Convert the repeating decimal 0.252525… into a fraction in lowest terms Identify the correct simplified fraction for the recurring block 25.

Introduction / Context:
Purely recurring decimals with a block of r digits convert to fractions whose denominator is r copies of 9. For 0.252525…, the repeating block is “25” with length r = 2.

Given Data / Assumptions:

• Number: x = 0.(25) = 0.252525…
• Repeating length r = 2.
• We want the simplified fraction.

Concept / Approach:
For x = 0.(AB) where AB is two digits, x = AB/99. Here AB = 25, hence x = 25/99 directly. This is a standard result from shifting the decimal by powers of 10 and subtracting to eliminate the repeat.

Step-by-Step Solution:
Let x = 0.252525…100x = 25.252525…Subtract: 100x − x = 25.2525… − 0.2525… = 25So 99x = 25 ⇒ x = 25/99.

Verification / Alternative check:
Compute 25 ÷ 99 ≈ 0.252525…, confirming the correct repeating block length and digits.

Why Other Options Are Wrong:
25/90 and 25/999, 25/9999 use incorrect denominators for a two-digit repeating block; 5/18 = 0.277… represents a single-digit repeating 7, not “25”.

Common Pitfalls:
Using 9, 99, 999 indiscriminately without matching the block length; forgetting that the decimal is purely repeating with no non-repeating prefix.

Final Answer:
25/99