Repeating Decimals to Fractions Game
Converting repeating decimals (like 0.333… or 0.121212…) into fractions is a standard algebra trick. It relies on the power of multiplication and subtraction to cancel out the repeating part. This method works because the subtraction step eliminates the infinite tail of the decimal, leaving you with a clean integer numerator and a multiple of 9 or 90/99/etc. in the denominator.
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Repeating Decimal Converter

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How to Play the Repeating Decimals to Fractions Game
Here’s how to play:

1. Start: Each Quiz consists of 10 questions. Select the difficulty: Simple Repeating, Delayed Delayed or Mixed Challenge.
   
2. Look at the Problem: Use the multiplication and subtraction algebra trick to calculate the fraction.
   
3. Select Your Answer: Select the correct fraction.
   
4. Check Your Work: If you selected the right answer, it will be highlighted in green. If you are wrong, it will be highlighted in red and the correct answer will be highlighted in green.
   
5. Get a New Problem: Click “Next Decimal” for a new problem.
   Your score is tracked, showing how many you’ve gotten right.
   
6. Finish Game When you have completed 10 questions, click “Finish Game” to get your final score.
    
   

Converting Repeating Decimals to Fractions
Converting a repeating decimal (a rational number) back into its fractional form (\(\frac{a}{b}\)) is done using a simple algebraic method that isolates and eliminates the infinitely repeating digits.

The process has two main cases: simple repeats (where the repeat starts immediately after the decimal point) and mixed repeats (where there are non-repeating digits before the repeating block starts).

Case 1: Simple Repeating Decimals
This applies when the repeating block starts immediately after the decimal point (e.g., \(0.\overline{3}, 0.\overline{12}\)).
Example A: Single Repeating Digit (\(0.\overline{3}\))
Convert 0.3333… to a fraction.
Step 1: Set the decimal equal to a variable (x).
\(x = 0.3333… \quad \text{(Equation 1)}\)
Step 2: Multiply x by 10^d, where d is the number of repeating digits.
Since only one digit (3) is repeating, d=1. We multiply by 10¹ = 10.
\(10x = 3.3333… \quad \text{(Equation 2)}\)
Step 3: Subtract Equation 1 from Equation 2.
The goal is to align the repeating decimals so they cancel out completely upon subtraction.
\(\begin{array}{rcl} 10x &=& 3.3333… \\ - \quad x &=& 0.3333… \\ \hline 9x &=& 3 \end{array}\)
Step 4: Solve for x and simplify the fraction.
\(x = \frac{3}{9}\)
\(x = \frac{1}{3}\)

Example B: Multiple Repeating Digits (\(0.\overline{12}\))
Convert 0.121212… to a fraction.
Step 1: Set the decimal equal to x.
\(x = 0.121212… \quad \text{(Equation 1)}\)
Step 2: Multiply x by 10^d.
Since two digits (12) are repeating, d=2. We multiply by 10² = 100.
\(100x = 12.121212… \quad \text{(Equation 2)}\)
Step 3: Subtract Equation 1 from Equation 2.
\(\begin{array}{rcl} 100x &=& 12.121212… \\ - \quad x &=& 0.121212… \\ \hline 99x &=& 12 \end{array}\)
Step 4: Solve for x and simplify.
\(x = \frac{12}{99}\)
We can divide the numerator and denominator by 3:
\(x = \frac{4}{33}\)

Case 2: Mixed or Delayed Repeating Decimals
This applies when there are non-repeating digits before the repeating block starts (e.g., \(0.1\overline{6}, 0.52\overline{79}\)).
Example C: Mixed Repeat (\(0.1\overline{6}\))
Convert 0.16666… to a fraction.
Step 1: Set the decimal equal to x.
\(x = 0.16666… \quad \text{(Equation 1)}\)
Step 2: Create a second equation where the non-repeating part is moved to the left of the decimal.
One non-repeating digit (1). Multiply by 10¹ = 10.
\(10x = 1.6666… \quad \text{(Equation 2)}\)
Step 3: Create a third equation where one full repeating block is moved to the left of the decimal.
One repeating digit (6). We need to multiply the original x by \(10^{1 \text{ (non-repeat)} + 1 \text{ (repeat)}} = 10^2 = 100\).
\(100x = 16.6666… \quad \text{(Equation 3)}\)
Step 4: Subtract Equation 2 from Equation 3.
Subtracting Equation 2 from Equation 3 ensures only the repeating parts align and cancel.
\(\begin{array}{rcl} 100x &=& 16.6666… \\ - \quad 10x &=& 1.6666… \\ \hline 90x &=& 15 \end{array}\)
Step 5: Solve for x and simplify.
\(x = \frac{15}{90}\)
We can divide the numerator and denominator by 15:

This video gives a clear, step-by-step approach to learn how to use the Repeating Decimals to Fractions Formula.

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