Converting mixed numbers to improper fractions

Write 5 and 1/4 as an improper fraction. So just as a reminder, an improper fraction is one where the numerator is greater than or equal to the denominator. So improper fraction means that the numerator-- and actually, I should say the absolute value of the numerator-- is greater than or equal to the absolute value of the denominator. I could have just shortened it, but you get the idea. The numerator is greater than or equal to-- or the absolute value of the numerator is greater than or equal to the absolute value of the denominator. In this situation right here, it's a mixed number. We have it as a combination of 5, which is just a whole number, and 1/4, which is a proper fraction, because the numerator is less than the denominator. The absolute value of the numerator is less than the absolute value of the denominator. So to write it as an improper fraction-- and I'll show you kind of the methodology for doing it-- and then we'll talk a little bit about why that methodology works. So 5 and 1/4, the methodology is pretty straightforward. You say, OK, 5, the whole number 5, that is the same thing as 20 over 4. And so 20 over 4 plus 1 over 4 is 21 over 4. Or another way to think about it is, 5 times 4 is 20, plus 1 is 21. 21 over 4. So that's the methodology. Pretty straightforward. And I'm going to draw it out so it makes a little bit more sense. So once again, you take the mixed number, or you take the whole part of the mixed number, you multiply it times the denominator, you get 20. And then you add that 20 to 1, you get 21 over 4. And now it's an improper fraction. The absolute value of our numerator is greater than the absolute value of our denominator. Now I do want to show you why this works. So to see why this works, let's think about what 5 and 1/4 means. It means we have 5 wholes. So let's say that this is a whole. So that's 1 whole. And let me copy and paste it five times. So that's 2, that's 3, that's 4, and then that is 5. So we have five wholes. So that's this stuff that I've drawn in green. That's 5 holes right there. And then we have 1/4. So I'll do 1/4 of a whole. And just to be clear that it's part of a whole, I could have drawn the whole in a dotted line, to say that we don't have that whole, we have only 1/4 of it. And that's that 1/4. So this is 5 and 1/4. So to write it as an improper fraction, you can really view this 5 as some fraction over 4. And to think of it that way, divide each of these into fourths. Or this is one way to think about it. So this right over here is 4/4, this is another 4/4, this is another 4/4-- I should have copied and pasted this-- this is another 4/4, and this is another 4/4. So now how many fourths do we have? How many fourths? We have 4/4 here, 4/4 here, 4/4 here, 4/4 here, and 4/4 here. So just what we have in green. We have 20 fourths. That's what we have in green, right over here. That is the same thing as 5. Each of these are 4/4, so you could view this as 5 times 4/4, right? 4/4 is 1. 5 times 4/4 is 20 over 4. That's what we have right over here. And now we can add it to that 1/4. And you will get 21. We have the same denominator. So we can just add the numerators. 21 over 4. So that's the conceptual understanding of why it works. But when you see any mixed number, it's a pretty straightforward process. Multiply the 5 times 4, you get 20. 20 plus 1 is equal to 21 over 4. And actually, let me just write that out. Just so you get clear on what I'm doing. So 5 and 1/4, that is the same thing-- that's equal to 5 times 4 plus 1, over 4. And this is why.