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].
