Chain of ratios across four variables: Given A : B = 2 : 3, B : C = 4 : 5, and C : D = 6 : 7 for the same set of quantities, determine A : D. Show the alignment of the repeated term to combine the ratios consistently.

Introduction / Context: Chained ratios must be combined by matching the repeated term. Here, B connects A to C, and C connects to D. The task builds ratio-composition skills and careful scaling to get an exact A : D relationship.

Given Data / Assumptions:

• A : B = 2 : 3.
• B : C = 4 : 5.
• C : D = 6 : 7.

Concept / Approach: To combine A : B and B : C, equalize the value of B in both ratios. Then include C : D similarly by equalizing C. Scale to integers to avoid fractions and then read off A : D.

Step-by-Step Solution:

Verification / Alternative check: Confirm intermediate ratios: A : B = 16 : 24 = 2 : 3; B : C = 24 : 30 = 4 : 5; C : D = 30 : 35 = 6 : 7.

Why Other Options Are Wrong: 2 : 7 and 4 : 13 do not preserve the combined scaling; 7 : 8 reverses order; only 16 : 35 satisfies all three conditions.

Common Pitfalls: Adding or multiplying ratios directly without aligning the repeated variable; forgetting to scale consistently when introducing the third ratio.

Final Answer: 16 : 35