Three real numbers a, b and c satisfy a + b = 5, b + c = 7.5 and c + a = 8.5. What is the average value of a, b and c?

Introduction / Context:
This question deals with three unknown numbers a, b and c where only pairwise sums are given. Instead of individual values, we know a + b, b + c and c + a. The task is to find the average of a, b and c. This is a good example of how to use symmetry and simple algebra to work with partial information about several variables.

Given Data / Assumptions:

• a + b = 5.
• b + c = 7.5.
• c + a = 8.5.
• We are asked to find the average of a, b and c, that is (a + b + c) / 3.

Concept / Approach:
The key idea is to add the three pairwise equations. When we add (a + b), (b + c) and (c + a), each variable a, b and c appears exactly twice. This means the sum of the three equations equals 2(a + b + c). From that, we can solve for a + b + c directly, without needing each variable separately. Finally, the average is that total divided by 3. This approach is efficient and avoids unnecessary algebra.

Step-by-Step Solution:
Step 1: Write out the three given equations clearly. a + b = 5. b + c = 7.5. c + a = 8.5. Step 2: Add all three equations. (a + b) + (b + c) + (c + a) = 5 + 7.5 + 8.5. Left side becomes 2a + 2b + 2c = 2(a + b + c). Right side is 5 + 7.5 + 8.5 = 21. So 2(a + b + c) = 21. Step 3: Solve for a + b + c. a + b + c = 21 / 2 = 10.5. Step 4: Compute the average. Average of a, b and c = (a + b + c) / 3 = 10.5 / 3. 10.5 / 3 = 3.5.
Verification / Alternative check:
We could find individual values if needed. From a + b = 5 and a + b + c = 10.5, we get c = 10.5 − 5 = 5.5. From b + c = 7.5, b = 7.5 − 5.5 = 2. From a + b = 5, a = 5 − 2 = 3. Check: c + a = 5.5 + 3 = 8.5, consistent with the data. Average of 3, 2 and 5.5 is (3 + 2 + 5.5) / 3 = 10.5 / 3 = 3.5.
Why Other Options Are Wrong:
Values 1.5, 3, 2.5 and 4.5 do not match the exact total 10.5 when multiplied by 3. Any other average would imply a different value for a + b + c and would break at least one of the pairwise equations.
Common Pitfalls:
Some learners attempt to solve for a, b and c individually before finding the average, which is longer and more error prone. Another typical mistake is adding the right hand sides incorrectly or forgetting that the left side becomes 2(a + b + c) rather than a + b + c.
Final Answer:
The average of a, b and c is 3.5.