Ordering rational numbers: Out of 4/7, 5/13, 6/11, 3/5, and 2/3, which is the second smallest fraction?

Introduction / Context:
Comparing fractions efficiently is essential. Converting all to a common denominator is possible but slow; instead compare via decimal values or cross multiplication.

Given Data / Assumptions:

• Fractions: 4/7, 5/13, 6/11, 3/5, 2/3.
• Goal: identify the second smallest.

Concept / Approach:
Estimate each fraction as a decimal or compare pairs using cross multiplication. Sort the approximate values.

Step-by-Step Solution:

Verification / Alternative check:
Use cross multiplication to compare neighbors: for example, compare 6/11 and 4/7 by 6*7 = 42 versus 4*11 = 44, so 6/11 < 4/7, matching the order above.

Why Other Options Are Wrong:
5/13 is the smallest, not second smallest. 4/7 and 3/5 are larger than 6/11. 2/3 is the largest here.

Common Pitfalls:
Rounding too coarsely or misordering fractions that are close in value, especially 6/11 and 4/7.

Final Answer:
6/11