WEBVTT mathematics/basic-math/pyo
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Welcome back to Educator.com.
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This lesson, we are going to add and subtract fractions with different denominators.
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Before we begin with that, let's review over the lesson on least common multiple, the LCM.
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This I believe was a few lessons ago.
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If you want, you can go back to that lesson.
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We are just going to do a brief example here.
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To find the LCM of 6 and 4, I am going to take the two numbers.
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I am going to do the factor tree method on each of them.
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For 6, a factor pair of 6 is going to be 2 and 3.
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They are both prime; I am going to circle them both.
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For 4, it is going to be 2 and 2; I am going to circle them.
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Here, to find the LCM, I am going to look at what they have in common.
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I know that I have a 2 here; I also have a 2 here.
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I am going to write that 2 by itself.
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This pair cancels out one of them; I am going to write a 2.
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The other numbers, 3 and 2, the other remaining numbers are going to go tag along with it.
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Again to find the LCM, I just find what they have in common.
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Since there is a 2 here and a 2 here, one of those 2s get cancelled.
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It is going to be 2 times 3 times 2.
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2 times 3 is 6; 6 times 2 is 12; my LCM is 12.
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The LCM of 6 and 4 is going to be 12.
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LCM is the same thing as LCD.
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LCM stands for least common multiple; LCD is least common denominator.
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When I am using those two numbers as my denominators, then it is going to be called LCD.
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But I am still going to find the LCM between those two denominators.
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The reason for this, whenever I add fractions or subtract fractions,
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I have to make sure that these denominators are the same.
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In order to make them the same, I need to find the LCM or the LCD between the two numbers.
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Here 6 and 4, just like what we did, the example, we know that the LCM is 12.
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The LCM, 1/6, what I am going to do is I am going to make 1/6 the same fraction with the denominator becoming 12.
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Same thing here, 3/4, the denominator is going to change to 12.
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I want to figure out what these top numbers are going to be, my numerators.
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How do I go from a 6 to a 12?--what do I multiply it by?
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I multiplied this by 2; or I can do 12 divided by 6; I get 2.
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Since I multiplied the 6 by 2 to get 12, I need to also multiply the top number by 2.
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1 times 2 is 2; the fraction 1/6 became 2/12.
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These are the same fractions; 1/6 is the same thing as 2/12.
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Same thing here, 3/4; to go from 4 to 12, I have to multiply it by a 3.
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Then I have to multiply the top number by 3; 3 times 3 is 9.
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3/4, because I multiplied the top and the bottom by the same number, these fractions become the same.
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They are the same fraction; 3/4 is the same thing as 9/12.
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Now since I know that 2/12 is the same thing as 1/6 and 9/12 is the same thing as 3/4,
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I can add these two fractions, this fraction and this fraction.
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If I add these two, then my answer will be the same as if I add these two.
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I have to do that because these denominators are different.
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I have to make them the same by converting these fractions,
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by changing these fractions so that the denominators will be the same.
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Now that they are, I am going to take my numerators and add them together.
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2 plus 9 is 11; here the denominator is 12; 12.
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Then my denominator here has to stay the same as a 12.
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2/12 plus 9/12 is 11/12; or I can say that 1/6 plus 3/4 is 11/12.
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Let's do another example; here I am going to subtract.
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But before I do that, I have to check my denominators.
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This denominator is an 8; this one is a 4; they are different.
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I have to find the LCD or the LCM between 8 and 4 so that I can make the denominators the same.
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I am going to take 8 and 4; you could do the factor tree.
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4 and 2; circle that one; that is a prime.
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2 and 2; this is 2 and 2; look at this one.
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There is a common one here; I am going to cancel out one of them.
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I have another common between these two so I am going to cancel out one of them.
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Then I just multiply all the remaining circled numbers.
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It is 2 times 2 times 2 which is 8.
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If I look at these, I can just look at them and figure out what the LCM is by looking at the multiples.
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Multiples of 8 would be 8, 16, 24, and so on.
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For 4, it would be 4, 8, 12, 16, and so on.
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You are going to find the smallest common multiple between them which is 8.
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Here I am going to change this fraction and this fraction so that their denominators will be the same.
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For this fraction, 7/8, my LCM is already 8.
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My LCM or my LCD, it is already 8.
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For that one, I can just keep it the way it is.
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For this one however, 1/4, I have to convert it; I have to change it.
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I need a top number; 4; to get 8, I multiply it by 2.
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Again I have to multiply the same number to the top which is 2.
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Whenever you are converting fractions, as long as you multiply the top
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and the bottom by the same number, then your fraction will stay the same.
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Even if you change the numbers, it is still the same fraction; 1/4 became 2/8.
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Now I am going to rewrite my problem, 7/8 minus 2/8.
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Make sure the denominators are the same.
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If they are not the same, then you did something wrong.
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Go back and check your work.
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Since they are the same, I can go ahead and subtract them.
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7 minus 2 which is 5; then my denominator, 8.
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8 here; it stays an 8 there; 7/8 minus 1/4 is going to equal 5/8.
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Let's add this next problem, 9/10 plus 3/15.
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Again I have to check my denominators; they are not the same.
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I have to find the least common denominator with them.
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I am going to take 10; do the factor tree which is 5 and 2.
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Circle them if they are prime; only circle them if they are prime.
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Then 15, this becomes 5 and 3.
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If you are confused about how to find the LCD or LCM,
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then you can go back and look at the lesson on that one before continuing.
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My LCM or I am just going to call it the LCD since they are my denominators.
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I look for any common numbers between them; they have a 5; 5 is common.
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Whenever they have something in common, just cancel one of them out.
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That is all they have in common.
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Then for my LCD, I am just going to write out the remaining circled numbers.
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Remember they can only be circled; 5 times 2 times 3.
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5 times 2 is 10; 10 times 3 is 30; my LCD is going to be 30.
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I have to change this fraction so that my denominator will become 30.
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Same thing here, change this fraction so my denominator will be 30.
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9/10, going to convert it; I can take 30 divided by 10; that is 3.
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I know that I did 10 times 3 to get 30.
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Again you have to do it to both the top and the bottom, the same number.
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That is the only way you are going to have the same fraction because you don't want to change your fraction.
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Even if you are changing the numbers, it is still the same fraction.
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9 times 3 is 27.
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I am going to do the same thing for the other fraction.
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15 times 2 was 30; 3 times 2... again multiply it by the same number.
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It is going to be 6.
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Since 9/10 is the same thing as 27/30 and 3/15 is the same as 6/30, I need to add my new fractions.
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Again double check your denominators; make sure they are the same.
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It is going to be 27 plus 6.
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27 plus 6 is 33 over... your denominator will stay the same.
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It is 33/30; let's look at this fraction.
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This is your answer; this is a solution to this problem.
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But I have an improper fraction because the top number, the numerator, is bigger than the denominator.
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You can either leave it like this; this is still the correct answer; or I can simplify it.
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I know that a 3 goes into 33 and a 3 goes into 30.
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I can take that number, the common number, the common factor between 33 and 30,
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divide it to both the top and the bottom.
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Remember as long as you are doing the same thing to the top and to the bottom of the fraction,
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you are not changing it; you are just simplifying it.
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33 divided by 3 is 11; 30 divided by 3 is 10.
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This is your new improper fraction, 11/10.
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Since it is an improper fraction, we can change it to a mixed number.
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Or we can just leave it like that; that is fine too.
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But if I do want to change it to a mixed number,
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then this 10 fits into the top number 11 only one time.
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10 fits into 11 only one time.
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How many left over do I have?--only one.
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My denominator always has to stay the same.
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11/10 is the same thing as 1 and 1/10.
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Another example, we are going to take 11/20 and subtract it to 11/30.
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My denominators are different; I have to find the common denominator.
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I can take 20; 5 is a prime number; I am going to circle it.
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4, 2, and 2; I circle those; and then 30.
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For this one, I can either do 3 and 10 or I can do 15 and 2, any factor pair.
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Let's do 3 and 10; here 3 is a prime number; I am going to circle it.
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10 is 5 and 2; they are both prime.
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I am going to look for any numbers they have in common.
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Here; I have a 2 here; and I have a 2 here.
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I am going to cancel one of them out.
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Here I have a 5; and I have a 5 here.
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I am going to cancel just one of them out.
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Any others?--nope, that is it.
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My LCD or my LCM is going to be 2 times 2 times 5 times 3.
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This is going to be 4 times 5 which is 20, times 3 which is 60; my LCD is 60.
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Then my next step is going to be to change each fraction so that their denominator will become 60.
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20, to figure out what you have to multiply to 20 to get 60,
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I can just take 60 and divide it by 20.
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This is going to be 3; 20 times 3 was 60.
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Again you have to multiply the top number by the same number.
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11 times 3 is 33.
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For the second fraction, 11/30, 30 times 2 is 60.
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Multiply the top number by that number; 22.
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11/20 is the same thing as 33/60; I am going to subtract.
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Then 11/30 is the same thing as 22/60.
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Again double check your denominators; make sure that they are the same.
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Since they are, now I can subtract; 33 minus 22 which is 11 over...
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Keep your denominator the same; do not add or subtract your denominators.
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11/20 minus 11/30 became 11/60.
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Let's do another example; this example, 23/95 plus 4/5.
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In order for me to add these two fractions, I have to make sure they have a common denominator.
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In this case, they don't; 95 is this denominator; 5 is the other one.
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I have to look for the common denominator.
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For 95, I can either look for the LCM, the least common denominator or least common multiple, between 95 and 5.
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Or I can list all the multiples out and see the smallest common multiple.
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I know that 95 is divisible by 5 because any number that ends in a 5 or 0 is divisible by 5.
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In this case, a 5, if this number is divisible by this number,
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then this becomes the new common denominator, the least common denominator.
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Or if you want to just do the factor tree to find the least common denominator, then you can do that too.
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95 is going to be 5 times 19; these are both prime numbers.
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I am going to circle them; 5 is just 5 and 1.
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To find the LCD, I am going to look for any factors they have in common.
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Here, there is a 5 here and a 5 here.
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I am going to cancel only one of them out.
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Whenever they have something in common, just cancel only one of them out.
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Then I am going to write all the circled numbers again; 5 times 19.
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This is just a 1 so I don't have to write that.
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5 times 19 I know is 95.
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My LCD, my least common denominator, is going to be 95.
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For this fraction here, since the denominator is already 95, I don't have to change it.
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This one can stay as it is.
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This one however, I have to change that 5 to make it a 95 so they will have a common denominator
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because that is the only way I can add these fractions, if their denominators are the same.
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For this fraction right here, I need to change it so that the denominator will become 95.
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I am going to take this 95, divide it by 5 to see what I have to multiply this by.
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That is 19; here I am going to take this and multiply it by 19.
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This will become 76; 4/5 became 76/95.
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Make sure you multiply it by the same number.
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You have to multiply the top and the bottom number by the same number.
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That way you are not changing the fraction.
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You are just changing the numbers; but they are still equal fractions.
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Now I am going to do 23/95 plus 76/95.
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Again I have to make sure the denominators are the same.
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If they are not the same at this point, then there is something wrong.
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Go back and check your work.
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But since they are the same, I can go ahead and add the fractions.
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23 plus 76, I am going to add the numerators together.
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If I add them, it is going to be 99.
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Here denominator stays the same; it is 95 here; 95 here.
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My denominator is going to become 95; 23/95 plus 4/5 is 99/95.
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That is it for this lesson; thank you for watching Educator.com.