Question

Convert 0.27 (27 is repeated) to a rational number in simplest form.

3/11
27/99
1/27
27/100

Answer 1

To convert the repeating decimal \(0.27\overline{27}\) into a rational number, we can denote it as \(x\):

\[
x = 0.27272727\ldots
\]

Multiplying both sides of the equation by 100 (to move the decimal point two places to the right), we get:

\[
100x = 27.27272727\ldots
\]

Now, if we subtract the original equation from this new equation, we can eliminate the repeating part:

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

This simplifies to:

\[
99x = 27
\]

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

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

Next, we simplify the fraction \(\frac{27}{99}\). Both the numerator and the denominator can be divided by 9:

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

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

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

The correct answer is \( \frac{3}{11} \).