If a + b = 2c for distinct real numbers a, b and c, then what is the value of the expression a / (a − c) + c / (b − c)?

Introduction / Context:
This question tests algebraic manipulation and substitution using a linear relation between three variables. It shows how expressions can simplify dramatically when the given condition is used correctly. Such problems are typical in algebraic simplification sections of quantitative aptitude tests.

Given Data / Assumptions:

a + b = 2c.
a, b, and c are distinct real numbers, so denominators are non zero in the given expression.
The required value is a / (a − c) + c / (b − c).

Concept / Approach:
The key idea is to express one variable in terms of the others using the relation a + b = 2c. It is convenient to write b in terms of a and c. Once b is written as 2c − a, we substitute this into the expression and simplify. Because the structure of the expression is symmetric in a and c in a certain way, terms nicely cancel to give a constant result independent of the particular values of a, b, and c.

Step-by-Step Solution:
From a + b = 2c, express b as b = 2c − a.Now consider the expression a / (a − c) + c / (b − c).Substitute b = 2c − a into the second fraction: b − c = (2c − a) − c = c − a.Thus the expression becomes a / (a − c) + c / (c − a).Note that c / (c − a) = c / (−(a − c)) = −c / (a − c).So the expression simplifies to a / (a − c) − c / (a − c).Combine over a common denominator (a − c): (a − c) / (a − c) = 1.Therefore a / (a − c) + c / (b − c) = 1.

Verification / Alternative check:
We can verify this with specific numbers that satisfy a + b = 2c. For example, take c = 3 and a = 1, then b = 2c − a = 6 − 1 = 5. Compute the expression: a / (a − c) = 1 / (1 − 3) = 1 / (−2) = −1/2, and c / (b − c) = 3 / (5 − 3) = 3/2. Their sum is −1/2 + 3/2 = 1, confirming the algebraic simplification.

Why Other Options Are Wrong:
The values −1, 0, 1/2, and 2 would require the combined numerator after simplification to be a different multiple of (a − c). However, as shown, the expression simplifies exactly to (a − c) / (a − c), which is 1 for all allowed values of a and c. Therefore only the option 1 is consistent with the identity for all valid triples (a, b, c) satisfying a + b = 2c.

Common Pitfalls:
Learners sometimes substitute incorrectly for b or mishandle the sign when changing c / (c − a) to an equivalent form. Forgetting that c − a is the negative of a − c can easily lead to incorrect cancellation. Always check denominators carefully and keep track of signs when factoring out a minus sign.

Final Answer:
The value of a / (a − c) + c / (b − c) is the constant 1.