Question

Write the following infinite decimal expansion as a fraction. You do not need to simplify. Click for Help Video



0.21‾0.\overline{21}0.
21

Answer 1

To express the repeating decimal \(0.\overline{21}\) as a fraction, we can set:

\[
x = 0.\overline{21}
\]

This means:

\[
x = 0.21212121\ldots
\]

Next, we notice that the digits "21" repeat every two decimal places, so we can multiply both sides of the equation by \(100\) (since there are two digits in the repeating part):

\[
100x = 21.212121\ldots
\]

Now, we can write our system of equations:

1. \( x = 0.212121\ldots \)
2. \( 100x = 21.212121\ldots \)

Next, we subtract the first equation from the second:

\[
100x - x = 21.212121\ldots - 0.212121\ldots
\]

This simplifies to:

\[
99x = 21
\]

Now, we can solve for \(x\):

\[
x = \frac{21}{99}
\]

So, the infinite decimal \(0.\overline{21}\) can be expressed as the fraction \(\frac{21}{99}\).

While simplification is not required, note that \(21\) and \(99\) have a common factor of \(3\):

\[
\frac{21 \div 3}{99 \div 3} = \frac{7}{33}
\]

Thus, the fraction \(\frac{21}{99}\) is equivalent to \(\frac{7}{33}\), but since we are not required to simplify, the answer is:

\[
\frac{21}{99}
\]