Adding Fractions with Unlike Denominators - Part 1 (Fraction Addition) - By Math and Science
Transcript
| 00:00 | Hello , welcome back . The title here is called | |
| 00:02 | adding fractions with unlike denominators , this is part one | |
| 00:06 | . Really excited to teach this because this is where | |
| 00:09 | for a lot of students , for some students , | |
| 00:11 | the wheels start to come off the train . A | |
| 00:13 | lot of students understand what a fraction is . Kind | |
| 00:15 | of , how to simplify a fraction . But when | |
| 00:17 | you start adding fractions it becomes very difficult for some | |
| 00:20 | students were going to make it very easy for you | |
| 00:23 | to understand . In the beginning , I'm gonna show | |
| 00:25 | you how to add fractions with . Unlike denominators , | |
| 00:28 | I'm going to use the magnets here to kind of | |
| 00:30 | draw pictures for you to understand what's really going on | |
| 00:33 | . And then after the first few problems , we | |
| 00:35 | won't be drawing any more pictures , we'll just be | |
| 00:37 | going through it because you'll understand how to solve the | |
| 00:39 | problems and you'll know what the process is really doing | |
| 00:42 | . Now , if you remember I taught you that | |
| 00:44 | when you add fractions , the denominator is the bottom | |
| 00:47 | numbers must be the same . I'll say it again | |
| 00:50 | when you add fractions , the denominator is the bottom | |
| 00:53 | numbers must be the same number . Third time when | |
| 00:56 | you add fractions or subtract fractions the bottom numbers , | |
| 01:00 | the denominators must be the same numbers . So we've | |
| 01:02 | done that before . Here we have one more step | |
| 01:05 | because here the fractions we're going to add will not | |
| 01:07 | have the same denominators . So step one , when | |
| 01:11 | we do this is we must change the fractions so | |
| 01:14 | that they have the same denominator because you can't add | |
| 01:16 | them unless they do unless they have the same denominator | |
| 01:19 | , you can't add them . So the first step | |
| 01:20 | is we will change these fractions so that the denominators | |
| 01:24 | are the same After that we add them as we | |
| 01:27 | usually have been doing . So let's go over here | |
| 01:30 | And begin the process with a problem like this , | |
| 01:33 | let's take a look at the fraction 1/10 . And | |
| 01:36 | we're going to add this to the fractions 3/5 . | |
| 01:40 | Now first notice the denominator of this is 10 and | |
| 01:44 | the denominator of this , one is three . So | |
| 01:47 | if these were denominators were the same , then I | |
| 01:50 | would just keep the denominator the same denominator in the | |
| 01:52 | answer and I would add the top numbers enumerators , | |
| 01:55 | but you cannot do that here because they are not | |
| 01:58 | the same . A lot of students will start trying | |
| 02:00 | to add five plus 10 or they'll start trying to | |
| 02:02 | add three plus one . You cannot do that when | |
| 02:04 | you see that the bottom numbers are different . The | |
| 02:07 | very first thing you must do is change them . | |
| 02:09 | Now , we know that we can change fractions , | |
| 02:12 | right ? We can do anything , we can multiply | |
| 02:14 | or divide a fraction by any number we want , | |
| 02:17 | as long as we do it to the top and | |
| 02:19 | the bottom of the fraction . So what we're going | |
| 02:21 | to do is change one or both of these fractions | |
| 02:24 | so that the new fraction has a common denominator , | |
| 02:27 | so um or has the same denominator before we get | |
| 02:30 | to that point , let's take a look . What | |
| 02:32 | does 1/10 Actually represent ? Well , if I have | |
| 02:36 | a pizza cut into 10 pieces , I could I | |
| 02:38 | could put these all the way around and you only | |
| 02:40 | had 1/10 it would look like a slice of pizza | |
| 02:42 | . That big . Right ? What does the second | |
| 02:46 | one here ? 3/5 . If I built a pizza | |
| 02:48 | with cut into five pieces , so there's 1234 and | |
| 02:52 | five , there will be five and you have three | |
| 02:55 | of those pieces . 123 That is what this fraction | |
| 02:58 | represents . So , you have a very small amount | |
| 03:01 | of pizza here and a very large amount of pizza | |
| 03:04 | or a much larger amount of pizza down below . | |
| 03:05 | Now , when we add them together , what we're | |
| 03:07 | really doing is we're taking this pizza and we're adding | |
| 03:10 | it there were putting it there and we're saying , | |
| 03:12 | okay , now we have this much pizza . So | |
| 03:14 | the idea of adding these fractions together is literally like | |
| 03:18 | taking slices of pizza and just putting them together and | |
| 03:20 | seeing how much you have . But the problem is | |
| 03:23 | because this part of the pizza is expressed in fifth | |
| 03:26 | and this part of the pizzas express intense . We | |
| 03:29 | don't know how to write the answer down . So | |
| 03:31 | what we do is we say , we're going to | |
| 03:33 | add these and we're going to first get a common | |
| 03:36 | denominator , that means you have to have the same | |
| 03:39 | number on the bottom . Now , here's the thing | |
| 03:42 | , I don't care what denominator you use , I | |
| 03:44 | really don't because you're going to get the same answer | |
| 03:47 | no matter what I will show you as we do | |
| 03:49 | problems That you it doesn't matter what denominator you pick | |
| 03:52 | , you will always get the same answer . But | |
| 03:55 | if we look at what we have here , we | |
| 03:56 | have a five and we have a 10 . Okay | |
| 03:59 | , remember I can multiply this fraction by anything I | |
| 04:02 | want as long as I do it to the top | |
| 04:03 | and the bottom . I can also multiply this fraction | |
| 04:06 | by anything I want as long as I do it | |
| 04:08 | to the top and the bottom . So since this | |
| 04:09 | is five and this is 10 , all I have | |
| 04:11 | to do is multiply this fraction by to multiply two | |
| 04:15 | on the top , two on the bottom , That | |
| 04:17 | will make a 10 on the bottom . And I | |
| 04:18 | already have a 10 here . So I'm going to | |
| 04:21 | find an equivalent fraction . I'm gonna change this fraction | |
| 04:24 | by multiplying the top and the bottom . Now this | |
| 04:27 | fraction on the top , the 1/10 is not going | |
| 04:29 | to change at all , I'm going to leave it | |
| 04:31 | there , but this fraction the 3/5 . What I'm | |
| 04:35 | going to do is I'm going to multiply the top | |
| 04:38 | of that fraction by two , and I'm also gonna | |
| 04:40 | multiply the bottom of the fraction of that fraction by | |
| 04:42 | two . So what do I actually get ? This | |
| 04:45 | fraction becomes what it becomes ? Three times two is | |
| 04:48 | six and five times two is what ? 10 ? | |
| 04:52 | 6/10 . So just to keep things tidy , I'm | |
| 04:56 | gonna erase this and I'm going to kind of move | |
| 04:58 | it over here . I'm gonna change the problem . | |
| 05:01 | So it's going to be 1/10 And I'm going to | |
| 05:03 | be adding it to 6/10 . I'm gonna be adding | |
| 05:05 | these guys together right here . So you see what | |
| 05:07 | I have actually done is I have taken this fraction | |
| 05:12 | and I have multiply top and bottom by two , | |
| 05:15 | just so I can change the way this fraction looks | |
| 05:17 | and change it into 6/10 and now the denominators are | |
| 05:21 | the same . So let's just make sure that we | |
| 05:23 | understand now . This 1/10 of course is represented by | |
| 05:26 | 1/10 and this 6/10 is 1/10 . 2/10 3 10th | |
| 05:32 | , 4/10 . Here's 5/10 here's 6/10 . Now I'm | |
| 05:37 | hoping that you can kind of let me kind of | |
| 05:40 | turn it a little bit . I'm hoping that you | |
| 05:41 | can see that this amount of pizza is exactly the | |
| 05:44 | same as this amount of pizza . The 3/5 that | |
| 05:47 | we started with , we just change the way that | |
| 05:49 | the fraction looks , so it looks different , but | |
| 05:52 | since I changed the top and the bottom by the | |
| 05:54 | same thing , I multiplied by double the top and | |
| 05:56 | the bottom . Then all of my slices are smaller | |
| 06:00 | now , but I have more slices , so it's | |
| 06:02 | actually the same amount of pizza . So adding these | |
| 06:04 | together is exactly the same thing as adding these together | |
| 06:07 | . And that's why we find the common denominator because | |
| 06:11 | we change the way things look , but we really | |
| 06:13 | aren't changing the amount of stuff we're adding together , | |
| 06:16 | We're still adding the same amount of stuff . So | |
| 06:18 | what is going to happen here when we add these | |
| 06:20 | together ? Well 10 and 10 or my common denominator | |
| 06:23 | ? So I'll just put that as a 10 and | |
| 06:25 | in the top we have one plus 61 plus six | |
| 06:28 | because we add the numerator as we've talked about that | |
| 06:30 | before . What do we get ? 7/10 ? And | |
| 06:33 | then we ask ourselves , can we simplify this fraction | |
| 06:36 | any further ? Can we divide top and bottom by | |
| 06:39 | anything we want to make this simpler ? And we | |
| 06:41 | can't we can't divide anything top and bottom to change | |
| 06:45 | these numbers because this is an odd number and this | |
| 06:47 | is an even number , so we can't remember . | |
| 06:49 | You can multiply numbers by any fractions by anything you | |
| 06:51 | want , as long as you do it to the | |
| 06:53 | top and the bottom , it doesn't change the fraction | |
| 06:55 | . You can also divide a fraction by anything you | |
| 06:58 | want , as long as you do it to the | |
| 06:59 | top and the bottom . Right ? So the answer | |
| 07:01 | to this is 7/10 . So the answer is the | |
| 07:05 | following . Here's 1/10 here's 2/10 here's 3/10 here's 4/10 | |
| 07:10 | here's 5/10 here's 6/10 and here is 7/10 . Now | |
| 07:15 | let me kind of rotate this and kind of Line | |
| 07:17 | it up a little bit , kind of more like | |
| 07:18 | what we have had before . So essentially what we | |
| 07:21 | have done is we're saying we're adding this plus this | |
| 07:23 | , so we go ahead and put it down here | |
| 07:25 | and you get what you have here , 1234567 notice | |
| 07:30 | this is the amount of pizza I have in my | |
| 07:31 | final answer , it's the same thing as if I | |
| 07:33 | had just taken this and put it down here . | |
| 07:35 | This amount of pizza , if you pick this up | |
| 07:38 | and put it here , it's exactly the correct amount | |
| 07:40 | of pizza . So we started with something that looks | |
| 07:43 | totally different . 1/10 of a pizza and 3/5 of | |
| 07:45 | a pizza . We know that if we add this | |
| 07:48 | to this , then this is the total amount of | |
| 07:50 | pizza in the box , we know that . But | |
| 07:52 | in order to do it with math , we have | |
| 07:55 | to change one or both of these fractions to have | |
| 07:57 | the same bottom . Once we have the same bottom | |
| 08:00 | , then we can just add the tops . Get | |
| 08:02 | 7/10 . This is the amount of pizza which is | |
| 08:05 | exactly the correct amount of pizza . As if I | |
| 08:07 | had just done it this way . So it looks | |
| 08:10 | very different when it's out of tents or whatever . | |
| 08:12 | But that that is the way we're going to be | |
| 08:13 | doing all these problems . We must change the denominators | |
| 08:17 | so that they are the same . All right , | |
| 08:20 | So let's go on to the next problem And we'll | |
| 08:23 | get a little bit more practice as we go . | |
| 08:25 | Let's say that we have the fraction 1/2 And then | |
| 08:30 | we have the fraction 1/4 that we're going to add | |
| 08:34 | to that . So , again notice we have a | |
| 08:36 | four on the bottom and the two on the bottom | |
| 08:38 | . So we cannot add these at least we can't | |
| 08:40 | do it mathematically , we can we can put a | |
| 08:42 | model up there but we cannot really add them together | |
| 08:45 | . So the one half I'll do it this way | |
| 08:48 | . The one half is represented by half of a | |
| 08:50 | pizza like that , and the fourth is represented by | |
| 08:53 | half the pizza like that . So of course I | |
| 08:55 | know if I'm going to add them together , I | |
| 08:57 | just do it like this . Now , you might | |
| 08:59 | be able to tell immediately what that looks like , | |
| 09:01 | what does it look like in terms of fractions of | |
| 09:03 | a pizza , we'll save it to the end and | |
| 09:06 | just kind of line it up with what we get | |
| 09:07 | is our answer . But for now notice that we | |
| 09:08 | have a half of the pizza and I Fourth of | |
| 09:10 | a pizza , but we can't add them mathematically without | |
| 09:13 | using a model because they don't have the same denominator | |
| 09:16 | . So what I'm going to do is I'm gonna | |
| 09:18 | change this 1/4 . Now , if I take a | |
| 09:20 | look , I have a four on the bottom and | |
| 09:22 | the two on the bottom , if I multiply this | |
| 09:24 | fraction by two on the top and two on the | |
| 09:26 | bottom , two times two is four , then it | |
| 09:28 | would match with this denominator . So I only have | |
| 09:30 | to mess around with this one fraction . So what | |
| 09:33 | I will do is I will say , well let | |
| 09:34 | me start out with 1/2 and I can multiply by | |
| 09:37 | whatever I want . So I'm gonna multiply by a | |
| 09:40 | two on the top and a two on the bottom | |
| 09:42 | . And what's that , what's that going to give | |
| 09:43 | me ? It's going to give me 2/4 it's going | |
| 09:48 | to give me 2/4 so 2/4 of a pizza , | |
| 09:51 | two out of four pieces is exactly the same as | |
| 09:53 | one half of the pizza , right ? And then | |
| 09:56 | I'm going to say , well I'm going to add | |
| 09:58 | it to the 1/4 I started with Plus 1/4 and | |
| 10:01 | this is what I'm going to now add . So | |
| 10:03 | I'm going to change the problem , I'm going to | |
| 10:05 | change it from adding these fractions to adding these fractions | |
| 10:09 | because these fractions are exactly equivalent to these . Let's | |
| 10:13 | make sure we understand that here , we've changed this | |
| 10:17 | into 2/4 here . If I cut this pizza here | |
| 10:20 | into force , but I have two pieces , look | |
| 10:23 | at what I have , it's exactly the same thing | |
| 10:25 | as half the pizza . And then of course I | |
| 10:27 | have 1/4 for here as well . So now I | |
| 10:30 | can add these guys together and what I'm going to | |
| 10:33 | get , since I have a four and a four | |
| 10:35 | that stays on the bottom , I don't add them | |
| 10:37 | , I just keep it and then I have two | |
| 10:40 | plus one . You add the new operators two plus | |
| 10:45 | one . So what do you get ? You get | |
| 10:46 | 3/4 and then you ask yourself , can you simplify | |
| 10:51 | this by dividing top and bottom by some number and | |
| 10:55 | you can this is odd and even , so there's | |
| 10:56 | no real way to divide them . So the answer | |
| 10:58 | is 3/4 of a pizza . Let's see if this | |
| 11:00 | makes sense . This answer is basically 1/4 2/4 3/4 | |
| 11:06 | . Of course , if I had 4/4 of the | |
| 11:08 | pizza would be the whole thing . But I only | |
| 11:09 | have 3/4 of the pizza . So this is I'll | |
| 11:13 | just do it like this . This is the amount | |
| 11:15 | of pizza that I actually have there . So if | |
| 11:18 | I take one half of a pizza and 1/4 of | |
| 11:20 | a pizza , we already did it before . Look | |
| 11:22 | at what this is . This is 3/4 of a | |
| 11:24 | pizza , exactly the same as what we get here | |
| 11:26 | . So from using the models or from drawing pictures | |
| 11:29 | , you can kind of figure out what it's supposed | |
| 11:31 | to be . But ultimately , we're going to drop | |
| 11:34 | the drawings here . We're not going to be doing | |
| 11:36 | that for every problem because it just slows us down | |
| 11:39 | . What we need to do is just find a | |
| 11:41 | common denominator and then add add the fractions after we | |
| 11:45 | get a common denominator , which is what we've done | |
| 11:47 | , we've done here , and then we get the | |
| 11:49 | answer and simplify it if we can't . All right | |
| 11:52 | , So this is our third problem . This will | |
| 11:54 | be the last one where I'm going to use the | |
| 11:57 | magnets . I may use it later , but probably | |
| 11:59 | not . I think it's good to get some practice | |
| 12:01 | without let's add the fractions one third and we'll add | |
| 12:06 | to that 1/6 . So before we do anything , | |
| 12:10 | let's see what one third and 16 actually looks like | |
| 12:13 | . So one third of a pizza is something like | |
| 12:15 | this , because this will be two thirds , and | |
| 12:17 | then the three thirds will be down there . So | |
| 12:18 | here is what one third actually looks like , And | |
| 12:21 | then 1/6 of the Pizza looks like this . Of | |
| 12:24 | course , I could have six pieces all the way | |
| 12:26 | around , but I only actually have one of them | |
| 12:28 | , so I'm adding this to this . So of | |
| 12:31 | course , I know that if I actually add them | |
| 12:33 | together , I can line them up and see that | |
| 12:35 | it's going to be look at that , you can | |
| 12:37 | see the answer . It's actually half of a pizza | |
| 12:39 | . Look at that . It's actually exactly equal to | |
| 12:41 | half of pizza . So you can kind of see | |
| 12:42 | the answer . But you got to realize that whenever | |
| 12:45 | you're doing this for larger and larger fractions , you're | |
| 12:47 | not gonna be able to easily draw a picture of | |
| 12:50 | it . So perfectly see here , I have magnets | |
| 12:52 | and I can line them up and it makes it | |
| 12:53 | easy to see . But when you're on your paper | |
| 12:56 | you can't draw pictures of all these fractions forever . | |
| 12:58 | So what we have to do is get a common | |
| 13:01 | denominator . Here , we have a three here , | |
| 13:03 | we have a six . How can I get the | |
| 13:05 | same bottom number ? Well , if I just take | |
| 13:07 | this and multiply by two , two times three or | |
| 13:10 | six and I'll have the same denominator as this fraction | |
| 13:13 | . So what I'm going to do is work with | |
| 13:14 | the top fraction here , I'm gonna say one third | |
| 13:17 | . I can multiply top and bottom by whatever I | |
| 13:20 | want . So this is going to be multiplied by | |
| 13:22 | two , multiplied by two . What do I get | |
| 13:25 | ? One ? Times two is two and two times | |
| 13:28 | three or three times two is six . All right | |
| 13:31 | . So , what I have found here is that | |
| 13:33 | actually to sixth is exactly the same thing as one | |
| 13:37 | third . Right . So what I'm going to do | |
| 13:40 | is take this to six . Let's see where am | |
| 13:43 | I going to do this ? I guess I'll do | |
| 13:44 | it . I'll just say you know what , I'll | |
| 13:47 | do it like this . Let's erase this a little | |
| 13:49 | bit . Let me give myself a little more room | |
| 13:50 | . I'll say this is equal to 26 Like this | |
| 13:54 | . All right . And then this fraction I'm not | |
| 13:57 | going to change at all . I'm gonna leave . | |
| 13:58 | It is 16 Now , I need to add these | |
| 14:00 | fractions together . So all I have done is I've | |
| 14:03 | taken this fraction . I have changed it into this | |
| 14:05 | one , but actually it's the same amount of material | |
| 14:08 | . Let's actually see if that actually makes sense . | |
| 14:11 | To sixth . There's 1/6 there's 2/6 and if it's | |
| 14:15 | hard to see , I'll actually pick them up and | |
| 14:18 | put them right on top of here , and you | |
| 14:19 | can see that this is exactly the same amount of | |
| 14:21 | pizza to six , is exactly the same as the | |
| 14:25 | one third from before . And then of course I'm | |
| 14:27 | still adding it to the 16 here . Right , | |
| 14:30 | So what am I actually going to get ? I | |
| 14:33 | have a bottom number of six , so that'll stay | |
| 14:35 | and then two plus one . And what do I | |
| 14:38 | get an answer of ? 3/6 36 Right . And | |
| 14:44 | then I'm going to say , can I simplify this | |
| 14:47 | fraction ? 36 Can we make it any simpler ? | |
| 14:50 | And the answer is yes , I can divide top | |
| 14:52 | and bottom by three , so I can start here | |
| 14:55 | with 36 and I can divide the top by three | |
| 14:59 | in the bottom by three . I'm always looking for | |
| 15:00 | a way to make it simpler . At the end | |
| 15:02 | , Divide by three divided by three . Three divided | |
| 15:05 | by three is one and six divided by three is | |
| 15:08 | too . So what do I get one half of | |
| 15:11 | a pizza ? So let's make sure we understand when | |
| 15:14 | we add this to this , what do we get | |
| 15:16 | ? We already kind of know because we used the | |
| 15:18 | magnets but here's the two from here , we're gonna | |
| 15:20 | add one more . This is what 3 , 6 | |
| 15:23 | is actually equal to which is exactly the same thing | |
| 15:27 | as one half . So we'll do it this way | |
| 15:29 | and we'll see now that this 36 is the answer | |
| 15:34 | . It is three out of six slices . That | |
| 15:36 | is a very valid way of writing down the answer | |
| 15:39 | . There's many different ways of writing down the answer | |
| 15:41 | . Basically you have this much pizza plus this much | |
| 15:44 | pizza . You can put it there right and you | |
| 15:46 | can express the answer in a couple of ways . | |
| 15:48 | You can say that it's +36 of a pizza , | |
| 15:50 | which is this amount , which is exactly the same | |
| 15:53 | as what you have here . This is half of | |
| 15:54 | a pizza also , right ? But what we always | |
| 15:57 | try to do is make the numbers as small as | |
| 15:59 | we can . So one half is the smallest simplest | |
| 16:01 | way of writing this amount of pizza . And this | |
| 16:03 | is really the best answer that we're going to write | |
| 16:06 | . So the final answer is one half . So | |
| 16:08 | now what we're gonna do is crank through the rest | |
| 16:10 | of our problems and we're not going to use the | |
| 16:13 | magnets anymore because they're very good to visualize what's happening | |
| 16:17 | in the beginning to understand why we're finding a common | |
| 16:19 | denominator , because it allows us then to just add | |
| 16:21 | two pieces , right ? That's what we're doing . | |
| 16:23 | And then we simplify the result and get it into | |
| 16:26 | lowest terms . But after the first few problems , | |
| 16:28 | we need to get some practice , you know , | |
| 16:31 | without , without doing that , without having that . | |
| 16:33 | Uh for every problem . So let's take a look | |
| 16:37 | at the fraction two tens and we'll add it to | |
| 16:41 | the fraction 31 hundred's . Now this is a good | |
| 16:44 | example , we want to add these fractions together . | |
| 16:47 | But the problem is that this is two out of | |
| 16:50 | 10 and this is three out of 100 . So | |
| 16:52 | if I wanted to put magnets on the board , | |
| 16:53 | I would need really small magnets . So the circle | |
| 16:56 | was sliced into 100 pieces . It would be really | |
| 16:59 | , really hard to do . I don't have magnets | |
| 17:00 | cut . And even if you tried to draw it | |
| 17:02 | , you can't draw 100 slices in a pizza very | |
| 17:05 | easily . But what we know is that we just | |
| 17:08 | need to change one of these or both of these | |
| 17:10 | , two have the same bottom number . What happens | |
| 17:12 | if I change this by multiplying by 10 ? If | |
| 17:16 | I multiply by 10 , 10 times 10 is 100 | |
| 17:19 | , then it would match this denominator . That is | |
| 17:21 | the strategy we're going to use here . We're going | |
| 17:24 | to take the top number , two Tents and we're | |
| 17:27 | going to change it by multiplying the top and the | |
| 17:30 | bottom by 10 . Top of the mountain by 10 | |
| 17:34 | . What do we get ? Two times 10 is | |
| 17:36 | 20 and 10 times 10 is 100 . So now | |
| 17:40 | we have a common denominator which which matches . And | |
| 17:43 | then we could just add three over 100 to it | |
| 17:46 | . We have to make the denominators are the same | |
| 17:49 | . Yeah , we have 100 as the common denominator | |
| 17:53 | . So we put it on the bottom . What | |
| 17:54 | do we do for the numerator ? 20 plus 3 | |
| 17:56 | , 20 plus three . What do we get ? | |
| 17:59 | 23 out of 123 . 1 hundred's is another way | |
| 18:04 | to say that . And we try to say well | |
| 18:06 | can we simplify this by dividing top and bottom by | |
| 18:09 | some number ? And we really can't . There's not | |
| 18:11 | a clean way to do that . You can't divide | |
| 18:13 | by two . For instance , you can't divide by | |
| 18:14 | three . There's nothing I can divide the top and | |
| 18:18 | the bottom by to make both of those numbers any | |
| 18:20 | simpler ? So the answer is 23 1 hundreds . | |
| 18:23 | So if I start with 3100 and I start with | |
| 18:26 | 2/10 I can express the 2/10 as 2100 and then | |
| 18:30 | I add them and then 23 100 says the final | |
| 18:32 | answer . And that is how we do these problems | |
| 18:35 | . So let me go . I think I skipped | |
| 18:38 | around a little bit . Let's move along to the | |
| 18:42 | to the next problem , which is The problem . | |
| 18:47 | The next problem is a very similar problem to this | |
| 18:50 | one . What about 1/10 ? And we'll add to | |
| 18:54 | it 1100s . We have the same problem , We | |
| 19:00 | have 10 and 100 . So we want to find | |
| 19:03 | a common denominator and all we have to do is | |
| 19:05 | multiply this by 10 and that will be 100 . | |
| 19:07 | It'll match the other fraction . So let's start out | |
| 19:10 | with 1/10 and let's multiply top and bottom by what | |
| 19:14 | , by 10 ? Because that's going to give us | |
| 19:17 | 100 . So one times 10 is 10 and then | |
| 19:20 | 10 times 10 is 100 just like this . And | |
| 19:24 | we're adding to it 11 , 1 hundreds , 1100s | |
| 19:29 | . And now the denominators match completely so we can | |
| 19:32 | add these fractions together . The 100 is in the | |
| 19:35 | bottom in both cases . So we have 100 then | |
| 19:38 | 10 plus 11 , 10 plus 11 . So what | |
| 19:42 | happens when you add 10 plus 11 , you get | |
| 19:44 | 21 121 out of 100 . Now , if you | |
| 19:49 | try to simplify this answer and divide top and bottom | |
| 19:53 | by some number to make it simpler , you can't | |
| 19:55 | do it . So that is the final answer . | |
| 19:56 | 21 1 hundreds . So I hope you see the | |
| 20:01 | the plan here , the way that we're going about | |
| 20:03 | it here is very , very consistent . It's very | |
| 20:07 | much the same Every time we do this , let's | |
| 20:10 | take a look at the next problem . Let's say | |
| 20:12 | we have 2/8 , and we're going to add to | |
| 20:16 | that 1/4 2/8 and we're going to add 1/4 to | |
| 20:21 | it . All right , So what are we going | |
| 20:23 | to do ? We have an eight and we have | |
| 20:25 | a four . Right ? So we want to add | |
| 20:27 | those guys together , we have a four and eight | |
| 20:29 | . Now , if I take this bottom fraction and | |
| 20:31 | just multiply it by 22 on the top and bottom | |
| 20:34 | two times four is eight , and it will match | |
| 20:35 | this one . So , what I'm going to do | |
| 20:38 | is take the bottom one here , the bottom fraction | |
| 20:40 | , the 1/4 . And I'm going to multiply top | |
| 20:43 | and bottom of this fraction by two . And it's | |
| 20:46 | going to give me on that . It's gonna be | |
| 20:48 | 28 So two eights is exactly the same as 1/4 | |
| 20:53 | . And so I'm going to add to that . | |
| 20:55 | Two eights from the top . I guess I should | |
| 20:57 | put the plus sign probably down here somewhere I guess | |
| 21:00 | I'll put the plus sign over there so have to | |
| 21:03 | eight plus 2/8 now what do I get for the | |
| 21:05 | result ? I have a common denominator of eight and | |
| 21:10 | the numerator is 22 plus two here , so two | |
| 21:12 | plus two Is what ? 4 ? So I'm gonna | |
| 21:15 | have a 4/8 , 4/8 . And then I ask | |
| 21:21 | myself can I simplify 4/8 ? I'll put a 4/8 | |
| 21:24 | here and the answer is yes I can I can | |
| 21:26 | divide by two if I want to but I actually | |
| 21:29 | see that I can divide by four . Remember you're | |
| 21:32 | going to save steps if you divide by the biggest | |
| 21:34 | thing you can That will fit , that will divide | |
| 21:36 | evenly . So I will divide the top by four | |
| 21:39 | and the bottom I will also divided by four . | |
| 21:42 | So four divided by four is 18 divided by four | |
| 21:45 | is two because I'm sorry four divided by four . | |
| 21:48 | I said it was one and of course wrote the | |
| 21:49 | wrong thing down should be a one and then a | |
| 21:51 | divided by four is two . So you get one | |
| 21:53 | half , two times four is eight and one times | |
| 21:56 | four is four . So the answer is one half | |
| 21:59 | . So it doesn't look like it can be one | |
| 22:00 | half because it's 2/8 And you're adding 1/4 . And | |
| 22:04 | then we're saying somehow through all of this it's half | |
| 22:06 | of a pizza . Well I said I wasn't going | |
| 22:08 | to use the magnets anymore , but actually it's kind | |
| 22:10 | of irresistible . I do want to use them this | |
| 22:12 | one last time . So what I'm going to do | |
| 22:14 | is say , well this fraction this two eights is | |
| 22:17 | if I cut a pizza into eight equal slices and | |
| 22:20 | have two of them . This is how much pizza | |
| 22:22 | I actually have . And then I'll go over here | |
| 22:24 | and grab my fourths and say 1/4 of the pizza | |
| 22:28 | , I'll just put it down here is equal to | |
| 22:30 | this . So I'm adding this to this . So | |
| 22:32 | let's just for giggles come over here and actually add | |
| 22:34 | them together . You can see that it's going to | |
| 22:36 | be exactly half of a pizza , which is what | |
| 22:39 | we actually got . But let's kind of keep them | |
| 22:41 | separate for now and let's kind of map out how | |
| 22:44 | we did our solution , we change the two aides | |
| 22:46 | . We know we kept the two eights the same | |
| 22:49 | , right ? So we can kind of move these | |
| 22:50 | over here so we have to aids and then we | |
| 22:53 | said that 1/4 was also equal to two eights , | |
| 22:56 | right ? Which makes sense because if I take two | |
| 22:58 | eights and stick them in here , it's exactly the | |
| 23:00 | same thing . So we're basically saying this these two | |
| 23:03 | plus these two and we said that the answer was | |
| 23:06 | four eights , that's one two , 3 4/8 . | |
| 23:11 | Because we add this , we add this , we | |
| 23:12 | get four eights , look that's half of a pizza | |
| 23:14 | . So I grabbed my little half , show you | |
| 23:16 | that it's exactly the same thing and then that's exactly | |
| 23:19 | the same thing . So I know I'm kind of | |
| 23:20 | like shuffling things around the board here . But the | |
| 23:23 | point is is that when you add these things that | |
| 23:25 | look different you get the common denominators then you can | |
| 23:27 | very easily see how the pieces will go together . | |
| 23:30 | The answer is 48 which is a perfectly fine way | |
| 23:34 | of telling me what the answer is . But this | |
| 23:36 | one half is a simpler way with smaller numbers to | |
| 23:39 | tell me the exact same amount of the pizza . | |
| 23:43 | Alright , the next problem is 3/6 and we're going | |
| 23:47 | to be adding that to one third right now . | |
| 23:51 | I know I said I wouldn't use the magnets anymore | |
| 23:52 | but it's kind of irresistible because I do want you | |
| 23:55 | to visualize this . So 36 is represented by 162636 | |
| 24:01 | , write something like this which looks exactly like half | |
| 24:04 | of a pizza , right ? And then the one | |
| 24:06 | third is represented by one third . Of course 52 | |
| 24:10 | 3rd and three thirds . It would be cut into | |
| 24:11 | three pieces . I have one third . So how | |
| 24:14 | do I get a common denominator ? I have a | |
| 24:15 | six and a three . So all I have to | |
| 24:17 | do is multiply this fraction by two over to multiply | |
| 24:21 | by two and then I will have six on the | |
| 24:24 | bottom , so I will take this fraction . Let | |
| 24:27 | me scoot this over a little bit and I'll say | |
| 24:29 | 1/3 . Let me multiply this fraction on the top | |
| 24:34 | by two and on the bottom by two , what | |
| 24:37 | am I going to get ? One times two is | |
| 24:39 | two and three times two is six . And then | |
| 24:42 | of course I'm gonna have 36 here and I'm gonna | |
| 24:46 | add these guys together . So you can see it's | |
| 24:48 | a little bit clunky the way I'm doing this , | |
| 24:50 | I'm trying to show you that I'm changing the fraction | |
| 24:53 | and then I'm adding 36226 here . Okay , now | |
| 24:57 | what's gonna happen is I have the same bottom number | |
| 25:00 | , so the same denominator is a six and then | |
| 25:03 | three plus 23 plus two is going to be equal | |
| 25:06 | to 56 Can I simplify this fraction ? I cannot | |
| 25:11 | divide top and bottom to make it any simpler ? | |
| 25:13 | So I just say that it's basically finished . So | |
| 25:17 | what do I have here ? This of course we | |
| 25:19 | already just said is 36 to arrange it like this | |
| 25:22 | , that one actually didn't change the bottom fraction was | |
| 25:25 | one third , but we changed it to to six | |
| 25:28 | , which is this much pizza . We're saying the | |
| 25:30 | answer is 16263646 56 which is exactly what you would | |
| 25:38 | get . If you just stick these guys together , | |
| 25:40 | you get 56 So it's almost a complete pizza . | |
| 25:43 | Right ? And that's why it makes sense because if | |
| 25:45 | you add this much pizza to this much pizza , | |
| 25:49 | look at what you have . This is the amount | |
| 25:51 | of pizza that you have which exactly matches this . | |
| 25:53 | Of course I spent it and rotated to make it | |
| 25:55 | line up . But you get the idea , you | |
| 25:57 | add these guys together , this is what you get | |
| 26:00 | and this is what's happening . You're changing these things | |
| 26:02 | which look different into something that I know how to | |
| 26:04 | add and then I can get the final answer down | |
| 26:07 | below . All right , Let's take a look at | |
| 26:12 | problem # eight . So what I have here is | |
| 26:16 | 6/10 and I'm adding it to 71 hundred's . I | |
| 26:21 | don't have fractions with 100 slices , so I can't | |
| 26:24 | really use a fraction here , a magnet here , | |
| 26:27 | but I have 10 and 100 . So I know | |
| 26:29 | I can multiply this by 10 . So give me | |
| 26:31 | 100 on the bottom for both . So I'll start | |
| 26:33 | with 6/10 and I already know that I can multiply | |
| 26:38 | top and bottom by whatever I want . So , | |
| 26:39 | I'm gonna multiply 10 on the top 10 on the | |
| 26:42 | bottom , so I'll get 60 on the top and | |
| 26:45 | 100 on the bottom . And I'm going to add | |
| 26:48 | it to the 71 hundred's that I already had in | |
| 26:52 | the original problem . So now the denominators are the | |
| 26:56 | same . 100 is on the bottom and I add | |
| 26:58 | 60 plus seven . And what do I get , | |
| 27:02 | 67 67 . 100 . And then you say , | |
| 27:07 | can I divide top to bottom by something to make | |
| 27:09 | this simpler ? And you really can't . This is | |
| 27:11 | an odd number . This is an even number . | |
| 27:13 | There's really no way that I can no number I | |
| 27:15 | can think of to divide the top and the bottom | |
| 27:17 | by the same exact number to make it any simpler | |
| 27:22 | . So that is the final answer to that problem | |
| 27:25 | . Yeah . All right . Only two more . | |
| 27:27 | Let's take a look at to 12th and we'll add | |
| 27:31 | to that 16 Now , this will be the last | |
| 27:35 | time , I promise for a while then I'll use | |
| 27:37 | the magnets . But again , it's kind of irresistible | |
| 27:39 | . I'd like you to visualize with me what we | |
| 27:41 | have here . Pizza cut into 12 slices . You | |
| 27:44 | only have two of the slices . This is how | |
| 27:46 | much pizza you have to 12th . This is a | |
| 27:49 | pizza cut into eight equal slices , but you only | |
| 27:52 | have one slice , one out of 6 , 1/6 | |
| 27:56 | . Now , what you wanna do is basically add | |
| 27:58 | these together like this , that's what's going to happen | |
| 28:00 | in the final answer . But how do I do | |
| 28:01 | it with math and figure out exactly what the final | |
| 28:04 | answer is . I need the bottom numbers , the | |
| 28:06 | denominators to be the same . I have a six | |
| 28:09 | and 12 . If I multiply this by 22 times | |
| 28:12 | six is 12 , it'll match this fraction . So | |
| 28:14 | the top fraction I'll keep the same . 2 , | |
| 28:16 | 12 . The bottom fraction I will say . Um | |
| 28:20 | actually I probably shouldn't have written that right there , | |
| 28:22 | let's say 16 and I'll multiply top and bottom by | |
| 28:27 | two . And to what am I going to get | |
| 28:30 | ? One times two is two and six times 2 | |
| 28:33 | is 12 . And I'm going to add these guys | |
| 28:35 | together like this actually put the plus on on the | |
| 28:37 | other side just so we don't confuse ourselves , so | |
| 28:39 | that's equal to that . We'll add these guys together | |
| 28:43 | . So now I have a 12 and 12 , | |
| 28:45 | 12 on the bottom , add two plus to add | |
| 28:49 | enumerators . What do I get ? Four out of | |
| 28:52 | 12 ? We think this is the correct answer , | |
| 28:54 | but we always ask can we make it simpler ? | |
| 28:57 | Of course they're both even numbers . I can divide | |
| 28:59 | by two . That's fine , divide by two . | |
| 29:02 | I'll get some other fraction but then I'll see that | |
| 29:04 | I can divide it by two again . But then | |
| 29:06 | I noticed instead of dividing by two I can divide | |
| 29:09 | both of these by four . You always save a | |
| 29:11 | little time . If you can pick the largest number | |
| 29:13 | you can think of to divide by . So I | |
| 29:15 | have here 4/12 . I will divide the top and | |
| 29:20 | the bottom both . I will divide the top by | |
| 29:23 | four and I will divide the bottom by four . | |
| 29:25 | Also I know they're both divisible by four . Four | |
| 29:28 | divided by four is one and 12 divided by four | |
| 29:31 | is three . So I think the answer is one | |
| 29:34 | third of a pizza . That's what I think the | |
| 29:36 | answer is . Let's see if it makes sense . | |
| 29:38 | Yes . Okay Up here was to 12th which is | |
| 29:42 | exactly what we started out with And then we change | |
| 29:45 | the 16 . We changed it also into 212 . | |
| 29:49 | Right so now we have a common denominator and I | |
| 29:51 | want you first of all to make sure you understand | |
| 29:53 | the 16 is exactly equal to the to 12th . | |
| 29:56 | It's the same amount of pizza when I add them | |
| 29:58 | . I get four slices when it's cut into 12 | |
| 30:02 | pieces , 1 12 to 12 3/12 4 12 . | |
| 30:06 | So this is the answer . And then I simplify | |
| 30:09 | that to 1/3 , which is one third of a | |
| 30:12 | pizza . And so we think this is the answer | |
| 30:14 | . Notice that the one third is exactly equal to | |
| 30:16 | the 4 12 . This is the same amount of | |
| 30:18 | pizza . This is a simpler way of writing it | |
| 30:20 | . If I take the original fractions and add them | |
| 30:23 | together , it's exactly equal to one third right there | |
| 30:27 | , exactly the same thing . So I'm just showing | |
| 30:29 | you visually how it doesn't often look in your mind | |
| 30:33 | , you can't figure out what's going to happen . | |
| 30:35 | You have to put your mind to work by changing | |
| 30:38 | these denominators , putting them into the same denominator so | |
| 30:42 | that you can add the slices together , get the | |
| 30:45 | answer and then simplify it . This is going to | |
| 30:47 | be the process will use for every one of these | |
| 30:49 | problems . Now we only have one more left . | |
| 30:53 | I'm not gonna use magnets on this one so it'll | |
| 30:55 | go a little faster Because what I have here is | |
| 30:58 | 3/10 and I have uh c 3/10 and then 31 | |
| 31:03 | hundreds . I want to add these guys together . | |
| 31:07 | I have a 110 . I can just change this | |
| 31:10 | by multiplying by 10/10 . So what I will do | |
| 31:13 | is I'll say 3/10 , I will change this fraction | |
| 31:17 | by multiplying the top by 10 and the bottom by | |
| 31:20 | 10 . And that will give me three times 10 | |
| 31:22 | is 30 And then 10 times 10 is 100 . | |
| 31:28 | So this , 3100 is the same thing as 3/10 | |
| 31:31 | . And then I'm gonna add it to 31 hundreds | |
| 31:35 | . Now I have the same bottom number , the | |
| 31:37 | same denominator in the bottom , so the same denominator | |
| 31:41 | means that stays in my final answer . 30 plus | |
| 31:45 | three is equal to 33 out of 100 slices , | |
| 31:50 | 33 1 hundreds , And that's the final answer . | |
| 31:54 | Now , I don't really care who you are . | |
| 31:56 | I don't think it's possible for most people to look | |
| 31:59 | at this and say , oh yeah , the answer | |
| 32:00 | is 33/100 . I just don't think it's possible . | |
| 32:03 | It isn't possible for me to do it . I | |
| 32:05 | mean , there's a lot of really smart people out | |
| 32:07 | there , so maybe you can look at that and | |
| 32:08 | figure it out . But most of us can't just | |
| 32:10 | by looking at it . So when you look at | |
| 32:12 | a fraction problem and you think I have no idea | |
| 32:14 | what to do , I have no idea what the | |
| 32:16 | answer is , that's okay . Nobody knows what the | |
| 32:19 | answer is . By just looking at these problems . | |
| 32:20 | I don't So what you have to do is take | |
| 32:23 | one step at a time to add fractions together , | |
| 32:26 | we have to have a common denominator , so if | |
| 32:29 | we don't have a common denominator , change one or | |
| 32:32 | sometimes we'll have to change both fractions here . We | |
| 32:34 | didn't have to but sometimes we have to change both | |
| 32:36 | fractions to get a common denominator . I don't care | |
| 32:39 | what denominator you get . Once you have a denominator | |
| 32:42 | you can add the numerator is keep the denominator , | |
| 32:45 | whatever answer you get , try to simplify it and | |
| 32:48 | what's happening is you're just changing this so that it | |
| 32:51 | is the same amount of pizza . But it's in | |
| 32:53 | terms of the same number of slices , that's why | |
| 32:55 | we get a common denominator because to add things together | |
| 32:58 | we have to have the same size slice and then | |
| 33:01 | we add them simplify and then we can teach ourselves | |
| 33:03 | with drawing pictures and models that actually it all works | |
| 33:07 | out to give me the same amount of pizza when | |
| 33:09 | I add these together . It's exactly the same as | |
| 33:12 | what we answer . Said that the answer should be | |
| 33:14 | and that's how I want you to do every one | |
| 33:16 | of these problems . So I'd like you to grab | |
| 33:18 | a piece of paper , solve these yourself , follow | |
| 33:20 | me on to the next lesson . We're going to | |
| 33:22 | get a lot more practice with adding and then later | |
| 33:24 | on subtracting fractions that have unlike denominators . |
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Adding Fractions with Unlike Denominators - Part 1 (Fraction Addition) is a free educational video by Math and Science.
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