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