To convert the repeating decimal \(3.12\overline{12}\) into a rational number, we can break it down as follows:
Let \(x = 3.121212...\)
Now, to eliminate the repeating part, we can multiply \(x\) by \(100\) (which shifts the decimal two places to the right):
\[ 100x = 312.121212... \]
Next, we can set up an equation by subtracting the original \(x\) from this new equation:
\[ 100x - x = 312.121212... - 3.121212... \] \[ 99x = 312 - 3 \] \[ 99x = 309 \]
Now, we solve for \(x\):
\[ x = \frac{309}{99} \]
To simplify \(\frac{309}{99}\), we find the greatest common divisor (GCD) of 309 and 99. Both numbers can be divided by 3:
\[ 309 \div 3 = 103 \] \[ 99 \div 3 = 33 \]
Thus, the simplified form is:
\[ x = \frac{103}{33} \]
Now, we need to rewrite this in the form of a mixed number. We divide \(103\) by \(33\):
\[ 103 \div 33 = 3 \quad \text{(with a remainder of } 4\text{)} \]
Now we write it as:
\[ \frac{103}{33} = 3 + \frac{4}{33} \]
So, the rational number equivalent to \(3.12\overline{12}\) is:
\[ 3 \frac{4}{33} \]
However, looking at your provided answer choices, the equivalent that fits closest is:
3 and 4 over 33
This matches our calculation.