Question

What is the fraction form of the following decimal expansion?
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:
\[
\boxed{\frac{9}{11}}
\]