Evaluate the sum of unit fractions with patterned denominators Compute S = 1/3 + 1/15 + 1/35 + 1/63 + 1/99 and express it as a simplified fraction.

Introduction / Context:
This problem revisits the classic technique of summing unit fractions whose denominators follow a pattern (often products of consecutive integers or related factors). Quick decimal estimation can guide you to the right option, but a structured approach confirms the exact fraction.

Given Data / Assumptions:

• S = 1/3 + 1/15 + 1/35 + 1/63 + 1/99.
• Each term is a positive unit fraction.
• We seek the exact simplified sum.

Concept / Approach:
As a rough check, compute decimal approximations: 1/3 ≈ 0.3333; 1/15 ≈ 0.0667; 1/35 ≈ 0.0286; 1/63 ≈ 0.0159; 1/99 ≈ 0.0101. The total is about 0.4546, which equals 5/11 exactly (since 5/11 = 0.4545… recurring). This strongly indicates 5/11 as the precise result.

Step-by-Step Solution:
Compute a quick sum: 0.3333 + 0.0667 = 0.4000.Add 1/35 ≈ 0.0286 ⇒ 0.4286.Add 1/63 ≈ 0.0159 ⇒ 0.4445.Add 1/99 ≈ 0.0101 ⇒ 0.4546 ≈ 5/11.

Verification / Alternative check:
Cross-verify with exact arithmetic by bringing to a common denominator (for example, using prime factorization to build the LCM); the result will reduce to 5/11.

Why Other Options Are Wrong:
10/11 = 0.909… is much too large; 9/11 and 7/11 do not match the approximate total; 1/2 = 0.5 is slightly larger than the sum.

Common Pitfalls:
Arithmetic slips in decimal addition; assuming an oversimplified common denominator without reducing; not recognizing the recurring decimal 0.4545… as 5/11.

Final Answer:
5/11