Question

Convert 0.27 repeated to a rational number in simplest form

Answer 1

To convert the repeating decimal \( 0.27\overline{27} \) to a rational number, we can follow these steps:

1. Let \( x = 0.272727...\).

2. Multiply both sides of the equation by 100, since the repeating part has two digits:
\[
100x = 27.272727...
\]

3. Now, we have two equations:
- \( x = 0.272727... \)
- \( 100x = 27.272727... \)

4. Subtract the first equation from the second:
\[
100x - x = 27.272727... - 0.272727...
\]
This simplifies to:
\[
99x = 27
\]

5. Solve for \( x \):
\[
x = \frac{27}{99}
\]

6. Now simplify the fraction \( \frac{27}{99} \). The greatest common divisor (GCD) of 27 and 99 is 9. Dividing the numerator and the denominator by 9, we get:
\[
x = \frac{27 \div 9}{99 \div 9} = \frac{3}{11}
\]

Therefore, the repeating decimal \( 0.27\overline{27} \) as a rational number in simplest form is:
\[
\boxed{\frac{3}{11}}
\]