`1.bar27` in the form `(q)/(p)` 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 a fraction in the form \( \frac{q}{p} \), we can follow these steps:

### Step 1: Let \( x = 1.\overline{27} \)
We start by defining our repeating decimal as a variable:
\[ x = 1.27272727\ldots \]

### Step 2: Multiply by 100 to shift the decimal point
Since the repeating part "27" has two digits, we multiply \( x \) by 100:
\[ 100x = 127.27272727\ldots \]

### Step 3: Set up the equation
Now, we have two equations:
1. \( x = 1.27272727\ldots \)
2. \( 100x = 127.27272727\ldots \)

### Step 4: Subtract the first equation from the second
Subtract the first equation from the second to eliminate the repeating part:
\[ 100x - x = 127.27272727\ldots - 1.27272727\ldots \]
This simplifies to:
\[ 99x = 126 \]

### Step 5: Solve for \( x \)
Now, divide both sides by 99 to solve for \( x \):
\[ x = \frac{126}{99} \]

### Step 6: Simplify the fraction
To simplify \( \frac{126}{99} \), we find the greatest common divisor (GCD) of 126 and 99. The GCD is 9.
So, we divide both the numerator and the denominator by 9:
\[ x = \frac{126 \div 9}{99 \div 9} = \frac{14}{11} \]

### Final Answer
Thus, the repeating decimal \( 1.\overline{27} \) in the form \( \frac{q}{p} \) is:
\[ \frac{14}{11} \]

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