What is the rational number (fraction in simplest form) that is equivalent to the recurring decimal 0.125125125... ?

Introduction / Context:
This problem involves converting a recurring decimal with a repeating block of three digits into an equivalent fraction. Understanding this method is useful whenever you see repeating patterns like 0.125125125..., which can be represented exactly as a rational number rather than approximated.

Given Data / Assumptions:

• The decimal given is 0.125125125..., where the block "125" repeats indefinitely.
• We must express this as a fraction in simplest form.
• No rounding is involved, since the repeating pattern is exact.

Concept / Approach:
For a decimal where a block of three digits repeats, we multiply the number by 1000 to shift the decimal three places to the right. Subtracting the original number from this new value cancels the repeating part. This yields a simple linear equation whose solution is the required fraction. Finally, we check whether the fraction can be simplified further.

Step-by-Step Solution:
Step 1: Let x = 0.125125125... Step 2: Multiply both sides by 1000: 1000x = 125.125125... Step 3: Subtract the original x: 1000x - x = 125.125125... - 0.125125... Step 4: The recurring parts cancel, giving 999x = 125. Step 5: Solve for x: x = 125 / 999. Step 6: Check if 125 / 999 can be simplified. The prime factors of 125 are 5 × 5 × 5, and 999 = 3 × 3 × 3 × 37. There is no common factor, so the fraction is already in simplest form.

Verification / Alternative check:
Divide 125 by 999 using long division. You will obtain 0.125125125..., where "125" repeats continuously, confirming that 125 / 999 is the correct rational representation of the decimal.

Why Other Options Are Wrong:
125/99 corresponds to a decimal with a shorter repeating block and a different value.
125/9999 would represent a much smaller decimal since the denominator is larger.
125/9 is approximately 13.888..., clearly not between 0 and 1.
25/198 simplifies to 25 / (2 × 99) = about 0.12626..., which is not equal to the repeating 0.125125125....

Common Pitfalls:
One common mistake is choosing 125/99 by confusing the number of digits in the repeating block. Since "125" has three digits, you must multiply by 1000, not 100, leading to the denominator 999 instead of 99. Forgetting to simplify the fraction when possible is another error, though in this case 125 / 999 is already in lowest terms.

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