Final answer:
The original fraction of the repeating decimal 0.148148148... is found using algebra and the concept of a geometric series, resulting in the fraction 148/999.
Step-by-step explanation:
The original fraction of the repeating decimal 0.148148148... can be found by recognizing it as a geometric series and using algebra to solve for the fraction.
Step 1:
Let x = 0.148148148...
Step 2:
Multiply x by 1000 because the repeating part has three digits: 1000x = 148.148148...
Step 3:
Subtract the original x from this result to get: 1000x - x = 148.148148... - 0.148148148...
Step 4:
This simplifies to: 999x = 148, now divide both sides by 999 to find x.
Step 5:
The fraction is x = 148/999.
Therefore, the original fraction of 0.148148148... is 148/999.