Adding fractions with unlike denominators khan academy

Video transcript

- What I want to do in this video is really make sure that we feel comfortable manipulating algebraic expressions that involve fractions. So we'll start with some fairly straightforward ones. So let's say that I had, let's say I had A over B plus C over D, and if I actually wanted to add these things, so it is just one fraction, how would I do that? Well, what we could do is, we could find a common denominator. Well, over here, we don't know what B is, we don't know what D is, but we know a common denominator is just going to be B times D. That is going to be a common multiple of B and D. So we could rewrite this as two fractions, with the common denominator BD, so, plus, BD, actually, let me color code it a little bit. So A over B is going to be the same thing as what over BD? Well, to get BD, I multiplied the denominator by D, so let me multiply the numerator by D as well, then I haven't changed the value of the fraction, I'm just multiplying by D over D. So this is going to be A times D over B times D. Notice I could divide the numerator and the denominator by D, and I'm going to get back to A over B. And then we can look at the second fraction, C over D, to go from D to BD, we multiplied by B and so, if I multiply the denominator by B, if I don't want to change the value of the fraction, I have to multiply the numerator by B as well. So let's multiply the numerator by B as well, and it's going to be BC, BC. BC over BD. This is C over D. So what I have here in magenta, this fraction is equivalent to this fraction. I just multiplied it by D over D, which we can assume is one, if we assume that D is not equal to zero, and then, if we just multiply C over D times one, which is the same thing as B over B, if we assume that B is not equal to zero, then this fraction and this fraction are equivalent. Now, why did I go through all of this trouble? Well, now, I have a common denominator, so I can add these two fractions. So what's this going to be? Well, common denominator is BD, so let me just, so the common denominator is BD, and I could just add the numerators, just like you would've done if these were numbers, if this wasn't an algebraic expression. So this is going to be, this is going to be AD plus BC, all of that over BD.

Video transcript

Cindy and Michael need 1 gallon of orange paint for the giant cardboard pumpkin they are making for Halloween. Cindy has 2/5 of a gallon of red paint. Michael has got 1/2 a gallon of yellow paint. If they mix their paints together, will they have the 1 gallon they need? So let's think about that. We're going to add the 2/5 of a gallon of red paint, and we're going to add that to 1/2 a gallon of yellow paint. And we want to see if this gets to being 1 whole gallon. So whenever we add fractions, right over here we're not adding the same thing. Here we're adding 2/5. Here we're adding 1/2. So in order to be able to add these two things, we need to get to a common denominator. And the common denominator, or the best common denominator to use, is the number that is the smallest multiple of both 5 and 2. And since 5 and 2 are both prime numbers, the smallest number's just going to be their product. 10 is the smallest number that we can think of that is divisible by both 5 and 2. So let's rewrite each of these fractions with 10 as the denominator. So 2/5 is going to be something over 10, and 1/2 is going to be something over 10. And to help us visualize this, let me draw a grid. Let me draw a grid with tenths in it. So, that's that, and that's that right over here. So each of these are in tenths. These are 10 equal segments this bar is divided into. So let's try to visualize what 2/5 looks like on this bar. Well, right now it's divided into tenths. If we were to divide this bar into fifths, then we're going to have-- actually, let me do it in that same color. So it's going to be, this is 1 division, 2, 3, 4. So notice if you go between the red marks, these are each a fifth of the bar. And we have two of them, so we're going to go 1 and 2. This right over here, this part of the bar, represents 2/5 of it. Now let's do the same thing for 1/2. So let's divide this bar exactly in half. So, let me do that. I'm going to divide it exactly in half. And 1/2 literally represents 1 of the 2 equal sections. So this is one 1/2. Now, to go from fifths to tenths, you're essentially taking each of the equal sections and you're multiplying by 2. You had 5 equals sections. You split each of those into 2, so you have twice as many. You now have 10 equal sections. So those 2 sections that were shaded in, well, you are going to multiply by 2 the same way. Those 2 are going to turn into 4/10. And you see it right over here when we shaded it initially. If you Look at the tenths, you have 1/10, 2/10, 3/10, and 4/10. Let's do the same logic over here. If you have 2 halves and you want to make them into 10 tenths, you have to take each of the halves and split them into 5 sections. You're going to have 5 times as many sections. So to go from 2 to 10, we multiply by 5. So, similarly, that one shaded-in section in yellow, that 1/2 is going to turn into 5/10. So we're going to multiply by 5. Another way to think about it. Whatever we did to the denominator, we had to do the numerator. Otherwise, somehow we're changing the value of the fraction. So, 1 times 5 is going to get you to 5. And you see that over here when we shaded it in, that 1/2, if you look at the tenths, is equal to 1, 2, 3, 4, 5 tenths. And now we are ready to add. Now we are ready to add these two things. 4/10 plus 5/10, well, this is going to be equal to a certain number of tenths. It's going to be equal to a certain number of tenths. It's going to be equal to 4 plus 5 tenths. And we can once again visualize that. Let me draw our grid again. So 4 plus 5/10, I'll do it actually on top of the paint can right over here. So let me color in 4/10. So 1, 2, 3, 4. And then let me color in the 5/10. And notice that was exactly the 4/10 here, which is exactly the 2/5. Let me color in the 5/10-- 1, 2, 3, 4, and 5. And so how many total tenths do we have? We have a total of 1, 2, 3, 4, 5, 6, 7, 8, 9. 9 of the tenths are now shaded in. We had 9/10 of a gallon of paint. So now to answer their question, will they have the gallon they need? No, they have less than a whole. A gallon would be 10 tenths. They only have 9 tenths. So no, they do not have enough of a gallon. Now, another way you could have thought about this, you could have said, hey, look, 2/5 is less than 1/2, and you could even visualize that right over here. So if I have something less than 1/2 plus 1/2, I'm not going to get a whole. So either way you could think about it, but this way at least we can think it through with actually adding the fractions.

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