Convert 0.72¯¯¯¯¯0.72¯ to a rational number in simplest form.

Answer 1

Let \( x = 0.727272...\) where the digits "72" repeat indefinitely. To express this repeating decimal as a fraction, we can use the following steps:

1. Multiply \( x \) by 100: Since the repeating part has two digits, we multiply by \( 100 \) to shift the decimal point two places to the right:

   \[ 100x = 72.727272... \]

2. Set up an equation: We can now write two equations:

   \[ x = 0.727272... \] \[ 100x = 72.727272... \]

3. Subtract the first equation from the second:

   \[ 100x - x = 72.727272... - 0.727272... \] This simplifies to:

   \[ 99x = 72 \]

4. Solve for \( x \):

   \[ x = \frac{72}{99} \]

5. Simplify the fraction: Next, we need to simplify \( \frac{72}{99} \). First, find the greatest common divisor (GCD) of 72 and 99.

   • The prime factorization of \( 72 \) is \( 2^3 \times 3^2 \).
   • The prime factorization of \( 99 \) is \( 3^2 \times 11 \).

   The GCD is \( 3^2 = 9 \).

   Now, divide the numerator and denominator by their GCD:

   \[ \frac{72 \div 9}{99 \div 9} = \frac{8}{11} \]

Thus, the rational number in simplest form that corresponds to \( 0.727272...\) is

\[ \boxed{\frac{8}{11}}. \]