The periodic decimal `0.272727... = 0.bar(27)` is the rational number

Options

A. `3/11`
B. `1/7`
C. `2/7`
D. `1/11`

Answer

To convert the periodic decimal \(0.272727...\) (which can be denoted as \(0.\overline{27}\)) into a rational number, we can follow these steps:

### Step 1: Let \(x\) equal the periodic decimal.
Let:
\[ x = 0.272727... \]

### Step 2: Multiply \(x\) by a power of 10 to shift the decimal point.
Since the repeating part "27" has 2 digits, multiply \(x\) by \(100\):
\[ 100x = 27.272727... \]

### Step 3: Set up an equation to eliminate the repeating part.
Now, we have two equations:
1. \(x = 0.272727...\)
2. \(100x = 27.272727...\)

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

### Step 4: Solve for \(x\).
Now, divide both sides by \(99\):
\[ x = \frac{27}{99} \]

### Step 5: Simplify the fraction.
To simplify \(\frac{27}{99}\), we can divide both the numerator and the denominator by their greatest common divisor, which is \(9\):
\[ x = \frac{27 \div 9}{99 \div 9} = \frac{3}{11} \]

### Conclusion
Thus, the periodic decimal \(0.272727...\) can be expressed as the rational number:
\[ \frac{3}{11} \]