Convert Repeating Decimal to Fraction

This solution explains the process of converting a repeating decimal into its equivalent fraction using algebraic methods. It focuses on demonstrating the calculation relevant to the provided answer options.

Steps for Decimal to Fraction Conversion

1. Represent the repeating decimal with a variable. Based on the structure of the options provided, we will demonstrate the conversion for the decimal $0.5\overline{6}$, which represents $0.5666...$

   Let $y = 0.5\overline{6}$.

2. To work with the repeating part, set up algebraic equations. Multiply $y$ by 10 to shift the decimal point just before the repeating sequence begins:

   $10y = 5.666...$

3. Multiply $y$ by 100 (which is $10 \times 10$) to shift the decimal point just after the first cycle of the repeating digits:

   $100y = 56.666...$

4. Subtract the equation from Step 2 ($10y$) from the equation in Step 3 ($100y$). This step eliminates the infinitely repeating part of the decimal:

   $100y = 56.666...$

   $- \quad 10y = \quad 5.666...$

   --------------------

   $90y = 51.000...$

   This simplifies to $90y = 51$.

5. Solve for $y$ by dividing both sides of the equation by 90:

   $y = \frac{51}{90}$

6. The resulting fraction is $\frac{51}{90}$. This fraction matches one of the provided options, confirming the conversion process.