What is the fraction form of the following decimal expansion?

Question

0.overline81

Answer 1

To convert the repeating decimal \( 0.\overline{81} \) to a fraction, we can use the following method.

Let: \[ x = 0.\overline{81} \]

This means: \[ x = 0.8181818181\ldots \]

To eliminate the repeating part, we can multiply by 100 (since the repeating part has two digits): \[ 100x = 81.8181818181\ldots \]

Now we have two equations:

1. \( x = 0.8181818181\ldots \)
2. \( 100x = 81.8181818181\ldots \)

Next, we subtract the first equation from the second: \[ 100x - x = 81.8181818181\ldots - 0.8181818181\ldots \] \[ 99x = 81 \]

Now, solve for \( x \): \[ x = \frac{81}{99} \]

Next, we can simplify the fraction \( \frac{81}{99} \). The greatest common divisor (GCD) of 81 and 99 is 9. We divide both the numerator and the denominator by their GCD: \[ \frac{81 \div 9}{99 \div 9} = \frac{9}{11} \]

Thus, the repeating decimal \( 0.\overline{81} \) in fraction form is: \[ \boxed{\frac{9}{11}} \]