If four real numbers A, B, C and D satisfy A + B = 2C and C + D = 2A, then which of the following relationships between these variables is always true?

Introduction / Context:
This question tests basic algebraic manipulation and understanding of how to combine two linear equations in four variables. By carefully substituting expressions from one equation into the other, we can identify which relationship between A, B, C and D must always be true, regardless of the particular values of the variables, as long as the given equations are satisfied.

Given Data / Assumptions:

• A, B, C and D are real numbers.
• A + B = 2C.
• C + D = 2A.
• We assume standard algebraic rules for equality and operations.

Concept / Approach:
The common strategy is to express B and D in terms of A and C and then substitute into each proposed option. If an identity holds for all allowed values of A and C (that is, without imposing any extra conditions such as A = C), then that relationship is always true and is the required answer. If an option is true only for special values, then it is not correct for this reasoning question.

Step-by-Step Solution:
From A + B = 2C, rearrange to B = 2C − A.From C + D = 2A, rearrange to D = 2A − C.Now compute B + D using these expressions: B + D = (2C − A) + (2A − C) = C + A.Observe that A + C on the left side is exactly equal to B + D on the right side, so A + C = B + D is an identity.Check the other options quickly: for example, A + C = 2D would imply A + C = 4A − 2C, which simplifies to A = C, so it holds only in a special case, not always.

Verification / Alternative check:
Choose simple numbers that satisfy the original equations. For example, take C = 0 and A = 0. Then from A + B = 2C we get B = 0, and from C + D = 2A we get D = 0. In this case A + C = 0 and B + D = 0, so A + C = B + D holds.Take another example: let C = 1 and A = 2. Then A + B = 2C implies B = 2*1 − 2 = 0, and C + D = 2A implies D = 2*2 − 1 = 3. Here, A + C = 3 and B + D = 3 again.In both cases A + C = B + D is satisfied, confirming the algebraic proof.

Why Other Options Are Wrong:
A + C = 2D would require A + C = 4A − 2C, which simplifies to A = C. Since A and C are not required to be equal, this is not always true.A + D = B + C becomes 3A − C = 3C − A, which simplifies to A = C again, so it is also only sometimes true.A + C = 2B leads to A + C = 4C − 2A, which also forces A = C, so it fails in general.

Common Pitfalls:
A common mistake is to plug in one convenient set of values, find that more than one option works for that specific case, and then wrongly assume they are always true.Another error is performing algebraic manipulation but not simplifying fully, which can hide the fact that an option requires a restrictive condition like A = C.

Final Answer:
The only relationship that holds for all real values of A, B, C and D satisfying the given equations is A + C = B + D.