To convert the decimal \( 3.242424... \) into a mixed number, we can first separate the whole number part from the decimal part.
The whole number part is \( 3 \).
Next, we focus on the decimal part \( 0.242424... \). This decimal is repeating, and we can express it as a fraction. We can let \( x = 0.242424... \).
To eliminate the repeating part, we can multiply \( x \) by 100:
\[ 100x = 24.242424... \]
Now, we can set up the equation by subtracting the original \( x \) from this new equation:
\[ 100x - x = 24.242424... - 0.242424... \]
This simplifies to:
\[ 99x = 24 \]
Now, we can solve for \( x \):
\[ x = \frac{24}{99} \]
Next, we can simplify the fraction \( \frac{24}{99} \). The greatest common divisor (GCD) of 24 and 99 is 3, so we divide both the numerator and the denominator by 3:
\[ x = \frac{24 \div 3}{99 \div 3} = \frac{8}{33} \]
Now we can express the entire number. Combining the whole part and the fractional part gives us:
\[ 3 + \frac{8}{33} = \frac{3 \cdot 33 + 8}{33} = \frac{99 + 8}{33} = \frac{107}{33} \]
To write it as a mixed number, we can see that \( 3 + \frac{8}{33} \) is indeed already a mixed number.
So the mixed number equivalent to \( 3.242424... \) is \( 3 \frac{8}{33} \).
Now, looking at the answer choices:
- \( 3833 \)
- \( 31225 \)
- \( 314 \)
- \( 315 \)
None of these seem to correspond directly. The closest answer appears to be \( 3 \frac{1}{4} \) (which is not an available option).
Thus, the equivalent mixed number to \( 3.242424... \) in common representation is not directly in the list, as it would need \( \frac{8}{33} \) represented.
If we needed to see if \( 3.24 \) might round to anything, \( 314 \) could be an approximation (but incorrect with \( 3 \frac{8}{33} \)), and \( 3.25 \) is closer to \( \frac{1}{4} \).
The answer you are looking for remains not explicitly listed but assuredly \( \frac{107}{33} \) or \( 3 \frac{8}{33} \) if asked for a conversion against choices.
So, your answer is that none of the given options accurately conveys \( 3.242424... \), but \textbf{it can be expressed as a mixed number in other forms however it will not directly match what's here}.