Order and select: third-highest value among 5/7, 7/13, 4/7, 4/15, and 9/14 Compare these fractions without a calculator and identify which one is the third largest.

Introduction / Context:
Ranking fractions efficiently is a core aptitude skill. Instead of converting to long decimals, use approximate values or cross-multiplication to compare pairs. Here we must find the third-highest value among five given fractions.

Given Data / Assumptions:

• Fractions: 5/7, 7/13, 4/7, 4/15, 9/14.
• We require only their relative order.
• No calculators mandated; exact ordering is possible mentally.

Concept / Approach:
Approximate each fraction with a simple decimal to see the order quickly. Alternatively, compare two fractions a/b and c/d by cross-multiplying: a*d vs. c*b. Repeat to build a sorted list.

Step-by-Step Solution:
Estimate each: 5/7 ≈ 0.7143; 9/14 ≈ 0.6429; 4/7 ≈ 0.5714; 7/13 ≈ 0.5385; 4/15 ≈ 0.2667.Descending order: 5/7 (1st), 9/14 (2nd), 4/7 (3rd), 7/13 (4th), 4/15 (5th).Therefore, the third-highest fraction is 4/7.

Verification / Alternative check:
Use pairwise cross-multiplication to confirm: compare 4/7 vs. 7/13 ⇒ 4*13 = 52 vs. 7*7 = 49, so 4/7 > 7/13; compare 4/7 vs. 9/14 ⇒ 4*14 = 56 vs. 9*7 = 63, so 4/7 < 9/14. The position between them is consistent with third place.

Why Other Options Are Wrong:
5/7 is the largest; 9/14 is second; 7/13 and 4/15 are smaller than 4/7. Hence only 4/7 fits the third-highest rank.

Common Pitfalls:
Assuming larger numerators imply larger values without considering denominators; misestimating 7/13 as more than 4/7 due to 7 > 4.

Final Answer:
4/7