To convert the repeating decimal \( 0.72\overline{72} \) (where "72" repeats indefinitely) into a rational number, we can use the following steps:
Let \( x = 0.727272\ldots \).
To eliminate the repeating part, we can multiply \( x \) by 100 (because the repeating part has 2 digits):
\[ 100x = 72.727272\ldots \]
Now we have two equations:
- \( x = 0.727272\ldots \)
- \( 100x = 72.727272\ldots \)
Next, we can subtract the first equation from the second:
\[ 100x - x = 72.727272\ldots - 0.727272\ldots \]
This simplifies to:
\[ 99x = 72 \]
Now, we can solve for \( x \):
\[ x = \frac{72}{99} \]
Next, we can simplify the fraction \( \frac{72}{99} \).
To do this, we find the greatest common divisor (GCD) of 72 and 99. The GCD is 9.
Now divide both the numerator and the denominator by their GCD:
\[ \frac{72 \div 9}{99 \div 9} = \frac{8}{11} \]
So, the repeating decimal \( 0.72\overline{72} \) as a rational number in simplest form is:
\[ \frac{8}{11} \]