To convert the repeating decimal \( 0.27\overline{27} \) to a rational number, we can use a mathematical approach.
Let's denote \( x = 0.27\overline{27} \).
This means: \[ x = 0.27272727\ldots \]
To eliminate the repeating part, we can multiply \( x \) by 100 (since the repeating portion is two digits long): \[ 100x = 27.272727\ldots \]
Now, we can set up an equation to eliminate the repeating decimal: \[ 100x = 27.272727\ldots \quad (1) \] \[ x = 0.272727\ldots \quad (2) \]
Next, we subtract equation (2) from equation (1): \[ 100x - x = 27.272727\ldots - 0.272727\ldots \] \[ 99x = 27 \]
Now, we can solve for \( x \): \[ x = \frac{27}{99} \]
We can simplify \( \frac{27}{99} \) by dividing both the numerator and the denominator by their greatest common divisor, which is 9: \[ x = \frac{27 \div 9}{99 \div 9} = \frac{3}{11} \]
Thus, the rational number in simplest form is: \[ \frac{3}{11} \]
So the correct response is: 3/11.