To convert \( 0.27\overline{27} \) (where the "27" is the repeating part) to a rational number, you can use the following method:
Let \( x = 0.27\overline{27} \).
This can be expressed as: \[ x = 0.272727... \]
To eliminate the repeating decimal, multiply both sides by 100 (since the repeating part has two digits): \[ 100x = 27.272727... \]
Now, we have two equations:
- \( x = 0.272727... \)
- \( 100x = 27.272727... \)
Next, subtract the first equation from the second: \[ 100x - x = 27.272727... - 0.272727... \] \[ 99x = 27 \]
Now, solve for \( x \): \[ x = \frac{27}{99} \]
We can simplify this fraction: \[ \frac{27 \div 9}{99 \div 9} = \frac{3}{11} \]
Thus, \( 0.27\overline{27} \) as a rational number in simplest form is: \[ \frac{3}{11} \]
So the correct response is: Start Fraction 3 over 11 end fraction.