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Course: 7th grade (Eureka Math/EngageNY) > Unit 2

Lesson 2: Topic B: Multiplication and division of integers and rational numbers

Simplifying complex fractions

Learn how to take complex looking fractions and make them much, much simpler.

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  • leaf yellow style avatar for user Joshua
    Why can't you swap the reciprocal fractions when multiplying?

    When you turn the division problem into a multiplication by transforming the fraction on the right into the reciprocal of the fraction on the left, why can't you switch them around and make the fraction on top or on the left the reciprocal of the fraction on the right?
    (7 votes)
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    • piceratops ultimate style avatar for user fr33d0g
      It is kind of like saying 9 divided by 3, why cant we switch the numbers around and say 3 divided by 9. You can, but these two ways mean different things and you will get different answers. Hope this helps :)
      (9 votes)
  • boggle purple style avatar for user omocat.lol
    how do we simplify normal fractions tho
    (7 votes)
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    • duskpin ultimate style avatar for user Emma
      It's defo a tricky thing to see until you get it like i just did lol but there's 2 ways you simplify fractions, first is the one where you Simplify by finding the Greatest Common Factor (GCF) for the numerator and denominator of your answer then divide them both by the GCF.

      Then the other is where you Simplify an improper fraction into a mixed number by dividing the numerator (top number) by the denominator (bottom number), and you'll get a number with a remainder so you write the number as your whole number then write the remainder next to it, draw a line under it and write the original denominator of the fraction you just simplified under it.

      You can find the GCF for both proper and improper fractions too to reduce the size of them, this is needed for bigger fractions especially when multiplying fractions and using prime factoring. That's if there is a GCF between the numerator and denominator, if it's not already at it's smallest possible fraction.

      It becomes too hard and starts taking too long writing all the common factors when the numbers in your fractions start to get bigger.
      (3 votes)
  • hopper cool style avatar for user Ralph Gifford 5
    from to , Sal explains how to simplify 2/2/3. However, I could reason differently. I could simplify the 2/2, resulting in 1, and end up with 1/3. Their are other ways to get 1/3 too. I could break all the fractions to get 2÷2÷3, which I think could be 3 or 1/3 depending on how I look at it. How do I know which way I should do?
    (6 votes)
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  • starky seedling style avatar for user 25aaronmeyers
    So how would you do something like 1 / 1/2? Would it just be 1/2?
    (2 votes)
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    • starky seed style avatar for user Soerenna Farhoudi
      if you mean (1/1)/2 then yes because 1/1 is 1 and that leaves 1/2.
      If you meant 1/(1/2) then the answer is different:
      with the last expression you are dividing 1 by 1/2, or in other words you are asking how many times does 1/2 fit into 1? Because we are dealing with very small numbers you can just count halves until you reach 1, in this case 2 halves go into 1 so your answer is 2. If you take a different approach that will work with any number you can use the rule that something divided by something else is the same as multiplying by its reciprocal. So 1/(1/2) = 1 x 2/1 = 2

      hope it helped :)
      (4 votes)
  • aqualine ultimate style avatar for user Jayden
    At why did he flip 3/7?
    (2 votes)
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  • starky sapling style avatar for user Toby Newman
    how do you do the simplify fraction work?
    (3 votes)
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  • mr pants teal style avatar for user skyla
    I get how to simplify, but do you guys have anything that could help me solve the complex fractions. All I can find is simplifying complex fractions. I just want to solve them.
    (2 votes)
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  • primosaur tree style avatar for user Aidan Nelson
    For 2x 3/2 why didn't he change 2 to 2/1?
    (1 vote)
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  • hopper cool style avatar for user Hanging_With_My_Gnomies
    This math is my greatest foe. It's really Hard.
    (3 votes)
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    • piceratops tree style avatar for user 🐉~SMJ~🐉
      Ill give you an idea about how to do it, I dont think its that effective though. It'll take time to understand

      Lets say...


      1. Find which times table both numbers occur in: 9
      2. What does 9 need to be multiplied with to get to 27/54: 3 and 6
      3. The answer is 3/6.
      4. WE NEED TO SIMPIFY IT FURTHER. I dont know how to use the first part of the strategy on this part(I think the fraction is not applicable due to the numbers being tool small) but let me know if you do. When you think about 3/6 its obviously going to remind you of A HALF. AND A HALF IS WRITTEN AS 1/2

      5. The answer to simplifying 27/54 IS 1/2. Finally we're done!

      PS: This took me 24 minutes to type: I had to adjust the numbers and put in variables
      (1 vote)
  • female robot ada style avatar for user vicente.garcia
    this dose not make any sense.
    (2 votes)
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Video transcript

- What I hope to do in this video is emphasize the relation, the connection, between fractions and division and then using that knowledge to help us simplify some hairy looking fractions. So, let's say, let's just do a little bit of a review. So, if I say two divided by, two divided by three, two divided by three, I could write this as, I could write this as two over three. Two divided by, two divided by three. So, this expression here, sometimes we might say, "Hey, this is 2/3. Think of it as a fraction." But you can also view this as two divided by three. So, if we went the other way around. If someone were to write four over, actually, let me just do it in a different color, if someone were to say four, 4/9, four, four over nine, we could interpret this, this could be interpreted as four divided by nine. Four divided by nine. In fact, if we wanted to, if wanted to convert this into a decimal that's exactly what we would do. We would calculate what four divided by nine is. We would divide nine into four. Four divided by nine. So, this is all review. So, let's just use this knowledge and I'll think more complex fractions. So, if someone were to say, actually, let me just take this example right over here, if someone were to say two over, instead of saying two over three. If they said two over, I don't know, let me say two over 2/3. Two over 2/3. Well, what would this simplify to? Well, we could go the other way around. This is the same thing as two divided by 2/3. So, let's write it like that. This is the same thing as two divided by, two divided by 2/3, two divided by 2/3, which is the same thing as, dividing by a fraction's the same thing as multiplying by its reciprocal. So, this is going to be the same thing as two times, two times the reciprocal of 2/3, which is, we swap the denominator and the numerator, two times three over two. So, two times three halves, if I have three halves twice, well, that's just going to be equal to, that's going to be equal to six halves, and six halves, you can view that as, well, two halves is a whole, so this is gonna be three wholes. Or you can say six divided by two. Well, that's just gonna be equal to three. You can view it either way. And then that is equal to three. Let's do a few more of these. And let's keep making them a little bit more complicated. Just let me get some good practice. And like always, pause the video. You should get excited when you see one of these things. And pause the video and see if you can do it on your own. All right, let's do something really interesting. Let's say negative 16 over nine over, over, I'll do this in a tan color, over, let's say, three over seven. What is this? Can you simplify this complex fraction? Can you simplify this expression over here? Well, once again, we can view this as negative 16/9 divided by 3/7. So, this could be rewritten as, and I can write it either as negative 16 over 9 or I could rewrite it as negative 16/9. So, I can put the negative in front of the whole, in front of the whole fraction like that, or I could say that it's negative 16 over nine, or I can even write this as negative 16 over negative nine. Those would all be equivalent. But right now I'm writing at negative 16/9 and I am going to now divide, we can interpret this complex fraction as dividing it by 3/7. Dividing by 3/7. And so, this is going to be equal to, this is going to be equal to negative 16/9, negative 16 over nine. Actually, let me just rewrite it as negative 16 over nine. Negative 16 over nine just to show that we can do it. Times the reciprocal of this. So, times 7/3. I just swap the numerator and the denominator. Times, do that same brown color, times, times 7/3, and now what am I going to get? In the numerator, I have negative 16 times seven. Let me think about it. 10 times seven is 70. Six times seven is 42. 70 plus 42 is 112. So, this, so this part right over here, this part right over here would be negative 112. Negative 112 over nine times three. Nine times three is 20, nine times three is 27. And there you have it. You can view this as negative 112 over 27 or you can put the negative up front and you can say, "Hey, this is hinting "as negative 112, 112/27." 112/27 and just to make everything come full circle you can view this as negative 112 divided by 27 or negative 112/27. Either way.