Algebra 1 Complex Fractions

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In this lesson we are going to take a look at some complex fraction.0002

We will first have to do a little bit about explaining what a complex fraction is0012

and then I’m going to show you two techniques and how you can take care of them.0015

In the first one we will write out these complex fractions as a division problem0018

and then we will go ahead and use the method of this common denominator in order to simplify them.0023

Each of these methods has their own advantages but they both should work when dealing with simplifying complex fraction.0029

What a complex fraction is, it is either the numerator or the denominator is also a fraction.0040

Here is a good example of a complex fraction using numbers.0047

You will notice that the main division bars is actually sitting right here.0050

But in the numerator I have 2/5 and in the denominator I have 1/7.0055

If I have fractions made up of other fractions.0061

These are exactly the types of things that we are looking to simplify.0064

Since we are on a lot of rational expressions, then we will not only look at just numbers but we will look at more complex fractions like this.0072

One of the first techniques that we can use to clean this up is to use division.0082

We will use that main division bar and write this as a division problem.0089

Here I have (x + ½) ÷ (6x + 3)/4x.0095

It is the main division bar right there that will turn into division.0101

That means we will have to use all of our tools for simplifying the left and right, and eventually be able to get them together.0106

In a previous lesson we learn that we need to flip the second rational expression then multiply across the top and bottom.0115

That is exactly what you will see with these.0120

Let us grab on these rational expressions and give it a try.0125

You have (t² u³) / r ÷ t⁴u/r².0127

We want to identify what is on the top and what is on the bottom?0135

We want to write those again as a division problem.0140

(t² u³) / r ÷ t⁴u/r²0144

It looks pretty good.0157

We want to turn this into a multiplication process by flipping that second rational expression t² u³/ r will now be multiplied by r² ÷ t⁴u.0158

That looks much better.0177

We can go ahead and multiply across the top.0178

I will just put all of these on the top and multiply across the bottom.0182

I will put of all of these on the bottom.0187

Now that we have this we would simply go through and cancel out our common factors.0190

We will get rid of t² on the top and t² in the bottom.0198

Making a t² in the bottom.0204

We can cancel out u, bring this down to u² and we can cancel out on r.0207

We have r in the top.0217

Let us write down everything that is left over.0220

u² r ÷ t² and then we could consider this one simplified.0222

It changes it into a problem that we have seen before.0231

You just have to do a lot of work with simplifying.0234

When it gets to simplifying a complex fraction, even that process is not necessarily the easiest to go through.0240

In fact, you will find that this next problem is quite lengthy.0248

We have (1/x + 1 + 2/y – 2) ÷ (2 /y – 2 – 1/x + 3).0251

Let us identify everything on the top and everything in the bottom.0262

That way we can simply rewrite this 1/x + 1 + 2/y – 2 all of this is being divided by everything on the bottom 2/y – 2 – 1/x + 3.0269

If I have any hope I'm doing this as a division problem that I need to normally flip that second fraction.0298

Notice how in this one I do not have a single fraction.0305

It is tempting to say, hey why we just flip both of them but that is not how division works.0309

We need to combine it into a single fraction before we can flip it and then do the multiplication.0313

Let us see if we can get these guys together with some common denominators.0322

The common denominators on this side are the x + 1 and y – 2.0327

In order to get those together, I would have to give this fraction, y - 2 on the top and bottom.0343

To make it workout over here we will give the top and bottom of that one x +1 and the 2 is still up there.0353

It looks like that first piece will turn into y – 2 + let us go ahead and distribute this guy in there (2x + 2) ÷ (x + 1) (y – 2).0365

That just takes that and crunches it down a little bit.0384

Let us focus on this other one.0388

We need a common denominator and I see there is a y - 2 and x + 3.0390

I will need to give the fraction on the left an additional x + 3.0408

We still have the 2 in there.0412

2 × x + 30414

Over on the other side let us give the top and bottom of that one y -2.0417

When those are put together, we will do a little bit of distributing here.0424

We will have (2x + 6) – (y + 2) ÷ (y – 2) (x + 3).0431

Let us go ahead and write this again and see if we can do the actual division.0447

It looks like I can cancel out a few things in here.0455

Let us save ourselves a little bit of work.0458

y + 2x ÷ (x + 1) (y -2), we are dividing it by the second fraction.0461

Here is when I’m going to flip and multiply it, multiplied by 2x - y + 8.0476

I have combined the 6 and the 2 together, y - 2 x + 3.0487

I can just combine the tops and bottoms.0498

Quite a lot of pieces in here that is okay, at least I see one piece will cancel out and that is the y – 2.0516

We are left with y + 2x and (x + 3) ÷ (x +1) and 2x - y + 8.0528

Even though it can be a lengthy process, by rewriting it as a division process and using our tools from before0543

you will see is that it is possible to reduce and simplify this complex fractions.0549

The other method which can often be a lot cleaner is using the least common denominator0556

in order to clear out all the fractions in the top and bottom.0561

In order for this method to work, you must find the least common denominator of all the little fractions present in your complex fraction.0566

Go ahead and look at your numerator and denominator and think of all the least common denominator for all those fractions.0573

Once you find that LCD, then you are going to multiply on the top and the bottom of the main fraction0580

or on the top and bottom of the main division bar.0586

This will clear up things immensely but you have to be careful on canceling out.0591

You do not want to accidentally cancel out something that you should not.0596

You will see that you will clear out a bunch of stuff and then you end up simplifying just as you would normally.0598

No matter what method you use, you should get the same answer.0608

Use the method that you are more comfortable with.0614

I like to recommend a method two because it is usually much cleaner than using the first method.0620

However, anything that is cleaner use less opportunities for mistakes.0627

One downside to the second method is usually happen so quickly it is hard to keep track of everything that was in there.0633

Let us try this second method with the following complex fraction.0643

I have (2/s² t + 3 /st²) ÷ (4/s² t² – 1/st).0647

Let us first see if we can identify the least common denominator.0656

Try and pick out all these little denominators here, see what the LCD would be.0661

They all have some s and the largest one I see there is an s².0668

They all have t’s but the largest one in there is a t².0675

I’m going to take this and I’m going to multiply it on the top and the bottom of this expression so I can rewrite it.0680

There is our main division bar.0698

Let us get to multiplying.0708

On the top of this entire main division bar I will multiply it by this LCD.0709

I will do the same thing on the bottom to keep things nice and balanced.0715

The top and bottom both have two terms, so we will definitely make sure that we distribute each of these parts.0719

We are going to write the result of this multiplication and then watch how many things will cancel out in the next step.0730

(2s² t² / s² t) + (3s² t² / st²) ÷ (4s² t / s² t² - s² t/st).0737

It looks messy and it looks I have actually made things even more complicated.0768

Watch what is going to go take a vacation here soon.0772

I have these s² they will go away.0776

Then I have a single t and t up top, they will go away.0779

On the next one, there is my s one of those will be gone and both my t² are gone.0783

Onto the bottom s² is gone, s² is gone.0790

I think I am missing some of my squares.0797

Our t² are gone and we can get rid of one of these t’s.0805

The denominators of all those little fractions, these guys that we are so worried about at the very beginning,0817

all of them have been cancel out in some way or another.0823

This means as soon as we write down our leftover pieces, this one is simplified.0826

2t + 3s, those are the only things that survived up here ÷ 4 - st and those are the only things that survived on the bottom.0831

This one is in its most simplified form.0847

It is nice, quick and easy method you just have to properly identify the LCD first.0852

You will know you are using the method right if all of these denominators end up going away.0858

If any of them are still in there double check to see what LCD you used.0862

Let us try this one more time but something a little bit more complicated.0869

This one is (2y + 3 / y – 4) ÷ (4y² – 9) + (y² – 16).0873

Let us examine these denominators so we can find our LCD.0882

We are looking at this one this is the same as y - 4 and y + 4.0888

Over here it already has the y -4 in it.0897

The only piece that I am missing is the y + 4.0900

My LCD will contain both of these parts.0905

I have the y – 4 and y + 4, both of those in there.0908

I’m going to take that and we are going to multiply it on the top and bottom of the original.0917

Let me just quickly rewrite this and I’m going to rewrite it with the factored form on the bottom.0923

We will take our LCD and we will multiply it on the top and on the bottom of our main fraction here.0939

y – 4 y + 4.0952

Let us go ahead and put everything together and let us see how this looks.0959

I have (2y + 3 ) (y – 4) (y + 4) ÷ y – 4.0965

Then comes our main division bar right there.0980

On the bottom is (4y² – 9) × (y – 4) × (y + 4) ÷ (y – 4) (y + 4).0985

Watch how many things will cancel in this next step.1005

y – 4 and y – 4 those are gone.1009

y – 4, y – 4, y + 4, y + 4, 4 those are gone.1013

All of these problems that we had at the very beginning, they are no longer problem.1017

They are gone.1022

We will simply write down all of the left over pieces.1023

2y + 3 y + 4 4y² – 9.1028

Be careful, there is still some additional reducing that you can do even after using your LCD like this.1044

One thing that I can see is that I can actually continue factoring the bottom.1050

Let us write that out.1056

2y + 3/ y + 4 and this will be over I have different squares on the bottom so, 2y + 3, 2y – 3.1058

I’m sure enough now we can more easily that I have an extra 2y – 3 in the bottom and that is gone as well.1074

This one finally reduces down to y + 4 / 2y -3 and now we are finally done.1081

The second method is definitely handy and clears up a lot of fractions very quickly.1092

Use whatever method you are more comfortable with.1097

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