numerator is greater than the denominator. This right here, it's not a pure fraction. We have a whole number mixed with a fraction, so we call this a mixed number. So let's think about what 5 and 1/4 represents, and let me rewrite it. So if we're talking about 5 and 1/4, and you can literally think of this as 5 and 1/4 or 5 plus 1/4, that's what 5 and 1/4 represents. So let's think about 5. Five is 5 wholes, or if you're thinking of pie, we could draw literally five pies. Let me just cut up the pies from the get go into four pieces since we're dealing with fourths. So let me just cut up the pies right over here. So that's one pie right over there. Let me copy and paste this. Copy and paste. So I have two pies, and then I have three pies, and then I have four pies, and then I have five pies. So this is what the 5 represents. 5 literally represents-- so let me circle all of this together. That is the 5 part right there. That is what 5 literally represents. It represents five whole pies. Now, I have cut up the pies into four pieces, so you can imagine each piece represents a fourth. Now, how many pieces do I have in these five pies? Well, I have four pieces per pie. Let me just right it here. 4 pieces per pie times 5 pies is equal to 20 pieces. Or another way to think of it, since each piece is a fourth, this is also equal to 20 times 1/4, or you could just write this as being equal to 20/4. So we have 5 whole pies is equal to 20 fourths. Let me write it like that. 20 fourths. Or we could write it as 20/4. I've kind of done the same thing twice. So that's what the five pies represent. 20/4 or 20 pieces, where each piece is 1/4. Now, the 1/4 right here represents literally one more fourth of a pie or one more piece of a pie, so let me draw another pie here. So that is another pie. Cut it into four pieces. But this 1/4 only represents one of these pieces, right? This is one of the four pieces. The denominator tells us how many pieces. The 1 tells us how many of those pieces we're dealing with, so it's just this one piece over here. That right there is the 1/4. Now, if we write 5 and 1/4, we just saw that the 5 right here is 20/4. So we could rewrite this. Let me write it like this. 5 and 1/4 can be rewritten as the same thing as 5 plus 1/4, which is the same thing as-- we just saw that five whole pies is the same thing as 20/4. And to see that these are the same thing, you literally just divide 4 into 20. You get 5, and nothing is left over. So 5 is the same thing as 20/4, and then this plus 1/4 is the same thing as plus 1/4. So if I have 20 fourths and I add one more fourth to it, how many fourths do I have? Well, I have 21. I have 21 fourths. Or another way of thinking about it, this 5 is-- so this right here is 20 pieces of pie. You can even count it. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20. But a quicker way is to say, well, we have five pies. Each of them have four pieces. 5 times 4 is 20. This 1/4 right here represents one piece plus one piece, so total we're going to have 21 pieces. So we have 21 pieces, where each piece is 1/4, so we could say we have 21 times 1/4 or 21 fourths pieces of pie. However you want to think of it, but we've solved the problem. We're at an improper fraction. We've written 5 and 1/4 as an improper fraction. Now, I've gone through great pains to give you the intuition of what 5 and 1/4 means, but there is a fairly straightforward process for getting straight to the improper fraction. Let me color code it. So if you have 5 and 1 over 4, to convert it into an improper fraction, you're going to keep the same denominator, so you're going to have the over 4 there. But your numerator is going to be your numerator of the fraction part before. So it's going to be 1 plus your whole number times your denominator. So 1 plus-- or actually, let me do it the way I tend to think of it. What I do is I take 4 times 5. So let me write that down and I want to color code it. 4 times 5, and then to that, I add this numerator. So I literally do 4 times 5 plus 1, which is-- so this is equal to 4 times 5 is 20, plus 1 is 21, and then that's over 4, so it's 21/4. And all of this is kind of a fast way to do it. We're literally doing the exact same thing that we did here in kind of a slower way. We're saying, OK, 5 wholes is the same thing as 20 fourths, so you take 5, and I figure that out, 5 times 4, and then I have one more fourth there, so 4 times 5 plus 1 gives 21.