In the last lesson, we learned how to work with equivalent fractions. For this lesson, we're going to learn how to multiply fractions using chocolate. If I have four chocolate bars, and I want to share them fairly between two people, not including myself, how many should each person get? You can write your answer here.
Each person should get two chocolate bars. We have four chocolate bars to start with, and we're going to divide them or split them up among two people. So 4/2 makes 2.
Okay, let's go back to our four chocolate bars. If I'm holding these four chocolate bars, what would one chocolate bar represent? In other words, what's the fraction for just one of these chocolate bars? You can enter your answer here.
If you said 1/4, nice work. I know that I have 1 candy bar, so my numerator is just bars. So I have 1 out of 4, or 1/4.
I know this is seeming pretty easy so far, but there's something else that's really interesting about 1/4. I can represent 1/4 in other ways like, if I had a clock. So if I start on the hour, I could go fifteen minutes. I could go another fifteen minutes. I could go another fifteen minutes and I could go another fifteen minutes. I know an hour is 60 minutes so I just need to go 1/4 of that way. So 1/4 is going to be one 15-minute period. This is represent part of an area of a square. Here's a square, notice that I split the square up into four parts. If I shade one of those parts I have 1 out of 4 or 1/4. Notice that in both cases I split up my whole part into four equal pieces, I split the square up into four equal pieces, and I split my hour up into four 15-minute time intervals. What's really important is that we always create equally sized pieces. What's also interesting is that I could have actually split up my square a little bit differently. I could have drawn this. So if I shaded in this area, I know that I would have 1/4 or one out of four equally divided pieces
We haven't covered this yet, but you might be wondering, what about 0's in fractions? How do they happen? And can they happen? Let's look at some squares to figure this out. Here I have a square broken up into four equal pieces. Notice that none of them are shaded. That's okay, I can still have 0 out of 4. For fractions, the numerator tells us the number of equally sized pieces we have. The denominator tells us how many pieces are whole. In this case, our square is split in 2. Notice that I have none of them shaded. So I have 0 out of 4. Since I have no pieces, it's really just 0 . 0 / 4 is 0. For this square, it doesn't make sense to split it up into 0 pieces. I can't take a square, and divide it by nothing. It just doesn't make sense. Fractions with 0 in the denominator are undefined.
Let's try your hand at a quiz. I'm going to draw a bunch of shapes and I want you to find the shape, or images, that have an area shaded as 1/5. Check all the images where the shaded area is 1/5 of the total area. Assume that the circle has been split up into equal parts, it might be hard to tell.
If you chose the circle, great work. I know I have one part of the circle shaded out of five equal parts, so that's definitely 1/5. The second one isn't 1/5. I do have five parts, but they're not all equal. I have one of them shaded, but remember, the parts need to be equal. This one is also correct. I have one out of five pieces, and all of the pieces are equally shaped. This triangle is also not correct. I do have one piece shaded, but I have one of four, so this one is really pieces of my entire rectangle and I have one of them shaded, so 1/5. This one is also not correct. I have five pieces, but they're not equally shaped. The area hasn't been divided into five equal pieces, so this can't be 1/5 of the area of the rectangle.
So far we've worked with the numerator being one. We've only had one piece out of five. Let's see what else we can do. Here is a square again, and I'm going to split it up into four parts. Now, instead of having one piece, I'm going to have three. So now, the area of this shaded square is three pieces, my denominator is four because I've split up my square into four equal pieces. Here's another square, and this time I'm going to have all the pieces. Let's see what that looks like. So now I have 4 out of 4. When I have 4/4, that's really 1. Now that we've seen that the numerator doesn't have to be 1, let's try an example where it's not. We'll try this out with a quiz.
So which of the following answer choices could represent 2/5 of an hour? Is it ten minutes, 12 minutes, 20 minutes, 24 minutes, or 25 minutes? Here is a clock to help you out. Remember thinking about splitting up your clock or your hour into equal pieces.
If you chose 24 minutes, nice work. I know that there's 60 minutes in an hour, so I need to split that 60 minutes up into five equal pieces. I know 60/5=12. So if I go 1/5 of an hour I've just gone double that. So to go 2/5 in an hour I need to go 212 minutes. So I need to go
Okay. This is a great review of fractions. But, what else can we do with fractions. With the counting numbers, we can add, subtract, multiply and divide. We can do the same operations with fractions. You might be asking yourself, what's an operation? An operation is just something that takes input values and produces an output value. Like addition, we could have the output. An operation just produces a new value. For now, we're going to continue working with fractions and look at multiplication. We're going to return to our chocolate bars. Here's my one chocolate bar. This time I'm going to split up my chocolate bar into pieces. I'm going to find a fraction of a fraction. So what is 1/3 of 1/2 of a chocolate bar? Let's find out. Here's a drawing of a chocolate bar, and now first I'm going to split it in 1/2. So I started with 1 bar, and now I have 1/2 a bar. I want to find into thirds, or three equal pieces. I've split my half into thirds now. And now, I just need 1/3 of the 1/2. So this piece right here represents 1/3. Of my 1/2. So I found 1/3 of 1/2. It's this shaped piece. But I'm wondering, what fraction is this piece, compared to the original candy bar? I can split up my original candy bar into the same sized pieces as this one. That will let me see what fraction is this of my candy bar. So visually, I can see that this piece is 1/6 of. My candy bar. So let's pair this with some math. 1/3 of 1/2 has to be equal to 1/6. But I need to figure out what did I do to these two fractions? How did I combine them to get quiz.
Okay. So, to find a fraction of a fraction, you have to do something with the numerators and the denominators. So, what do you do with the numerators and what do you do with the denominators? Do you add them, subtract them, multiply them or divide them? One of these operations will work for the numerator and the denominators, you choose.
You know you want 1/3 of 1/2 to be 1/6, so somehow I need to get 1. Well, 1 1 is 1, so I know I need to multiply the numerators, so if you chose multiplication, great. And then for the denominator, I need 6. Well, I know 3 2 is 6, so it must be multiplication as well. Whenever we take a fraction of a fraction, we multiply the numerators and multiply the denominators
Okay. Let's try your head at one more quiz. What's 1/3 of 1/4 of a chocolate bar? What fraction do you think that would be? Is it 1/12, 2/7, 2/12, 3/4, or 4/3? You also might want to draw some diagrams to help you out.
If you said 1/12, nice work. This time, I'm going to use some blocks to help me out. This full block represents one candy bar. I'm going to go ahead and section the candy bar into fourths. Okay, I have 1/4, together make up my whole candy bar. I have 1/4 of my candy bar here, but now I need 1/3 of it. I'm going to split this represents 1/3 of 1/4 of my candy bar. Let's see how it compares to the original whole candy bar. I can see that the one piece is one out of the 12 pieces of my candy bar, so 1/3 of 1/4 must be 1/12. And you probably did it really easily. You probably just said 1/3 times 1/4. Remember, when we multiply fractions, we multiply the numerators straight across, and the denominators straight across. So, I have 1 times 1 and 3 times 4, or 1/12. This piece is
I'm going to get you started on this problem, and then I want you to finish it out. I have 2/5 of a candy bar. I need to share it with three others and myself, so I'm really sharing it among four people. So I take 2/5 and divide it by 4. Remember, you need to change something here. When we divide by a number we could do the same thing as multiply by the reciprocal. So we're going to have 2/5 times 1/4. Okay. You finish this one now.
If you said 2/20, great work. We know 2 times 1 is 2 and 5 times 4 is 20. You might have also put a different answer. You might have reduced. You might have divided your fraction by 2/2, and you would have gotten Let's look at why. So here, I have two candy bars. This top one's been put up on into 10th. I know that my answer was supposed to be 2/20. Two out of 20 pieces. Visually, I can see that's 1/10. I have one out of the ten pieces down here. 2/20 and 1/10 are actually equal fractions. They represent the same amount. Oftentimes, you say that 1/10 is the reduced form or the simplified form of actually equal. They just look different.
To get to this answer of 1/10 faster, I actually didn't need to wait to simplify. I could have done it sooner. Let's see what that looks like. I can reduce any common factors that appear in the numerators and in the denominators. I know if I multiply these fractions, I would do I'm going to break up my four into factors of two. I know 2/2 makes one. So, whenever I see a 2/2, I can simplify that, then I would just multiply as usual. 11 is one and 512 is ten, so I still get 1/10. You might have heard of the term crosscanceling before, that could also work. I know that two goes in a two and two goes in a four. If I would write a factor in the numerator and at the denominator, I can cancel. 2/2 makes one, and 4/2 makes two. When I multiply across, I'm going to have 11 and 52. Either way, I still wind up with 1/10. The key thing is you want to make sure you're only cancelling that factor that appears in the numerator and in the denominator, because we know anything over itself is one.
So you're probably saying so what? Why would I want to reduce factors first? Well, what if I wanted to find 3/105/12? I could draw a diagram of the candy bar and split up the pieces before it. I could split up the 10's and then 12's. But that's kind of a lot of work and I know I just really need to multiply these 2 fractions. Multiplying these two fractions together I get 15/120. But now I have to figure out if I can simplify this, and how to simplify this fraction. This can be frustrating working with large numbers, so let's make our work easier and simplify our fractions first. I know 10 can be written as 25, and 12 can be written as 34. Remember any number divided by itself makes 1. Like 4/4 makes but we should keep this in mind when we're simplifying. Here I have 3/3 so that makes just multiply my numerators. So I have have 2 times 1 times 1 times 4. Which I know that's just really 24 or 8. So 15/120 is really the same thing as so you can really see why we can cancel factors. You can do ahead of the time and this is what it would look like. I know 5 and 10 share a common factor of 5, so I can divide them both by 5, 5/5 makes 1, of 3, so 3/3 makes 1, and 12/3 makes 4. Now, I multiply, just like before, 11 is this long way, or this shorter way with cross-cancelling, I get 1/8.
Now that we've seen how to multiply fractions and how to simplify them, let's try a quiz. So, what's 11/1624/55? You can put your answer in this box, the numerator in the top, the denominator in the bottom.
Okay, so to solve this problem, first I'm going to reduce my fractions. I want to take out, anything that equals 1. I know 8 goes into 24 and 16. So, I'll have an 8 over 8 in my answer. So lets get rid of that, 24/8 is 3, and 16/8 is 2. With 11 and 55, I know they both share a factor of in my numerator and 25 in my denominator, so I get 3/10. If you said 3/10, nice work.
Alright. You've already done a lot of math. You've identified fractions, you've done fractions of fractions, and you reduced fractions. Let's see what else we can do.
Thanks Chris for showing us how to multiply and divide fractions. Now let's practice what we learned. There will be several question videos at the end of each lesson. Let's try a multiplication of fractions question. The first question that we want to do is 3/4 5/7. You can put your answer here.
If you got 15/28, great job. Now, let's go ahead and do the problem. If I have fractions that we have to multiply the numerators together and we have to multiply the denominators together. So, need to check to see if we need to reduce the fraction but since there's no factors in common between 15 and 28, we're done.
Okay, let's try a slightly harder question. This time we're going to do 8/15
If you got 10/21, you got it right. Let's see how to get that answer. Since these numbers are large, we want to try to factor them before we multiply them and see if we can cancel any factors. So, we can write 8 as 42 and we can write 15 as For the 2nd fraction, we can write 25 as If you notice, there's a 4 in the numerator, and a 4 in the denominator, so we know that 4/4 = 1. Also, there's a 5 in the numerator, and a 5 in the denominator, and we know that 5/5 is equal to 1. So, after we cancel any factors that are in common between the numerator and the denominator, we simply multiply the numerators together, so we're left with together, 37. So we know 25=10 and we know that 37=21.