Question

Convert  0.27¯¯¯¯¯  to a rational number in simplest form.(1 point)Responses2799Start Fraction 27 over 99 end fraction27100Start Fraction 27 over 100 end fraction311Start Fraction 3 over 11 end fraction127Start Fraction 1 over 27 end fraction

Answer 1

To convert the repeating decimal \( 0.27\overline{27} \) to a rational number, we can use the following method:

Let \( x = 0.272727...\).

Then, to eliminate the repeating part, we can multiply by 100 (since two digits are repeating):

\[
100x = 27.272727...
\]

Now, we have:

\[
x = 0.272727...
\]
\[
100x = 27.272727...
\]

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

\[
100x - x = 27.272727... - 0.272727...
\]
\[
99x = 27
\]

Now, solve for \( x \):

\[
x = \frac{27}{99}
\]

Next, we simplify \( \frac{27}{99} \):

Both numbers can be divided by 9:

\[
\frac{27 \div 9}{99 \div 9} = \frac{3}{11}
\]

Thus, the repeating decimal \( 0.27\overline{27} \) can be expressed as the rational number:

\[
\frac{3}{11}
\]

So, the correct response is:

**\( \frac{3}{11} \)**