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- 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.