Combine two ratios that share a middle term: If A : B = 5 : 7 and B : C = 6 : 11, determine the three-term ratio A : B : C by correctly aligning the common term B.

Introduction / Context:
When two ratios share a variable, align the shared term to the same value to combine them into a single three-term ratio. This technique underlies many proportion problems in quantitative aptitude.

Given Data / Assumptions:

• A : B = 5 : 7.
• B : C = 6 : 11.

Concept / Approach:
Let A : B = 5x : 7x and B : C = 6y : 11y. Set 7x = 6y to make B consistent, then compute A and C accordingly and write A : B : C in integer form.

Step-by-Step Solution:

Verification / Alternative check:
Check subratios: A : B = 30 : 42 = 5 : 7; B : C = 42 : 77 = 6 : 11.

Why Other Options Are Wrong:
55 : 77 : 66 does not preserve B : C = 6 : 11; 35 : 49 : 42 mismatches both links; “None of these” is invalid since a correct option exists.

Common Pitfalls:
Adding numerators/denominators directly, or attempting to average ratios. Only matching the shared term yields a correct combination.

Final Answer:
30 : 42 : 77