The recurring decimal 0.125125125... has the block 125 repeating without end. What is the rational number (fraction in simplest form) that is exactly equal to this decimal?

Introduction / Context:
This problem again explores the connection between recurring decimals and rational numbers. Here, the repeating block has three digits, so you must be comfortable with the algebraic method for handling longer repeating patterns and converting them into fractions in lowest terms.

Given Data / Assumptions:

• The decimal is 0.125125125..., where the block 125 repeats indefinitely.
• The repetition is continuous with no extra digits breaking the pattern.
• We must find the exact fraction that represents this number.
• The final answer should be in simplest form.

Concept / Approach:
The method is similar to the earlier recurring decimal question but now uses a power of 10 that matches the length of the repeating block, which is three digits. We set the decimal equal to a variable, multiply by 1000 to shift one full block to the left of the decimal point, subtract the original variable, and solve the resulting linear equation. This reliably eliminates the repeating part and yields a clean fraction.

Step-by-Step Solution:
Let x = 0.125125125..., with 125 repeating.Because the repeating block length is three digits, multiply both sides by 1000: 1000x = 125.125125...Now subtract the original x: 1000x - x = 125.125125... - 0.125125...This simplifies to 999x = 125.Solve for x: x = 125 / 999.Check if 125 / 999 is in simplest form. The prime factors of 125 are 5 × 5 × 5, while 999 factors into 3 × 3 × 3 × 37. There is no common factor other than 1.Therefore, 0.125125125... = 125 / 999 in simplest form.

Verification / Alternative check:
To verify, perform long division of 125 by 999. The quotient will be 0.125125125..., with the block 125 repeating indefinitely. Any scientific calculator that can show repeating decimals or enough digits will support this, as the digits will follow the pattern 125 without deviation. This confirms that our algebraic procedure and simplification are correct.

Why Other Options Are Wrong:
Option 125/99 corresponds to a repeating block of two digits, not three, and its decimal expansion does not show the pattern 125. Option 125/9999 involves four nines in the denominator and would produce a different repeating cycle. Option 125/9 gives a very large repeating decimal with a different pattern. Option 25/198 is a reduced fraction with a completely different decimal representation. None of these match the exact pattern of 0.125125125....

Common Pitfalls:
Some learners mistakenly multiply by 10 or 100 instead of 1000, which fails to align the repeating blocks correctly. Others forget that the denominator must be 10^n - 1 for a repeating block of length n, and thus they guess denominators without algebraic justification. Forgetting to check whether the fraction can be simplified is another common oversight. Careful factorization ensures that the final answer is in lowest terms.

Final Answer:
The recurring decimal 0.125125125... is exactly equal to the fraction 125/999 in simplest form.