To convert the repeating decimal \( 0.27\overline{27} \) into a rational number, we can use the following method:
Let \( x = 0.27\overline{27} \).
This means: \[ x = 0.27272727\ldots \]
To eliminate the repeating part, we can multiply \( x \) by 100 (since the repeating part "27" has 2 digits):
\[ 100x = 27.272727\ldots \]
Now we have two equations:
- \( x = 0.272727\ldots \) (1)
- \( 100x = 27.272727\ldots \) (2)
Next, we can subtract equation (1) from equation (2):
\[ 100x - x = 27.272727\ldots - 0.272727\ldots \]
This simplifies to:
\[ 99x = 27 \]
Now, solve for \( x \):
\[ x = \frac{27}{99} \]
To simplify \( \frac{27}{99} \), we can find the greatest common divisor (GCD) of 27 and 99, which is 9.
Dividing both the numerator and denominator by 9:
\[ \frac{27 \div 9}{99 \div 9} = \frac{3}{11} \]
Therefore, \( 0.27\overline{27} \) as a rational number in simplest form is:
\[ \frac{3}{11} \]
So, the correct response is:
311 Start Fraction 3 over 11 end fraction.