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

Problem restatement
Express 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/Approach
For 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 calculation
Let x = 0.125125125…1000x = 125.125125…Subtract: 1000x − x = 125.125125… − 0.125125… = 125999x = 125 ⇒ x = 125/999

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

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

Final Answer
125/999