Question

Convert the following repeating decimal to a fraction in simplest form. 0.01 (01 is repeated)

Answer 1

To convert the repeating decimal \( 0.01\overline{01} \) (where \( 01 \) is the repeating part) into a fraction, we can use the following steps:

Let \( x = 0.01\overline{01} \).

This means:
\[
x = 0.0101010101\ldots
\]

To eliminate the repeating decimals, we can multiply \( x \) by 10000 (since there are 4 digits in each full cycle of the repetition):
\[
10000x = 101.0101010101\ldots
\]

Now, we have two equations:
1. \( x = 0.0101010101\ldots \) (the original equation)
2. \( 10000x = 101.0101010101\ldots \) (the equation after multiplying by 10000)

Next, we subtract the first equation from the second:
\[
10000x - x = 101.0101010101\ldots - 0.0101010101\ldots
\]
This simplifies to:
\[
9999x = 101
\]

Now, solve for \( x \):
\[
x = \frac{101}{9999}
\]

To simplify this fraction, we can check if 101 and 9999 have any common factors. Since 101 is a prime number, we can check if 9999 is divisible by 101. Dividing \( 9999 \) by \( 101 \):
\[
9999 \div 101 \approx 99
\]
Calculating \( 101 \times 99 = 9999 \), which confirms \( 9999 = 101 \times 99 \).

Thus, we can simplify:
\[
\frac{101}{9999} = \frac{1}{99}
\]

Therefore, the simplest form of the fraction that represents the repeating decimal \( 0.01\overline{01} \) is:
\[
\boxed{\frac{1}{99}}
\]