Repeating Decimal Conversion

To convert a repeating decimal like \(0.232323...\) into a fraction, we use an algebraic method. This method allows us to represent the repeating decimal as a rational number in the form \(\frac{p}{q}\).

Let's convert the given decimal \(0.232323...\) into a fraction step-by-step:

• Step 1: Set the given decimal equal to a variable. Let \(x\) represent the decimal:
  \(x = 0.232323...\) (Equation 1)
• Step 2: Identify the repeating pattern. The repeating block of digits is \(23\). There are \(2\) digits that repeat.
• Step 3: Multiply both sides of Equation 1 by \(10\) raised to the power equal to the number of repeating digits. Since there are \(2\) repeating digits, we multiply by \(10^2 = 100\):
  \(100x = 100 \times 0.232323...\)
  \(100x = 23.232323...\) (Equation 2)
• Step 4: Subtract Equation 1 from Equation 2. This step eliminates the repeating part of the decimal:
  \(100x - x = 23.232323... - 0.232323...\)
  \(99x = 23\)
• Step 5: Solve for \(x\) by dividing both sides of the equation by \(99\):
  \(x = \frac{23}{99}\)

So, the fraction equivalent of the repeating decimal \(0.232323...\) is \(\frac{23}{99}\).

Comparing this result with the provided options, we find:

• Option 1: \(\frac{23}{90}\)
• Option 2: \(\frac{23}{100}\)
• Option 3: \(\frac{23}{99}\)

Our calculated fraction matches Option 3.