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Math Antics - Adding and Subtracting Fractions

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- [Voiceover] Let's say that we have the fraction 9/10, and I want to add to that the fraction 1/6. What is this, what is this going to equal? So when you first look at this, you say, "Oh, I have different denominators here. It's not obvious how I add these." And you'd be right and the way to actually move forward is to find a common denominator, to convert both of these fractions into fractions that have a common denominator. So how do you think about a common denominator? Well, a common denominator's gonna have to be a common multiple of these two denominators of 10 and six. So what's a common multiple of 10 and six? And it's usually simplest to find the least common multiple, and a good way of doing that is start with the larger denominator here, 10, and say, okay is 10 divisible by six? No. Okay, now, is 20 divisible by six? No. Is 30 divisible by six? Yes. 30 is divisible by six. So I'm just going through the multiples of 10 and saying, "Well what is the smallest multiple of 10 that is divisible by six?" And that's going to be 30. So I could rewrite both of these fractions as something over 30. So nine over 10. How would I write that as something over 30? Well I multiply the denominator, I'm multiplying the denominator by three. So I've just multiplied the denominator by three. So if I don't want to change the value of the fraction, I have to do the same thing to the numerator. I have to multiply that by three as well because now I'm just multiplying the numerator by three and the denominator by three, and that doesn't change the value of the fraction. So nine times three is 27. So once again, 9/10 and 27/30 represent the same number. I've just written it now with a denominator of 30, and that's useful because I can also write 1/6 with a denominator of 30. Let's do that. So 1/6 is what over 30? I encourage you to pause the video and try to think about it. So what did we do go from six to 30? We had to multiply by five. So if we multiply the denominator by five, we have to multiply the numerator by five as well, so one times five, one times five is five. So 9/10 is the same thing as 27/30, and 1/6 is the same thing as 5/30. And now we can add, now we can add and it's fairly straightforward. We have a certain number of 30ths, added to another number of 30ths, so 27/30 + 5/30, well that's going to be 27, that's going to be 27 plus five, plus five, plus 5/30, plus 5/30, which of course going to be equal to 32/30. 32 over 30, and if we want, we could try to reduce this fraction. We have a common factor of 32 and 30, they're both divisible by two. So if we divide the numerator and the denominator by two, numerator divided by two is 16, denominator divided by two is 15. So, this is the same thing as 16/15, and if I wanted to write this as a mixed number, 15 goes into 16 one time with a remainder one. So this is the same thing as 1 1/15. Let's do another example. Let's say that we wanted to add, we wanted to add 1/2 to to 11/12, to 11 over 12. And I encourage you to pause the video and see if you could work this out. Well like we saw before, we wanna find a common denominator. If these had the same denominator, we could just add them immediately, but we wanna find a common denominator because right now they're not the same. Well what we wanna find is a multiple, a common multiple of two and 12, and ideally we'll find the lowest common multiple of two and 12, and just like we did before, let's start with the larger of the two numbers, 12. Now we could just say well 12 times one is 12, so that we could view that as the lowest multiple of 12. And is that divisible by two? Yeah, sure. 12 is divisible by two. So 12 is actually the least common multiple of two and 12, so we could write both of these fractions as something over 12. So 1/2 is what over 12? Well to go from two to 12, you multiply by six, so we'll also multiply the numerator by six. Now we see 1/2, and 6/12, these are the same thing. One is half of two, six is half of 12. And how would we write 11/12 as something over 12? Well it's already written as something over 12, 11/12 already has 12 in the denominator, so we don't have to change that. 11/12, and now we're ready to add. So this is going to be equal to six, this is going to be equal to six plus 11, six plus 11 over 12. Over 12. We have 6/12 plus 11/12, it's gonna be six plus 11 over 12, which is equal to, six plus 11 is 17/12. If we wanted to write it as a mixed number, that is what, 12 goes into 17 one time with a remainder of five, so 1 5/12. Let's do one more of these. This is strangely fun. Alright. Let's say that we wanted to add, We're gonna add 3/4 to, we're gonna add 3/4 to 1/5. To one over five. What is this going to be? And once again, pause the video and see if you could work it out. Well we have different denominators here, and we wanna find, we wanna rewrite these so they have the same denominators, so we have to find a common multiple, ideally the least common multiple. So what's the least common multiple of four and five? Well let's start with the larger number, and let's look at its multiples and keep increasing them until we get one that's divisible by four. So five is not divisible by four. 10 is not divisible by four, or perfectly divisible by four is what we care about. 15 is not perfectly divisible by four. 20 is divisible by four, in fact, that is five times four. That is 20. So what we could do is, we could write both of these fractions as having 20 in the denominator, or 20 as the denominator. So we could write 3/4 is something over 20. So to go from four to 20 in the denominator, we multiplied by five. So we also do that to the numerator. We multiply by three times five to get 15. All I did to go from four to 20, multiplied by five. So I have to do the same thing to the numerator, three times five is 15. 3/4 is the same thing as 15/20, and over here. 1/5. What is that over 20? Well to go from five to 20, you have to multiply by four. So we have to do the same thing to the numerator. I have to multiply this numerator times four to get 4/20. So now I've rewritten this instead of 3/4 plus 1/5, it's now written as 15/20 plus 4/20. And what is that going to be? Well that's going to be 15 plus four is 19/20. 19/20, and we're done.

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Adding fractions with unlike denominators (video) | Khan Academy

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Voiceover] Let's say that we have the fraction 9/10, and I want to add to that the fraction 1/6. What is this, what is this going to equal? So when you first look at this, you say, "Oh, I have different denominators here. It's not obvious how I add these."

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Subtracting fractions with unlike denominators (video) | Khan Academy

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Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. Khan Academy is a nonprofit with the mission of providing a free, world-class education for anyone, anywhere.

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Video transcript

let's think about what it means to take 8/3 and divide it by one-third so let me draw a number line here so there is my number line this is 0 this is 1 and this is 2 maybe this is 3 right over here and let me plot 8/3 so to do that I just need to break up my each each hole into thirds so let's see it's one-third two-thirds three-thirds 4/3 5/3 6 thirds 7/3 8/3 so right over here and then of course 9/3 would get us to 3 so this right over here is 8/3 8/3 now one way to think about 8/3 divided by 3 is what if we take this length and we say how many jumps would it take to get there if we're doing it in jumps of 1/3 or essentially we're breaking this up if we were to break up 8/3 into sections of 1/3 how many sections would I have or how many jumps what I have well let's think about that if we're trying to take jumps of 1/3 we're gonna have to go 1 2 3 4 5 6 7 8 jumps so we could view this as we could you let me do this in a different color in this orange so we took these 8 jumps right over here so we could view 8/3 divided by 1/3 as being equal to 8 now why is this actually make sense well when you're dividing things in 2/3 for every hole you're now going to have 3 jumps so whatever value you're trying to get to you're going to have that number times 3 jumps so another way of thinking about it is that 8/3 8/3 divided by divided by 1/3 is the same thing as 8/3 8/3 times 3 now we could either write it like this we could write x 3 like that or if we want to write 3 as a fraction we know that 3 is the same thing three over one and when we already know how to multiply fractions a multiply the numerators eight times three so you have eight let me do that that same color you have eight times three in the numerator now eight times three and then you have three times 1 in the denominator three times three times one in the denominator which would give you 24 thirds which is the same thing as 24 divided by three which once again is equal is equal to is equal to is equal to eight now let's see if this still makes sense instead of dividing by one-third if we were divided by two-thirds so let's think about what 8/3 divided by two-thirds is divided by two-thirds well once again this is like asking the question if we wanted to break up this section from zero to 8/3 into sections of 2/3 or jumps of two-thirds how many sections or how many jumps would I have to make well let's think about it one jump just in a different color we could make one jump that's the same color as my 8/3 we could do one jump my computer is doing something strange we could do one jump two jumps three jumps and four jumps so we see 8/3 divided by 2/3 is equal is equal to four now does this make sense in this world right over here well if we take 8/3 and we do the same thing saying hey look dividing by a fraction is the same thing as multiplying by a reciprocal well let's multiply by three halves let's multiply by the reciprocal of 2/3 so we swap the numerator and the denominator so we multiply it times three halves and then what do we get in the numerator once again we get eight times three which is 24 and in the denominator we get 3 times 2 which is 6 so now we get 24 divided by 6 is equal to four now does it make sense that we got half the answer if you think about the difference between what we did here and what we did here these are almost the same except here we really just didn't divide or you could say you divided by one while here you / - well does that make sense well sure because here you jumped twice as far you jumped twice as far so you had to take half the number of steps and so in the first example you saw why it makes sense to multiply by 3 when you divide by a fraction for every hole you're making 3 jumps so that's why when you divide by this fraction you multiply or when you divide whatever's in the denominator you multiply by it and now when the numerator is greater than 1 every jump you're going twice as far as you did in this first one right over here and so you would have to do half as many jumps hopefully that makes sense it's easy to think about just mechanically how to divide fractions dividing 8/3 divided by 1/3 is the same thing as 8/3 times 3 over 1 or 8/3 divided by 2/3 is the same thing as 8/3 times 3 over 2 but hopefully this this video gives you a little bit more of an intuition of why this is the case

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Understanding division of fractions (video) | Khan Academy

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Let's think about what it means to multiply 2 over 3, or 2/3, times 4/5. In a previous video, we've already seen how we can actually compute this. This is going to be equal to-- in the numerator, we just multiply the numerators. So it's going to be 2 times 4. And in the denominator, we just multiply the denominator. So it's going to be 3 times 5. And so the numerator is going to be 8, and the denominator is going to be 15. And this is about as simple as we can make it. 8 and 15 don't have any factors common to each other, than 1, so this is what it is. It's 8/15. But how, why does that actually makes sense? And to think about it, we'll think of two ways of visualizing it. So let's draw 2/3. I'll draw it relatively big. So I'm going to draw 2/3, and I'm going to take 4/5 of it. So 2/3, and I'm going to make it pretty big. Just like this. So this is 1/3. And then this would be 2/3. Which I could do a little bit better job making those equal, or at least closer to looking equal. So there you go. I have thirds. Let me do it one more time. So here I have drawn thirds. 2/3 represents 2 of them. It represents 2 of them. One way to think about this is 2/3 times 4/5 is 4/5 of this 2/3. So how do we divide this 2/3 into fifths? Well, what if we divided each of these sections into 5. So let's do that. So let's divide each into 5. 1, 2, 3, 4, 5. 1, 2, 3, 4, 5. And I could even divide this into 5 if I want. 1, 2, 3, 4, 5. And we want to take 4/5 of this section here. So how many fifths do we have here? We have 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. And we've got to be careful. These really aren't fifths. These are actually 15ths, because the whole is this thing over here. So I should really say how many 15ths do we have? And that's where we get this number from. But you see if 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15. Where did that come from? I had 3, I had thirds. And then I took each of those thirds, and I split them into fifths. So then I have five times as many sections. 3 times 5 is 15. But now we want 4/5 of this right over here. This is 10/15 right over here. Notice it's the same thing as 2/3. Now if we want to take 4/5 of that, if you have 10 of something, that's going to be 8 of them. So we're going to take 8 of them. So 1, 2, 3, 4, 5, 6, 7, 8. We took 8 of the 15, so that is 8/15. You could have thought about it the other way around. You could have started with fifths. So let me draw it that way. So let me draw a whole. So this is a whole. Let me cut it into five equal pieces, or as close as I can draw five equal pieces. 1, 2, 3, 4, 5. 4/5, we're going to shade in 4 of them. 4 of the 5 equal pieces. 3, 4. And now we want to take 2/3 of that. Well, how can we do that? Well, let's split each of these 5 into 3 pieces. So now we have essentially 15ths again. So 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15. We want to take 2/3 of this yellow area. We're not taking 2/3 of the whole section. We're taking 2/3 of the 4/5. So how many 15ths do we have here? We have 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12. So if you have 12 of something, and you want to take 2/3 of that, you're going to be taking 8 of it. So you're going to be taking 1, 2, 3, 4, 5, 6, 7, 8 or 8 of the fifteenths now. So either way, you get to the same result. One way, you're thinking of taking 4/5 of 2/3. Another way you could think of it as you're taking 2/3 of 4/5.

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Intro to multiplying 2 fractions (video) | Khan Academy

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Let's think about what it means to multiply 2 over 3, or 2/3, times 4/5. In a previous video, we've already seen how we can actually compute this. This is going to be equal to-- in the numerator, we just multiply the numerators. So it's going to be 2 times 4.

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In this lesson you will learn how to add decimals by using the standard algorithm.

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Adding & Subtracting Decimals Song | 4th & 5th Grade

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How Do You Subtract Decimals? | Virtual Nerd

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Virtual Nerd's patent-pending tutorial system provides in-context information, hints, and links to supporting tutorials, synchronized with videos, each 3 to 7 minutes long. In this non-linear system, users are free to take whatever path through the material best serves their needs. These unique features make Virtual Nerd a viable alternative to private tutoring.

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Check out Bas Rutten's Liver Shot on MMA Surge: http://bit.ly/MMASurgeEp1 In this video, Mahalo math expert Julie Clark explains how to divide with decimals in the divisor and dividend. How to Divide Decimals --------------------------------------------------------------------- Dividing with decimals can seem difficult at first, but it's important to treat them like any other long division problem.