Convert 0.27¯¯¯¯¯

Question

to a rational number in simplest form.(1 point)
Responses

2799
Start Fraction 27 over 99 end fraction

27100
Start Fraction 27 over 100 end fraction

127
Start Fraction 1 over 27 end fraction

311
Start Fraction 3 over 11 end fraction

Answer 1

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

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

1. Identify the repeating part: The decimal repeats every two digits ("27"). So, we can express \(x\) in terms of its repeating part.

2. Multiply by a power of 10: To eliminate the repeating part, we can multiply \(x\) by \(100\) (because the repeat is 2 digits). \[ 100x = 27.\overline{27} \]

3. Set up the equation: Now we also write the original \(x\) aligned with the new equation: \[ 100x = 27.\overline{27} \] \[ x = 0.27\overline{27} \]

4. Subtract the two equations: \[ 100x - x = 27.\overline{27} - 0.27\overline{27} \] \[ 99x = 27 \]

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

6. Simplify the fraction: The greatest common divisor of 27 and 99 is 9. \[ x = \frac{27 \div 9}{99 \div 9} = \frac{3}{11} \]

Thus, the repeating decimal \(0.27\overline{27}\) can be expressed as the rational number

\(\frac{3}{11}\).

Therefore, the answer is

\(\boxed{\frac{3}{11}}\).