The number 0.121212.... In the from `(p)/(q)` is equal to

Options

A. `(4)/(11)`
B. `(2)/(11)`
C. `(4)/(33)`
D. `(2)/(33)`

Answer

To convert the repeating decimal \(0.121212...\) into a fraction \(\frac{p}{q}\), we can follow these steps:

### Step 1: Let \(x\) be the repeating decimal.
Let \(x = 0.121212...\)

### Step 2: Multiply \(x\) by a power of 10.
Since the repeating part "12" has 2 digits, we multiply \(x\) by \(100\) to shift the decimal point two places to the right:
\[ 100x = 12.121212...\]

### Step 3: Set up an equation.
Now we have two equations:
1. \(x = 0.121212...\)
2. \(100x = 12.121212...\)

### Step 4: Subtract the first equation from the second.
Subtract the first equation from the second to eliminate the repeating part:
\[ 100x - x = 12.121212... - 0.121212...\]
This simplifies to:
\[ 99x = 12\]

### Step 5: Solve for \(x\).
Now, solve for \(x\) by dividing both sides by \(99\):
\[ x = \frac{12}{99}\]

### Step 6: Simplify the fraction.
Now, we simplify \(\frac{12}{99}\):
Both \(12\) and \(99\) can be divided by \(3\):
\[ \frac{12 \div 3}{99 \div 3} = \frac{4}{33}\]

### Final Answer:
Thus, the repeating decimal \(0.121212...\) can be expressed as:
\[ \frac{4}{33}\]

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