Recurring decimals into fractions

• What are recurring decimals?

  Numbers with a repeating sequence of digits
• What is the term for the group of digits that repeat in a recurring decimal?

  Repetend
• What is the repetend of the recurring decimal 0.666...?


  • Repetend: 6
• What is the process to set up an algebraic equation for a recurring decimal?

  1. Assign a variable (e.g., x).
  2. Write out the decimal clearly.
  3. Formulate the algebraic equation.
• What is the repetend of the recurring decimal 1.2555...?


  • Repetend: 5
• How do you identify the repeating pattern in a recurring decimal?

  Look for the group of digits that repeat
• If `x = 0.666...`, how would you set up the equation?

  x = 0.666...
• What happens to the patterns in recurring decimals?

  They keep going forever
• What is the first step to set up an algebraic equation for a recurring decimal?

  Assign a variable to the decimal
• What does simplifying the resulting equation achieve?

  Makes the equation easier to work with
• How does setting up the equation `x = 0.333...` help in manipulation?

  It makes it easier to manipulate algebraically
• What is the next step after multiplying both sides by 10?

  Subtract the original equation from the new one
• What is the significance of the repeating part in a recurring decimal?

  It indicates the decimal's infinite nature
• What happens to the decimal when multiplying `x = 0.333...` by 10?

  The decimal shifts one place to the right
• What does the equation `x = 0.333...` represent?

  The recurring decimal 0.333...
• How is the recurring decimal `0.333...` represented in an equation?

  x = 0.333...
• What is the repetend of the recurring decimal 3.141414...?


  • Repetend: 14
• What is the coefficient in the equation `9x = 3`?

  9
• What is the simplified form of $9x =$$3$?


  $x =$$\frac{3}{9}$
• Why is it important to align the equations when subtracting?

  To place like terms under each other
• What is the result of multiplying `x = 0.333...` by 10?

  10x = 3.333...
• What does writing out the decimal clearly help with?

  It shows the repeating part of the decimal
• What do you get when you divide the denominator 9 by the GCD 3?


  3
• What are the steps to subtract the original decimal equation from the new equation?

  1. Align the equations neatly
  2. Subtract the left and right sides separately
  3. Simplify the resulting equation
• How do you isolate the variable `x` in the equation `9x = 3`?

  Get `x` by itself on one side
• What do you get when you divide both sides of `9x = 3` by 9?

  x = $\frac{3}{9}$
• What does the fraction $\frac{3}{9}$ represent in this context?


  It converts the recurring decimal
• What is the significance of aligning the repeating digits in the equation?

  • Facilitates easier subtraction
  • Prepares for the next step in solving
• What do you get when you divide the numerator 3 by the GCD 3?


  1
• What is the purpose of multiplying both sides of the equation by a power of 10?

  To shift the decimal place for alignment
• What are the steps to simplify a fraction?

  1. Find the greatest common divisor (GCD) of the numerator and denominator.
  2. Divide both the numerator and denominator by the GCD.
• What is the next step after isolating the variable in `9x = 3`?

  Divide both sides by the coefficient
• What should you do after aligning the equations?

  Subtract the left and right sides separately
• What is the first step after subtracting the original equation from the new equation?

  You will have a simple equation
• What is the simplified fraction of 3/9?

  1/3
• Why is it easier to see the relationship between numerator and denominator in fractions?

  Fractions show proportional relationships clearly
• How do we convert recurring decimals into fractions?

  By identifying the proportional relationship
• How does dividing both sides of `9x = 3` by 9 affect the equation?

  It simplifies the equation to find x
• What does simplifying a fraction to its lowest terms mean?

  Finding the simplest equivalent form of the fraction
• What is the first step in subtracting the original decimal equation from the new equation?

  Align the equations neatly