Question

Convert
0.
8181818181

to a rational number

Answer 1

To convert the repeating decimal \( 0.8181818181\ldots \) into a rational number, we can follow these steps:

1. Let \( x = 0.8181818181\ldots \).

2. Since the decimal repeats every two digits, we can multiply \( x \) by \( 100 \) to move the decimal point two places to the right:
\[
100x = 81.8181818181\ldots
\]

3. Now, we can set up the equation:
\[
100x = 81 + x
\]

4. To isolate \( x \), subtract \( x \) from both sides:
\[
100x - x = 81
\]
\[
99x = 81
\]

5. Now, divide both sides by \( 99 \):
\[
x = \frac{81}{99}
\]

6. We can simplify \( \frac{81}{99} \) by finding the greatest common divisor (GCD) of 81 and 99. The GCD is 9:
\[
x = \frac{81 \div 9}{99 \div 9} = \frac{9}{11}
\]

Thus, the repeating decimal \( 0.8181818181\ldots \) can be expressed as the rational number \( \frac{9}{11} \).