Convert the recurring decimal 0.125125… (repeating block '125') into a fraction in lowest terms.

Problem restatementExpress the recurring decimal 0.125125… (where 125 repeats endlessly) as a rational fraction in simplest form.

Given data

• Decimal: 0.125125125…
• Repeating block length = 3 (the digits '125')

Concept/ApproachFor a repeating block of length 3, multiply by 10³ to align repeats, subtract to eliminate the repeating part, then solve for the original number.

Step-by-step calculationLet x = 0.125125125…1000x = 125.125125…Subtract: 1000x − x = 125.125125… − 0.125125… = 125999x = 125 ⇒ x = 125/999

Verification/AlternativeSince 999 = 3³ × 37 and 125 = 5³, there are no common prime factors, so the fraction is already in lowest terms.

Common pitfallsUsing denominator 990 instead of 999 (990 applies when there are two repeating digits).

Final Answer125/999