Question

Convert 0.09¯¯¯¯¯
to a rational number in simplest form.

tell me in a very short simple explanation give me an answer

Answer 1

Let \( x = 0.09\overline{9} \). Then, multiply by 100:

\[
100x = 9.99\overline{9}
\]

Now, subtract the original equation from this:

\[
100x - x = 9.99\overline{9} - 0.09\overline{9}
\]

This simplifies to:

\[
99x = 9.9
\]

Now divide by 99:

\[
x = \frac{9.9}{99}
\]

Convert \( 9.9 \) to a fraction:

\[
9.9 = \frac{99}{10} \quad \Rightarrow \quad x = \frac{99/10}{99} = \frac{1}{10}
\]

Thus, \( 0.09\overline{9} = \frac{1}{10} \).