Question

Convert 0.27¯¯¯¯¯ to a rational number in simplest form.(1 point) Responses 127 Start Fraction 1 over 27 end fraction 311 Start Fraction

Answer 1

To convert the repeating decimal \(0.27\overline{27}\) to a rational number, we will denote it as \(x\).

Let:

\[
x = 0.27272727\ldots
\]

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

\[
100x = 27.27272727\ldots
\]

Now we can set up a system of equations. We have:

1. \(x = 0.27272727\ldots\)
2. \(100x = 27.27272727\ldots\)

Next, we subtract the first equation from the second:

\[
100x - x = 27.27272727\ldots - 0.27272727\ldots
\]

This simplifies to:

\[
99x = 27
\]

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

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

Now we should simplify the fraction \(\frac{27}{99}\):

The greatest common divisor (GCD) of \(27\) and \(99\) is \(9\). Therefore, we divide both the numerator and the denominator by \(9\):

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

Thus, \(0.27\overline{27}\) as a rational number in simplest form is:

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

So, the final answer is:

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