`1.bar(27)` in the form `(p)/(q)` is equal to

Options

A. `(127)/(100)`
B. `(73)/(100)`
C. `(14)/(11)`
D. `(11)/(14)`

Answer

To convert the repeating decimal \(1.\overline{27}\) into the form \(\frac{p}{q}\), we can follow these steps:

### Step 1: Let \(x = 1.\overline{27}\)
We start by defining \(x\) as the repeating decimal.

\[ x = 1.27272727\ldots \]

### Step 2: Multiply \(x\) by 100
Since the repeating part "27" has two digits, we multiply \(x\) by 100 to shift the decimal point two places to the right.

\[ 100x = 127.27272727\ldots \]

### Step 3: Set up an equation
Now, we can set up an equation by subtracting the original \(x\) from this new equation.

\[ 100x - x = 127.27272727\ldots - 1.27272727\ldots \]

This simplifies to:

\[ 99x = 126 \]

### Step 4: Solve for \(x\)
Now, we can solve for \(x\) by dividing both sides by 99.

\[ x = \frac{126}{99} \]

### Step 5: Simplify the fraction
Next, we simplify the fraction \(\frac{126}{99}\). We can find the greatest common divisor (GCD) of 126 and 99, which is 9.

\[ \frac{126 \div 9}{99 \div 9} = \frac{14}{11} \]

### Final Result
Thus, \(1.\overline{27}\) in the form \(\frac{p}{q}\) is:

\[ \frac{14}{11} \]