Answer :

To convert the infinite decimal expansion 0.313131... to a fraction, follow these steps:

1. Identify the repeating decimal: Recognize that 0.313131... is a repeating decimal where the block "31" repeats indefinitely.

2. Set up the equation: Let [tex]\( x \)[/tex] represent the repeating decimal.
[tex]\[ x = 0.313131... \][/tex]

3. Shift the decimal point: To move the repeating block to the left of the decimal point, multiply [tex]\( x \)[/tex] by 100 (since the repeating block "31" has 2 digits).
[tex]\[ 100x = 31.313131... \][/tex]

4. Subtract the original [tex]\( x \)[/tex] from this equation: This removes the repeating part after the decimal point.
[tex]\[ 100x - x = 31.313131... - 0.313131... \][/tex]

5. Simplify the equation: By subtracting, the repeating parts cancel out.
[tex]\[ 99x = 31 \][/tex]

6. Solve for [tex]\( x \)[/tex]: Isolate [tex]\( x \)[/tex] by dividing both sides of the equation by 99.
[tex]\[ x = \frac{31}{99} \][/tex]

Therefore, the infinite decimal expansion 0.313131... can be written as the fraction [tex]\(\frac{31}{99}\)[/tex].

No further simplification is needed, so the final solution is [tex]\(\frac{31}{99}\)[/tex].