Question

Convert the following repeating decimal to a fraction in simplest form.
., start overline, 9, 8, end overline
.
98

Answer 1

Let \( x = 0.989898...\), where the digits "98" repeat indefinitely.

To eliminate the repeating decimal, we can multiply \( x \) by 100 (which shifts the decimal point two places to the right):

\[
100x = 98.989898...
\]

Now, we can set up a system of equations using these two equations:

1. \( x = 0.989898...\)
2. \( 100x = 98.989898...\)

Next, we'll subtract the first equation from the second:

\[
100x - x = 98.989898... - 0.989898...
\]

This simplifies to:

\[
99x = 98
\]

Now, we solve for \( x \):

\[
x = \frac{98}{99}
\]

The fraction \(\frac{98}{99}\) is already in its simplest form, as 98 and 99 have no common factors other than 1.

Thus, the repeating decimal \(0.989898...\) can be expressed as the fraction

\[
\boxed{\frac{98}{99}}.
\]