text: # How is multiplying by a fraction the same as dividing by the inverse of that fraction
  
  1. Aug 9, 2011
  
  ### jimgavagan
  
  Supposedly this proof answers my question.
  
  8 / 16 = .5
  8 / 8 = 1
  8 / 4 = 2
  8 / 2 = 4
  8 / 1 = 8
  8 / .5 = 16
  8 * 2/1 = 8 / .5
  
  I'm just wondering how this proof answers my question?
  
  2. Aug 10, 2011
  
  ### Dr. Seafood
  
  Re: How is multiplying by a fraction the same as dividing by the inverse of that frac
  
  It doesn't ... and that's not a proof o__O
  
  It follows kind of from "definition" of division: $x \over y$ actually means $x \cdot y^{-1}$ where $y^{-1}$ is the number such that $y \cdot y^{-1} = 1$, called the "multiplicative inverse" or "reciprocal" of y. This number is unique. Obviously $({a \over b})^{-1} = {b \over a}$ (since ${a \over b} \cdot {b \over a} = 1$) as long as we have $a, b \ne 0$. (Also of note: $0^{-1}$ does not exist!
  
  So it turns out that since ${x \over y} = x \cdot y^{-1}$, setting $y = {a \over b}$ we get
  
  "${{x} \over {a \over b}}$" $= {x} \cdot ({a \over b})^{-1} = {x} \cdot {b \over a}$ (again provided $a, b \ne 0$). I hope this explanation helps.
  
  Last edited: Aug 10, 2011
  3. Aug 10, 2011
  
  ### jimgavagan
  
  Re: How is multiplying by a fraction the same as dividing by the inverse of that frac
  
  ooooooooooooooooooooooooooh awesome! :D
  
  Very much appreciated!
explanation: The document contains a math question about the relationship between multiplying by a fraction and dividing by its inverse. It also includes a discussion and an explanation that answers the question.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: # How is multiplying by a fraction the same as dividing by the inverse of that fraction
  
  1. Aug 9, 2011
  
  ### jimgavagan
  
  Supposedly this proof answers my question.
  
  8 / 16 = .5
  8 / 8 = 1
  8 / 4 = 2
  8 / 2 = 4
  8 / 1 = 8
  8 / .5 = 16
  8 * 2/1 = 8 / .5
  
  I'm just wondering how this proof answers my question?
  
  2. Aug 10, 2011
  
  ### Dr. Seafood
  
  Re: How is multiplying by a fraction the same as dividing by the inverse of that frac
  
  It doesn't ... and that's not a proof o__O
  
  It follows kind of from "definition" of division: $x \over y$ actually means $x \cdot y^{-1}$ where $y^{-1}$ is the number such that $y \cdot y^{-1} = 1$, called the "multiplicative inverse" or "reciprocal" of y. This number is unique. Obviously $({a \over b})^{-1} = {b \over a}$ (since ${a \over b} \cdot {b \over a} = 1$) as long as we have $a, b \ne 0$. (Also of note: $0^{-1}$ does not exist!
  
  So it turns out that since ${x \over y} = x \cdot y^{-1}$, setting $y = {a \over b}$ we get
  
  "${{x} \over {a \over b}}$" $= {x} \cdot ({a \over b})^{-1} = {x} \cdot {b \over a}$ (again provided $a, b \ne 0$). I hope this explanation helps.
  
  Last edited: Aug 10, 2011
  3. Aug 10, 2011
  
  ### jimgavagan
  
  Re: How is multiplying by a fraction the same as dividing by the inverse of that frac
  
  ooooooooooooooooooooooooooh awesome! :D
  
  Very much appreciated!
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document contains a math question about the relationship between multiplying by a fraction and dividing by its inverse. It also includes a discussion and an explanation that answers the question.
  
  Label: question-and-answer