Math Antics - Simplifying Fractions - By mathantics
Transcript
00:05 | in a previous video , we learned all about factoring | |
00:08 | whole numbers and now we're going to learn how factoring | |
00:11 | can help us when working with fractions , we're going | |
00:13 | to learn how to simplify fractions simplifying a fraction means | |
00:17 | rewriting the fraction using the smallest top and bottom numbers | |
00:20 | we can without changing the value of the fraction . | |
00:24 | To help us understand what simplifying a fraction really means | |
00:27 | . Let's take a look at the simplest fraction , | |
00:29 | I can think of 1/2 . Now this is already | |
00:32 | as simple as it can possibly be . So let's | |
00:35 | go the other way . And complicated by dividing our | |
00:37 | rectangle here up into more parts , the amount of | |
00:41 | our rectangle that shaded is still the same . But | |
00:44 | now the numbers for our fraction are 3/6 . The | |
00:47 | numbers are bigger because our rectangle is now divided into | |
00:49 | more parts . The fraction we have now 3/6 is | |
00:53 | equivalent to our original fraction . 1/2 . That means | |
00:57 | they have the same value , they represent the same | |
00:59 | amount . So what if someone gives you the fraction | |
01:01 | 3/6 ? Like 3/6 of a candy bar ? Well | |
01:04 | we know from our picture that that means they're really | |
01:06 | giving you one half . But how can we show | |
01:09 | that using math and not pictures ? Well that's where | |
01:12 | factoring comes in . Let's take our complicated fraction 3/6 | |
01:16 | and factor both the top and bottom numbers . Now | |
01:19 | the bottom number six can be factored into two times | |
01:22 | three . The top number three is a prime number | |
01:26 | . It's only factors are one and itself so we | |
01:29 | can write that as one times three there we've rewritten | |
01:32 | our fraction using factoring and now it kind of looks | |
01:35 | like two fractions being multiplied together . 1/2 times 3/3 | |
01:40 | . Of course 3/3 is what I like to call | |
01:43 | a whole fraction since its value is equal to one | |
01:46 | . Now here's the interesting part since 3/3 equals one | |
01:50 | and multiplying by one has no effect on a number | |
01:53 | . We can just get rid of that . 3/3 | |
01:55 | , basically the three on the top and the three | |
01:58 | on the bottom cancel each other out . And once | |
02:01 | they're gone we're left with the fraction 1/2 . So | |
02:05 | that means that the fraction 3/6 simplifies to one over | |
02:08 | to another way of thinking about it is that we're | |
02:11 | trying to find any whole fractions that are hiding in | |
02:14 | the fraction . Were trying to simplify and if we | |
02:17 | find any we can just get rid of them . | |
02:19 | And the fraction we're left with is simpler than the | |
02:21 | one we started with . Now that we know the | |
02:23 | basics . Let's learn the procedure for simplifying fractions . | |
02:26 | First replace the top and bottom numbers of the fraction | |
02:29 | with their prime factors . Next look to see if | |
02:32 | any of the factors are the same on the top | |
02:34 | and bottom . If they are , then we call | |
02:37 | them common factors because there's something that both the top | |
02:39 | and bottom have in common . If you find a | |
02:42 | pair of common factors , you can cancel them out | |
02:44 | . Just draw a line through them like this . | |
02:47 | Yeah . And last once all the common factors have | |
02:49 | been canceled , you need to re multiply any factors | |
02:52 | that are left over on the top or bottom . | |
02:54 | This makes sure that you end up with only one | |
02:56 | number on the top and bottom of your simplified fraction | |
03:00 | . Oh , there's one important thing to remember . | |
03:02 | If you're ever able to cancel out all of the | |
03:04 | factors on the top or bottom of a fraction , | |
03:07 | don't be tempted to write in a zero , put | |
03:09 | a one in there instead . The reason you can | |
03:11 | write in a one is because one is always a | |
03:14 | factor of any number . It's just we usually don't | |
03:16 | write it in . For instance if you're going to | |
03:19 | factor the number 15 , you just say that it's | |
03:21 | five times three but you could also say that it's | |
03:24 | five times three times one . In fact you could | |
03:27 | even say it's five times three times one . See | |
03:32 | why there's always a one left over when you're canceling | |
03:35 | common factors . All right . So that's the basic | |
03:38 | idea behind simplifying fractions . And once you know the | |
03:40 | procedure it's really not that hard but you might want | |
03:44 | to re watch this video just to make sure you've | |
03:45 | got the idea . Now there aren't any exercises for | |
03:48 | this video because it's really just an introduction . But | |
03:51 | in part Two we'll see a couple more examples of | |
03:53 | how you can use the procedure to simplify fractions and | |
03:56 | then you'll get plenty of exercise is to do his | |
03:58 | homework . Oh yeah . Mhm . Uh huh . | |
04:03 | Mhm . Welcome to Part two of simplifying fractions . | |
04:06 | In Part one , we learned the procedure for simplifying | |
04:09 | fractions basically you just take the top and bottom numbers | |
04:12 | and factor them down to their prime factors . And | |
04:15 | then you see if there's any factors that are the | |
04:17 | same on the top and bottom , we call those | |
04:19 | common factors and if there are you just cancel them | |
04:22 | out and once you've cancelled out all the common factors | |
04:26 | , you re multiply whatever is left over to get | |
04:28 | your final answer in this video , we're going to | |
04:31 | see a couple examples of how we can use that | |
04:33 | procedure to simplify fractions . Let's start with an easy | |
04:36 | one . Let's simplify the fraction . 5/15 . Step | |
04:41 | one is to factor the top and bottom numbers so | |
04:44 | we know that 15 factors into five times three And | |
04:48 | five is a prime number . That means it's only | |
04:50 | factors are one and itself , but one is always | |
04:53 | a factor . So we don't need to write that | |
04:55 | down . Step two is to look for common factors | |
04:58 | and cancel them and we can see that there's a | |
05:00 | five on the top and there's a five on the | |
05:01 | bottom . There not directly over each other , but | |
05:04 | that doesn't matter . They still form a common factor | |
05:06 | pair and so we can cancel them out . Like | |
05:08 | this , Step three is to reorganize our answer . | |
05:12 | Now we don't have any factors that need to be | |
05:14 | recombined by multiplying we just have a three on the | |
05:17 | bottom and we don't have any factors left over on | |
05:19 | top . But you'll remember that there's always a factor | |
05:22 | of one . So 5/15 simplifies to 1/3 . Yeah | |
05:28 | . All right . I think we need to see | |
05:30 | another example , but a harder one this time . | |
05:33 | Let's simplify the fraction 30/36 . The procedure is the | |
05:37 | same . Step one is we factor the top and | |
05:40 | bottom numbers all the way down to their prime factors | |
05:43 | . Let's do the top number . 1st , 30 | |
05:46 | factors into five times six . Five is prime , | |
05:49 | But six can be factored into two times three , | |
05:53 | so are 30 on top becomes five times two times | |
05:57 | three . Now , the bottom number 36 can be | |
06:00 | factored into six times six and each of those six | |
06:03 | is can be factored into two times three . So | |
06:06 | our bottom number becomes two times three times two times | |
06:10 | three . Well , it looks like we do have | |
06:12 | some common factors . There's a two on both the | |
06:15 | top and bottom that will cancel each other out . | |
06:18 | And even though there's more than 12 on the bottom | |
06:20 | , we can only cancel one of them out because | |
06:22 | there's only 12 on top . Remember you always have | |
06:25 | to cancel common factors as pairs . Now we can | |
06:29 | see that there's another pair , we can cancel , | |
06:31 | there's a three on both the top and bottom so | |
06:34 | we can just cross those out . Okay , that's | |
06:36 | all the common factors we can cancel . So now | |
06:39 | all we have to do is see what's left over | |
06:41 | . We have a five on the top and at | |
06:43 | two times three on the bottom , we don't want | |
06:45 | to leave our problem looking like this so we need | |
06:47 | to recombine any factors that didn't cancel . That means | |
06:51 | multiplying together are two and three on the bottom , | |
06:53 | which gives us six . There were left with the | |
06:56 | fraction 5/6 . That's a simplified form of the fraction | |
07:00 | 30/36 . They both have the same value , but | |
07:03 | the simplified one is written using the smallest numbers possible | |
07:08 | . Now , some of you may have been taught | |
07:09 | that the way to simplify fractions is to find the | |
07:12 | greatest common factor of the top and bottom numbers and | |
07:14 | just cancel that . Basically , that's what we are | |
07:17 | doing when we cancel all of the common factors using | |
07:20 | our procedure . In fact , if you multiply all | |
07:24 | of the common factors together , you'll get the greatest | |
07:27 | common factor or G C F is I like to | |
07:30 | call it , you know , to to sound cool | |
07:32 | . All right . So that's how you simplify fractions | |
07:36 | . But I'll bet some of you are wondering why | |
07:37 | would we even want to simplify fractions ? That's a | |
07:40 | good question . Basically . It's to make life simpler | |
07:43 | . Well , at least for your teacher who has | |
07:45 | to grade all your homework for you , it just | |
07:48 | makes life more complicated . No , just kidding , | |
07:51 | simplified fractions . Make your life easier to because usually | |
07:55 | simplified fractions are much easier to work with . For | |
07:58 | example , if your friends said to you here , | |
08:01 | you can have 27 54th of my sandwich , it | |
08:03 | would have been much easier if they had just said | |
08:06 | that you could have one half of their sandwich , | |
08:08 | since one half is the simplified form of 27 54th | |
08:12 | . So now whenever you see a fraction , you | |
08:14 | can ask yourself , could that be any simpler ? | |
08:17 | And if so , you'll know just what to do | |
08:20 | . So get on out there , work on those | |
08:22 | exercises , start making the world a simpler place for | |
08:25 | assault , learn more at math antics dot com . |
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