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### Complex Fractions

- A complex fraction is where the numerator or the denominator of a rational expression contains fractions as well.
- To simplify a complex fraction you can combine the expressions in the numerator and denominator. Then transform it into a division problem. Remember when dividing to flip the second expression and multiply.
- An alternate way to simplify complex fractions is to multiply the numerator and denominator by the LCD of all the small fractions. Remember to distribute if using this method.
- Both methods for simplifying complex fractions should work, however using the LCD is often quicker and cleaner with fewer opportunities for a mistake.

### Complex Fractions

4e

^{2}− 3 − [(e + 1)/(e + 5)]

- [(( e + 5 ) ×( 4e
^{2}− 3 ))/(( e + 5 ) ×1)] − [(e + 1)/(e + 5)] - [(4e
^{2}− 3e + 20e^{2}− 15)/(e + 5)] − [(e + 1)/(e + 5)] - [(4e
^{2}− 3e + 20e^{2}− 15 − e − 1)/(e + 5)]

^{2}+ 20e

^{2}− 4e − 16)/(e + 5)]

7a

^{2}− 10 + [(a − 8)/(a + 6)]

- [(( a + 6 ) ×( 7a
^{2}− 10 ))/(( a + 6 ) ×1)] + [(a − 8)/(a + 6)] - [(7a
^{3}− 10a + 42a^{2}− 60 + a − 8)/(a + 6)]

^{3}+ 42a

^{2}− 9a − 68)/(a + 6)]

12b

^{2}+ 3 − [(b + 5)/(b − 1)]

- [(( b − 1 )( 12b
^{2}+ 3 ))/(( b − 1 )( 1 ))] − [(b + 5)/(b − 1)]

^{3}+ 3b − 12b

^{2}− 3 − b − 5)/(b − 1)]

[[([(x

^{2})/(y

^{2})])/([(3x + 1)/(4x − 5)])]]

- [([a/b])/([c/d])] = [ad/bc]
- [(x
^{2}( 4x − 5 ))/(y^{2}( 3x + 1 ))]

^{3}− 5x

^{2})/(3xy

^{2}+ y

^{2})]

[[([x/(y

^{3})])/([(7x − y)/(6x + 11y)])]]

- [(x( 6x + 11y ))/(y
^{3}( 7x − y ))]

^{2}+ 11xy)/(7xy

^{3}− y

^{4})]

[([(m

^{2})/n])/([(m + n

^{2})/(2m − n

^{2})])]

- [(m
^{2}( 2m − n^{2}))/(n( 2m − n^{2}))]

^{3}− m

^{2}n

^{2})/(2mn − n

^{3})]

[([(x

^{2}+ 3x − 4)/(x

^{2}+ 5x + 6)])/([(x

^{2}− 5x + 4)/(x

^{2}− 2x − 8)])]

- [(x
^{2}+ 3x − 4)/(x^{2}+ 5x + 6)] ×[(x^{2}− 2x − 8)/(x^{2}− 5x + 4)]

[([(x

^{2}+ 12x + 35)/(x

^{2}+ 8x + 12)])/([(x

^{2}− 25)/(x

^{2}− 3x − 10)])]

- [(x
^{2}+ 12x + 35)/(x^{2}+ 8x + 12)] ×[(x^{2}− 3x − 10)/(x^{2}− 25)] - [(( x + 7 )( x + 5 ))/(( x + 2 )( x + 6 ))] ×[(( x − 5 )( x + 2 ))/(( x + 5 )( x − 5 ))]

[([(x

^{2}− 64)/(x

^{2}+ 5x − 24)])/([(x

^{2}+ x − 72)/(x

^{2}+ x − 12)])]

- [(x
^{2}− 64)/(x^{2}+ 5x − 24)] ×[(x^{2}+ x − 12)/(x^{2}+ x − 72)] - [(( x − 8 )( x + 8 ))/(( x + 8 )( x − 3 ))] ×[(( x − 3 )( x + 4 ))/(( x − 8 )( x + 9 ))]

[([(x + 6)/(x

^{2}+ 12x + 20)])/([(x + 1)/(x

^{2}+ x − 2)])]

- [(x + 6)/(x
^{2}+ 12x + 20)] ×[(x^{2}+ x − 2)/(x + 1)]

^{2}+ x − 2 ))/(( x

^{2}+ 12x + 20 )( x + 1 ))]

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

Answer

### Complex Fractions

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

- Intro 0:00
- Objectives 0:09
- Complex Fractions 0:37
- Dividing to Simplify Complex Fractions
- Example 1 2:03
- Example 2 3:58
- Complex Fractions Cont. 9:15
- Using the Least Common Denominator to Simplify Complex Fractions
- Both Methods Lead to the Same Answer
- Example 3 10:42
- Example 4 14:28

### Algebra 1 Online Course

### Transcription: Complex Fractions

*Welcome back to www.educator.com.*0000

*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 is*0012

*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 problem*0018

*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 (t ^{2} u^{3}) / r ÷ t^{4}u/r^{2}.*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

*(t ^{2} u^{3}) / r ÷ t^{4}u/r^{2}*0144

*It looks pretty good.*0157

*We want to turn this into a multiplication process by flipping that second rational expression t ^{2} u^{3}/ r will now be multiplied by r^{2} ÷ t^{4}u.*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 t ^{2} on the top and t^{2} in the bottom.*0198

*Making a t ^{2} in the bottom.*0204

*We can cancel out u, bring this down to u ^{2} 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

*u ^{2} r ÷ t^{2} 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 + 3*0414

*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 before*0543

*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 denominator*0556

*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 fraction*0580

*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/s ^{2} t + 3 /st^{2}) ÷ (4/s^{2} t^{2} – 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 s ^{2}.*0668

*They all have t’s but the largest one in there is a t ^{2}.*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

*(2s ^{2} t^{2} / s^{2} t) + (3s^{2} t^{2} / st^{2}) ÷ (4s^{2} t / s^{2} t^{2} - s^{2} 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 s ^{2} 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 t ^{2} are gone.*0783

*Onto the bottom s ^{2} is gone, s^{2} is gone.*0790

*I think I am missing some of my squares.*0797

*Our t ^{2} 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) ÷ (4y ^{2} – 9) + (y^{2} – 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 (4y ^{2} – 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 4y ^{2} – 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

*Thank you so much for watching www.educator.com.*1101

1 answer

Last reply by: Professor Eric Smith

Sun Feb 22, 2015 4:16 PM

Post by Micheal Bingham on February 21, 2015

Hello Mr. Smith, I have a question pertaining to the third example in this video. When you find the LCD (s^2*t^2) to multiply both fractions by, why is it that you only multiply the numerator by aforementioned LCD? Sorry for the confusion, and thank you in advance.