If A : B = 2 : 5, B : C = 4 : 3 and C : D = 2 : 1, then what is the combined ratio A : C : D?

Introduction / Context:
This problem links several pairwise ratios A : B, B : C and C : D and asks for a combined three-term ratio A : C : D. Such questions are used in aptitude tests to check whether the student can systematically combine multiple ratios and express all quantities in terms of a single scale, then simplify the final ratio correctly.

Given Data / Assumptions:

• A : B = 2 : 5.
• B : C = 4 : 3.
• C : D = 2 : 1.
• We need to find A : C : D in simplest integer form.

Concept / Approach:
The strategy is to express A, B, C and D in terms of a single parameter by successively using each ratio. We start from A : B, then use B : C to connect C, and finally use C : D to include D. After writing all variables in terms of a single constant, we can read off the relative values of A, C and D and simplify them to get the final ratio.

Step-by-Step Solution:

Verification / Alternative check:
We can verify by choosing k = 8 for convenience. Then A = 16, B = 5k = 40, C = 30 and D = 15. Check original ratios: A : B = 16 : 40 = 2 : 5, B : C = 40 : 30 = 4 : 3, and C : D = 30 : 15 = 2 : 1. All conditions are satisfied, confirming that A : C : D = 16 : 30 : 15 is correct.

Why Other Options Are Wrong:

• 6 : 5 : 2, 7 : 20 : 10 and 8 : 30 : 15 do not match the combined relationships when checked against the original ratios.
• For example, if A : C : D were 8 : 30 : 15, then C : D = 30 : 15 = 2 : 1 is correct, but that set does not maintain the correct A : B or B : C ratios when B is reconstructed.

Common Pitfalls:
Some students mistakenly try to directly multiply or add the ratios instead of aligning the common terms. Others forget to maintain a consistent scale when switching from one ratio to another, leading to wrong relative values. The safe method is always to introduce variables, equate the common terms and simplify at the end.

Final Answer:
The combined ratio is 16 : 30 : 15.