Question

Convert the following repeating decimal to a fraction in simplest form.
., start overline, 2, 0, end overline
.
20

Answer 1

Let \( x = 0.20202020\ldots \), where the digits "20" repeat indefinitely.

To eliminate the repeating part, we can multiply \( x \) by \( 100 \) (since the repeating part consists of two digits):

\[
100x = 20.20202020\ldots
\]

Now, we have two equations:

1. \( x = 0.20202020\ldots \)
2. \( 100x = 20.20202020\ldots \)

Next, we can subtract the first equation from the second:

\[
100x - x = 20.20202020\ldots - 0.20202020\ldots
\]

This simplifies to:

\[
99x = 20
\]

Now, solve for \( x \):

\[
x = \frac{20}{99}
\]

We need to check if \( \frac{20}{99} \) is in simplest form. The greatest common divisor (GCD) of 20 and 99 is 1, since 20 (which factors to \( 2^2 \times 5 \)) and 99 (which factors to \( 3^2 \times 11 \)) share no common factors.

Thus, \( \frac{20}{99} \) is already in simplest form.

The final answer is:

\[
\boxed{\frac{20}{99}}
\]