Final answer:
Counterexamples to the theorem in question can be constructed by considering non-traditional mathematical systems where operation definitions differ from standard properties, or by considering ratios as symbolic representations that are not numerically comparable.
Step-by-step explanation:
The theorem that if ratios A:B and C:D are equivalent, then they have the same value, assumes that the operations are being performed in a conventional mathematical system where the properties of equality and equivalence hold true. However, to provide a counterexample, we must look at situations where these usual properties may not apply. An instance of this could involve non-traditional systems or contexts where the operations are defined differently.
Consider a mathematical system with different rules where the operation of 'division' is uniquely defined and doesn't follow the standard properties. For instance, we might define a context in which 'division' relates to some form of association or connection that does not equate to our standard understanding. In such a system, even if A:B is equivalent to C:D, they might not hold the same value according to the redefined operations. Therefore, while these situations largely exist in hypothetical or abstract mathematical constructs, they serve as counterexamples to the conventional theorem.
Another possible counterexample might involve a scenario where the ratios A:B and C:D could symbolically represent different concepts or quantities that aren't numerically comparable, even if they are 'equivalent' in some qualitative or contextual sense.