Convert 0.27¯¯¯¯¯ to a rational number in simplest form.(1 point

Answer 1

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

Let \( x = 0.27\overline{27} \).

This means that \( x = 0.27272727\ldots \).

Since the decimal repeats every 2 digits, we can multiply \( x \) by 100 to shift the decimal point two places to the right:

\[ 100x = 27.27272727\ldots \]

Now, we have two equations:

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:

\[ 99x = 27 \]

Solving for \( x \):

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

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

Both the numerator and denominator can be divided by 9:

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

Thus, the rational number in simplest form is:

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