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Hello.
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Right here we're looking at a complex fraction.
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The reason it's complex is you'll see that there's essentially four fractions there.
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You might say wait, i only see three.
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You can always take any whole number and put it over one to turn it into a fraction.
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So you've got these four fractions, seven nights, two thirds, three over one and one night.
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And all of these are even within another fraction.
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Kind of draw this second fraction bar.
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You've got fraction.
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In a numerator, fractions in a denominator, and then this red fraction bar separating the two.
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So it's a complex fraction, fractions within a fraction.
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And we want to go ahead and simplify this.
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And there's multiple ways that we can do this.
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One method that we're going to look at first, one method says go ahead and simplify your numerator, then simplify your denominator, then go ahead and just divide that top fraction by the the bottom fraction, kind of above and below the red line, by multiplying by the reciprocal.
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This red line you can think about is really just a division bar.
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So let's go ahead and see how this would work with a fraction such as this.
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So let's say i had seven nights minus two -thirds.
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You remember when you're subtracting fractions, you've got to have a common denominator.
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So i'm going to have to rewrite those with a common denominator.
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Nine would be a common denominator.
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Nine would be a common denominator because three and nine would both go into nine.
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And to convert two -thirds to be something over nine, three times three is nine.
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So i'd have to multiply the two times three as well.
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Two times three is six.
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So seven -ninths minus two -thirds is the same as seven -nights minus six -nights, which is just one nine.
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So there i simplified my numerator, using all of my skills i already know to subtract fractions.
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Now i need to do the same with the denominator.
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This time i'm adding a fraction 3 over 1 plus 1 over 9.
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Same idea.
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You've got to have a common denominator.
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One and nine.
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Let's go ahead and say that 9 could be my common denominator.
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So it's easy for the second term.
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One -ninth is just one -ninth.
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Three over one, well, i'd have to multiply.
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To get that over nine, i'd have to multiply it by nine.
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So i'd be multiplying by nine over nine.
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It's the same as one.
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Three times nine is 27.
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So then i've got 27 -nights plus one -nights, which is the same as 27 plus 1 is 28 -nights.
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So now my numerator is simplified.
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My denominator is simplified.
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So the next and last step is i need to just simply divide my fractions.
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So let's go ahead and see how that would look.
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My numerator was one ninth.
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My denominator simplified to be 28 ninths.
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And remember i told you that's just a division bar.
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That's the same as saying 1 9th divided by 28 9ths.
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And when we're dividing fractions, we can rewrite this as a multiplication problem.
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Keep the first one just as it is.
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Change it to multiplication.
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And then we need to go ahead and flip the second one on its head.
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We're taking the reciprocal.
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Which means we're just flipping it.
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Like the bottom, the top, and the top of the bottom.
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So now when i multiply, you could multiply straight across 1 times 9 and 9 times 28, but i don't know 9 times 28, but i can see here, i can simplify, if i can cross cancel just a little bit.
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Those nines and the nines would cancel and be one over one.
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So when i multiply this, i would just have one over 28.
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We're definitely testing out those fraction skills.
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So that's one way of doing this, using our method one.
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Now there's a second method.
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And the second method says, identify the least common denominator of all four of those little fractions.
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And then go ahead and multiply each one of them by that least.
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Common denominator and we're doing that because it's going to clear away fractions and then go ahead and simplify.
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So let's look and see how that would work.
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So i've got that same fraction and this time i'm going to go ahead and solve it using this second method.
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So if i'm looking at all of the denominators, the nine, the three, the one, and the nine, you need to identify that least common denominator.
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So between a nine to three, a one, and a nine, i need something that all of those numbers would evenly go into.
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And something they'd all go into is nine.
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And i kind of know that.
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When i say go into, i could say nine divided by three, three goes into nine three times.
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And there would be no remainder left over.
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So it goes in as a nice whole number.
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And we know one would go into nine as well.
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So nine would be my least common denominator.
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And you might want to go do a little refreshing on how to find a least common denominator.
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What i want to really show you here is once you know that, how can i use it to simplify these fractions? so i'm going to go ahead and i'm going to multiply every single one of these fractions.
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I'm going to multiply it by nine.
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And we can write that as nine over one if we'd like.
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So i have to multiply every one of these fractions.
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These fractions by nine.
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And you'll see when i'm multiplying by the least common denominator, something magical happens.
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So in this first case, those two nines go ahead and they cancel out, leaving me with a seven.
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Nine over one times seven ninths is just seven.
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On the next one, the three in the nine don't fully cancel, but i can simplify that a little bit.
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Nine, divided by three is just three.
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So then i have two times three is six.
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On the bottom part, let's see, oh, i don't really have anything here to cancel, but nine times three is 27 over one.
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It's just 27.
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And then this last one, you'll see those nines wonderfully cancel again.
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So one times one is just one...