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  • In a previous video, we learned all about factoring whole numbers.

  • And now were gonna learn how factoring can help us when working with fractions.

  • Were gonna learn how to simplify fractions.

  • Simplifying a fraction means re-writing the fraction using the smallest top and bottom numbers we can without changing the value of the fraction.

  • To help us understand what simplifying a fraction really means,

  • let’s take a look at the simplest fraction I can think of: 1 over 2

  • Now this is already as simple as it can possibly be.

  • So, let’s go the other way andcomplicateit by dividing our rectangle here up into more parts.

  • The amount of our rectangle that’s shaded is still the same, but now the numbers for our fraction are 3 over 6.

  • The numbers are bigger because our rectangle is now divided into more parts.

  • The fraction we have now (3 over 6) is equivalent to our original fraction (1 over 2)

  • That means they have the same value. They represent the same amount.

  • So what if someone gives you the fraction 3 over 6. (Like three-sixths of a candy bar)

  • Well we know from our picture, that means theyre really giving you one-half.

  • But how can we show that using math and not pictures?

  • Well, that’s where factoring comes in!

  • Let’s take ourcomplicatedfraction (3 over 6) and factor both the top and bottom numbers.

  • Now the bottom number (6) can be factored into 2 × 3.

  • The top number (3) is a prime number. Its only factors are ‘1’ and itself, so we can write that as 1 × 3.

  • Therewe have re-written our fraction using factoring,

  • and now it kinda looks like two fractions being multiplied together: (1 over 2) times (3 over 3)

  • Of course, 3 over 3 is what I like to call a “whole fractionsince its value is equal to 1.

  • Now here’s the interesting part...

  • Since 3 over 3 equals 1, and multiplying by 1 has no effect on a number, we can just get rid of that 3 over 3.

  • Basically, the 3 on the top and the 3 on the bottom cancel each other out.

  • And once theyre gone, were left with the fraction: 1 over 2.

  • So that means that the fraction 3 over 6 simplifies to 1 over 2.

  • Another way of thinking about it is that were trying to find any whole fractions that arehidingin the fraction we are trying to simplify.

  • And if we find any, we can just get rid of them and the fraction were left with is simpler than what we started with.

  • Now that we know the basics, let’s learn the procedure for simplifying fractions.

  • First: Replace the top and bottom numbers of the fraction with their prime factors.

  • Next: Look to see if any of the factors are the same on the top and bottom.

  • If they are, we call themcommon factorsbecause theyre something that both the top and bottom have in common.

  • If you find a pair of common factors, you can cancel them out. Just draw a line through them like this.

  • And Last: Once all the common factors have been canceled,

  • you need to re-multiply any factors that are left over on the top or bottom.

  • This makes sure that you end up with only one number on the top and bottom of your simplified fraction.

  • Ohand there’s one important thing to remember.

  • If youre ever able to cancel out ALL of the factors on the top or bottom of a fraction,

  • don’t be tempted to write in a zero. Put a ‘1’ in there instead!

  • The reason you can write in a ‘1’ is because ‘1’ is ALWAYS a factor of ANY number.

  • It’s just we usually don’t write it in.

  • For instance, if youre gonna factor the number 15, you just say that it’s 5 × 3.

  • But you could also say that it’s 5 × 3 × 1.

  • In fact, you could even say it’s 5 × 3 × 1 × 1 × 1 × 1 × 1 × 1…..

  • see why there’s always a ‘1’ left over when youre canceling common factors?

  • Alright, so that’s the basic idea behind simplifying fractions.

  • And once you know the procedure, it’s really not that hard.

  • But you might want to re-watch this video just to make sure youve got the idea.

  • Now there aren’t any exercises for this video cuz it’s really just an introduction.

  • But in Part 2, well see a couple more examples of how you can use the procedure to simplify fractions,

  • and then youll get plenty of exercises to do as homework. [cheering] Oh yeah!

  • Welcome to Part 2 of Simplifying Fractions. In Part 1, we learned the procedure for simplifying fractions.

  • Basically, you just take the top and bottom numbers and factor them down to their prime factors.

  • And then you see if there’s any factors that are the same on the top and bottom.

  • We call thosecommon factors”. And if there are, you just cancel them out.

  • And once youve canceled out all the common factors, you re-multiply whatever is left over to get your final answer.

  • In this video were gonna see a couple examples of how we can use that procedure to simplify fractions.

  • Let’s start with an easy one. Let’s simplify the fraction: 5 over 15.

  • Step 1 is to factor the top and bottom numbers, so

  • We know that 15 factors into 5 × 3

  • And 5 is a prime number. That means its only factors are ‘1’ and itself.

  • But ‘1’ is always a factor, so we don’t need to write that down.

  • Step 2 is to look for common factors and cancel them.

  • And we can see that there’s a 5 on the top and there’s a 5 on the bottom.

  • Theyre not directly over each other, but that doesn’t matter.

  • They still form a common factor pair, so we can cancel them out like this.

  • Step 3 is to re-organize our answer.

  • Now we don’t have any factors that need to be re-combined by multiplying.

  • We just have a 3 on the bottom, and we don’t have any factors left over on top.

  • But youll remember that there’s always a factor of ‘1’.

  • So, 5 over 15 simplifies to one-third.

  • Alright, I think we need to see another example, but a harder one this time.

  • Let’s simplify the fraction 30 over 36.

  • The procedure is the same:

  • Step 1 is we factor the top and bottom numbers all the way down to their prime factors.

  • Let’s do the top number first: 30 factors into 5 × 6.

  • 5 is prime, but 6 can be factored into 2 × 3.

  • So our 30 on top becomes 5 × 2 × 3.

  • Now the bottom number

  • 36 can be factored into 6 × 6. And each of those sixes can be factored into 2 × 3.

  • So our bottom number becomes 2 × 3 × 2 × 3.

  • Well, it looks like we do have some common factors.

  • There’s a 2 on both the top and the bottom that will cancel each other out.

  • And even though there’s more than one ‘2’ on the bottom, we can only cancel one of them out because there’s only one ‘2’ on top.

  • Remember, you always have to cancel common factors as pairs.

  • Now we can see that there’s another pair we can cancel.

  • The there’s a ‘3’ on both the top and bottom, so we can just cross those out.

  • Okay, that’s all the common factors we can cancel. So now all we have to do is see what’s left over.

  • Weve got a ‘5’ on the top and a ‘2 × 3’ on the bottom.

  • We don’t want to leave our problem looking like this, so we need to re-combine any factors that didn’t cancel.

  • that means multiplying together our 2 and 3 on the bottom, which gives us 6.

  • There, were left with the fraction 5 over 6. That’s the simplified form of the fraction 30 over 36.

  • They both have the same value, but the simplified one is written using the smallest numbers possible.

  • Now some of you may have been taught that the way to simplify fractions

  • is to find the greatest common factor of the top and bottom numbers, and just cancel that.

  • Basically, that’s what we ARE doing when cancel all of the common factors using our procedure.

  • In fact, if you multiply all of the common factors together, youll get the greatest common factor

  • or GCF as I like to call it. Ya knowto sound cool

  • Alright, so that’s how you simplify fractions. But, I’ll bet some of you are wondering,

  • Why would we even want to simplify fractions?”

  • That’s a good question. Basically it’s to make life simpler!

  • Wellat least for your teacher who has to grade all your homework.

  • For you it just makes life more complicated!

  • No, just kidding! [laughter] Simplified fractions make YOUR life easier too!

  • Cuz usually, simplified fractions are much easier to work with.

  • For example, if your friend said to you,

  • Here, you can have 27/54 of my sandwich.”

  • it would have been much easier if they had just said that you could have 1/2 of their sandwich,

  • since 1/2 is the simplified form of 27/54.

  • So now, whenever you see a fraction you can ask yourself,

  • Hmmmcould that be any simpler?”

  • And if so, youll know just what to do!

  • So get on out there, work on those exercises, and start making the world a simpler place for us all! [Cheers]

  • Learn more at www.mathantics.com

In a previous video, we learned all about factoring whole numbers.

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数学アンチックス - 分数の簡略化 (Math Antics - Simplifying Fractions)

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    Yassion Liu に公開 2021 年 01 月 14 日
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