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Adding and subtracting Fractions
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  1. Same Denominator
  2. Change one fraction -- examples 1, 2
  3. examples 3, 4
  4. Change both Fractions -- examples 1, 2
  5. example 3
  6. example 4
  7. example 5
  8. example 6
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(Instructor, Patrick Martin) Okay, Let's go over adding and subtracting fractions. We first need to start and look at when the denominators are the same (when the bottoms are the same). So, let's first think about money. If we had two quarters minus one quarter. Write down what that would be. Think about money: Two quarters, minus one quarter. Well, that's one quarter, isn't it? So, that's kinda funny because, this number on the bottom just stayed "4". But, we went: two minus one, and we got this 1 up here, and that's correct. So, why didn't we do this? Two minus one, over four minus four, to get one over zero? That's nonsense, isn't it? That is nonsense. And, the strange thing about fractions is: when these bottoms here are the same, we just leave them alone, and we just combine the tops. Let's have a look at inches. 7/8 inch, subtract 1/8 inch. Write down the answer. So, if a piece of wood is 7/8 inch long, and you want to plane off (cut off), 1/8 inch. What are you gonna be left with? Write it down, and remember, the pause button is down in that corner (on the left). That would be seven minus one over 8, wouldn't it? 6/8. And we can put that in lowest terms. Divide the top by 2, divide the bottom by 2, and we get ¾, right? So, again, when the bottoms are the same, we can combine the tops. And that's the strange thing about fractions, when you're adding and subtracting them. Let's have a look at baking: one third of a cup of sugar, plus 2/3 of a cup of sugar. What's that gonna be? One third of a cup of flour, plus two thirds of a cup of flour. Three thirds, isn't it? And, we can put that in lowest terms. Divide the top by 3, divide the bottom by 3, and get 1/1. Which is 1. Or 3/3 equals 1. Again, when the bottoms are the same, we combine the tops. That's how we add and subtract fractions. So, obviously, you do this then: If you had 2 dimes, plus 3 dimes. Think about money- what would that be? 5 dimes, right? Put it in lowest terms: 5 into 5 goes once. 5 into 10 goes twice, and that's 1/2 (half of a dollar). So, this is just like a thing we do in algebra called "adding like terms". Let's have a look at that: Adding Like Terms. Here's how adding like terms works: If you have two apples and you add three apples, how many apples is that? Five apples, isn't it? Just like the dimes.... 4 dimes plus 4 dimes would be 8 dimes. So, like terms. Or 5 bananas, (5b), plus 3b, would be 8b. Eight bananas. So, that's how we add and subtract fractions when the denominators are the same. Okay? adding and subtracting fractions when the denominators are different. We're gonna look at examples where we need to factor one denominator. That's one bottom of a fraction. Let's take an example with money: One half of a dollar, subtract three dimes. 3/10, right? What do you think that is? Write down the answer. Well, here's the procedure: We factorize one of the bottoms. In this case, 10. And then, it's two times five. Now, this is 2, this is 2 times 5. In the previous session we saw that we need to make the denominators the same, if we are to add or subtract fractions. So, to make this the same as this, we've got to multiply this by 5 over 5. And now our 1/2 becomes 5 over... (5 times 1 is 5, 5 times 2 is 10), it becomes 5/10. So, the interesting thing is, half of a dollar, has now been changed to 5 dimes. Which would you prefer? Half of a dollar or 5 dimes? Well, they're both the same quantity and that's OK. We can multiply the top and bottom of a fraction anytime we like. It does not change the quantity of the fraction. It does help us, because now we have 5/10 minus 3/10. Which is 2/10, right? And the lowest common denominator.... what do you think that is? The LCD is 10 in this case. Let's have a look at an example with inches: Three eighths of an inch, subtract one quarter of an inch. So, the procedure is, well, you can recognize that 4 goes into 8, and if we factorize 8, we get 2 times 4. Now, if we want to subtract fractions, the denominators need to be the same. So, 2 times 4. How do we change this bottom to be the same as this. Well, we multiply by something. What do you multiply it by? Well, you multiply it by 2. So, multiply this by 2/2, and this becomes (1 times 2 is 2, 4 times 2 is 8) so, the ¼ becomes 2/8. 1 quarter of an inch is the same as 2 eighths, right? So, 3/8 minus 2/8, is 1/8. And that's how that works. Let's go ahead and do this one then: One half of a tablespoon of nutmeg, plus 1/8 of a tablespoon of nutmeg.... Go ahead and do the procedure and add those together. Well, you need to make the bottoms the same: 8 is 4 times 2. So, change this also to 4 times 2. But you must multiply the top by 4 also. So, the 1/2 of a tablespoon becomes 4/8, which is the same quantity, it just looks different. And the denominator is the same as this denominator. So, we got 4/8 plus 1/8 which gives us 5/8, right? So, go ahead and do this one then: 5/27 minus 1/9. You'll see that 9 goes into 27. and 27, of course, is 9 times 3. So, if I factor 27, I get 9 times 3. Now make this bottom the same as this bottom. Make the bottoms the same. Make the denominators the same. Multiply that by 3/3, right? So, we have 5/27 minus 3/27. And, if the bottoms are the same, we can subtract the tops: 5 minus 3, is 2. 2/27, right? The reason we use this skill here, oh sorry... before we go on, looking at this example, the lowest common denominator is 8, right? Looking at this example, the lowest common denominator of the two fractions is 27. So, just to make a note of that. Let's see why we use this technique here: multiplying by 3/3, because you've learned a different technique before algebra, which is to find the lowest common denominator and put the fractions like this: you know, minus 1/9, get the lowest common denominator. That type of thing. Well, because it helps later on when we work with letters. For example, 1 over X squared, plus 3 over X. When we factorize X squared, we actually get X times X. Okay? To add these fractions, we need to make the bottoms the same. So, if l multiply that by X, now the bottoms are the same. But I must also multiply the top by X. So, X/X. Just like 3/3, right? So, I have: 1 over X squared, plus 3X over X squared, which is: 1 plus 3X over X squared. So you'll see this later. But the point is: this is a good skill to use, and you'll need to do this in your homework. because we've gotta learn how to do this technique with numbers, so we can then apply it to letters. Another quick example: 1 over ab, minus 7 over ab. Woops! Rather, make that a "c", okay? 1/ab minus 7/abc. Make the bottoms the same: multiply this by c over c. So, you've got abc, abc. And, c times 1 is c, so you've got: C minus 7, all over abc. So, you'll see that later. Let's look at adding and subtracting fractions when we have different denominators again, but this time we need to factor both denominators. So, one example, in a way, is 1/2 plus 1/3. What is that? A common mistake is to do things like this: people like to say, okay, that's 1 plus 1 over 2 plus three, which makes 2/5. Or does it? Let's check that common error, for example. A half of a square would look like this. Add a third of the square, would look like this. Here's the area of a third of a square, right? First of all, does that look like 2/5 to you? Or, does it look like more than 2/5? Because, 2/5 is... let's think about that. Here's a nice little trick. Watch this: the problem with adding a half and a third is that the bottoms are certainly not the same. And if you look at this shape, the shape of half an area of a square, and this, the shape of a third of the area of a square, they are completely different size shapes. So, it's very hard to add different size shapes, until we do this: split this also into thirds, and split this one in half. Now, how many rectangles, or squares, of equal size do we have? We have 1-2-3-4-5-6. And how many are shaded? 1-2-3. So, this area now, is in fact, three sixths. If you look at this area, we've got 1-2-3-4-5-6 squares (and they're all supposed to be the same size), and 2 of them are shaded. So, the shaded area is in fact 2/6, isn't it? So, 1/2 plus 1/3 is the same as 3/6 plus 2/6, which is 5/6 (and that would look like this....) 1-2-3-4-5 sixths. So, how do we do it by algebra? That's what we're here to learn, by math, basically. We make the bottoms the same again. Multiply this by 2 and then multiply this by 3. Now we have 6 on the bottom. But, don't forget you've got to multiply that by 2/2, and multiply this by 3/3. Three times one is three. Three times two is six. So, this is 3/6. One times two is two. Three times two is six. That's 2/6. So, if we have, for example, 2/3 -1/4 we can't subtract right away, because these bottoms are not the same. We need to make the bottoms the same. So, what do we put here and here to make the bottoms the same? You write it in. See if you can do it. Three here and a four here, right? Now the bottoms are the same. Four times three, four times three. But don't forget to multiply the top by three here, and multiply the top by four. So this 2/3 becomes [four times two], 8/12. Now, 2/3 is the same as 8/12. One quarter becomes [one times three] three. Four times three, 12. One quarter becomes 3/12. The same quantity. It looks different, but it's good because the bottoms are the same. 12 and 12. Eight minus three is five. So, that's 5/12. Okay? Oh, and just before we go, the lowest common denominator here, of course, is six. Lowest common denominator for here is 12. Let's have a look at mixed numbers. Three and 1/8, minus one and 2/3. First of all, we'll go and change them to improper factions. Eight times three plus one over eight. So, three and 1/8, that's 24 eighths. Plus, one eighth. That's 25/8. Minus three times one, plus two, over three. Because one and 2/3, is 3/3 and 2/3, which is 5/3. So, we've got eighths and thirds. And, if we want to subtract fractions, the bottoms have to be the same. So, put something here and put something here, so that these bottoms are the same. Put an eight here, and a three here. So, multiply this by 3/3, and this by 8/8. So, this becomes 75 over 24 minus 40, over 24. Which is 35 over 24. And, as a mixed number, 24 into that goes once, and remainder write it down 11. So, that's one and 11/24. Another thing that can happen is that you might need a factor both of the bottoms. So, let's have a look at one sixth plus five eighths. Now, in this case two goes into both bottoms, you'll see. If we factor both of the bottoms: Six can be factored to be two times three. Eight can be factored to two times four. Now, try and make the bottoms the same. Because, if you add fractions, the dominators need to be the same. So, put something here and put something here, to make the bottoms the same. So, we do the least amount of work to make the bottoms the same. If you put a four here. Now, over here we're including this two, and this four. Okay? Now, what would you put here, so that both bottoms are completely the same? Would you put a three? Right? So, you've got two times four times three. Two times four times three. Same bottom. Multiply this top by four though, of course, and multiply this top by three. And, now we've got four over, eight times 3, 4/24. So, 1/6 became 4/24. 5/8 became, five times three, 15 and that's 24. And, of course, the lowest common denominator here is 24. Four, plus 15, is 19. So, 19/24. Right? How about 9/10 minus 8/35? Factorize the bottoms, and then, make the bottoms the same. So, ten can be factored to five times two. 35 can be factored to five times seven. Now, put a number here and put a number here, so that both bottoms are the same. So, let's see, over here we've got a five. Same five here and here. But we don't have the seven. So, put the seven there. Now we've got the five and the seven. Five and seven. What do we need over here, so that both bottoms are the same? We need a two, right? So, multiply this one by 7/7, and this fraction by 2/2. and, seven times nine, this is 63. Over, multiply these, that's 35 times two, 70. Minus, two times eight, 16. Over, 35 times two, 70. Right? And then go: 63 minus 16, and we should get 47/70. Okay? So, really quickly, if you had a mixed number, that's not a big deal either. I mean, something like: 3 1/9 minus 5/6. What you wanna do, again, when you're subtracting fractions, is to turn these into improper fractions. So, nine times three, 27. Plus one, that's 28/9 minus 5/6. Now, factorize the bottoms. Nine, is three times three. Six, is three times two. Now, make the bottoms the same. Put a factor here, and a factor here, to make the bottoms the same. Now, this is a three, and another three. So, if we put a two, now we've got the three and the two (from here), right? So, to make the bottoms completely the same, we need to put the other three over here. So, multiply that by 2/2, and this by 3/3. And, go ahead and solve it. Let's see: two times that would be 56, over six times three, which is 18, minus 15/18. Which gives us 41/18. And, 18 into that goes two times remainder five, I think. So, 2 5/18. Now, why is this procedure, why do we do this: multiply it by 3/3 and 2/2? Well, again, if we had something like: three over a b, minus five over b c, which you'll see later on in algebra, the first step is to make the bottoms the same. So, multiply this by a, multiply this by c, and now you have bca, bca. Same thing. So, multiply this by c over c, and this by a over a. So, you got three c, minus 5 a, all over abc. So, that's why we use this method. And, in this case, the lowest common denominator of both fractions is abc. Of course, for this example, your lowest common denominator was 18. Right?