Express the mixed recurring decmal `1.bar(27)` in the form`p/q`

Options

A. `8/11`
B. `14/11`
C. `14/25`
D. `8/17`

Answer

To express the mixed recurring decimal \( 1.\overline{27} \) in the form \( \frac{p}{q} \), we can follow these steps:

### Step 1: Define the recurring decimal
Let \( x = 1.\overline{27} \). This means:
\[ x = 1.27272727\ldots \]

### Step 2: Eliminate the decimal
To eliminate the decimal, multiply both sides of the equation by 100 (since the recurring part has two digits):
\[ 100x = 127.27272727\ldots \]

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

### Step 4: Subtract the first equation from the second
Now, subtract Equation 1 from Equation 2:
\[ 100x - x = 127.27272727\ldots - 1.27272727\ldots \]
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
Now we need to simplify \( \frac{126}{99} \). We can find the greatest common divisor (GCD) of 126 and 99. The GCD is 9.
\[ \frac{126 \div 9}{99 \div 9} = \frac{14}{11} \]

### Final Answer
Thus, the mixed recurring decimal \( 1.\overline{27} \) can be expressed in the form \( \frac{p}{q} \) as:
\[ \frac{14}{11} \]

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