Difficulty: Medium
Correct Answer: 3.5
Explanation:
Introduction / Context:
This algebra question uses pairwise sums of three numbers to find their average. Rather than solving individually for a, b and c, you can use symmetry and summation to find the total directly and then compute the average.
Given Data / Assumptions:
- a + b = 5. - b + c = 7.5. - c + a = 8.5. - All three are real numbers. - We must find (a + b + c) / 3.
Concept / Approach:
By adding the three given equations, each variable a, b and c appears exactly twice in the total sum. Thus the left side becomes 2(a + b + c), and the right side becomes the sum of the three pairwise sums. This allows us to find a + b + c directly without solving for each variable separately. Dividing by 3 gives the required average.
Step-by-Step Solution:
Step 1: Add the three equations: (a + b) + (b + c) + (c + a). Step 2: Left side simplifies to 2a + 2b + 2c = 2(a + b + c). Step 3: Right side equals 5 + 7.5 + 8.5. Step 4: Compute right side: 5 + 7.5 = 12.5, 12.5 + 8.5 = 21. Step 5: So 2(a + b + c) = 21. Step 6: Therefore a + b + c = 21 / 2 = 10.5. Step 7: Average of a, b and c = (a + b + c) / 3 = 10.5 / 3 = 3.5.
Verification / Alternative check:
Optionally, you can solve for individual values. From a + b = 5 and a + c = 8.5, subtract to get (a + c) - (a + b) = 3.5, so c - b = 3.5. Combine with b + c = 7.5 to solve and you will again obtain a, b, c that sum to 10.5. This confirms that the average 3.5 is consistent.
Why Other Options Are Wrong:
- 1.5, 2.5, 3 and 4.5 do not satisfy the derived condition that a + b + c = 10.5. - Any average other than 3.5 would give a different total and violate one or more of the given equations.
Common Pitfalls:
Some learners attempt to guess values or solve three equations for three unknowns in a longer way. Others misadd the constants on the right side. Recognizing the symmetry and adding equations is the fastest and cleanest approach here.
Final Answer:
The average of a, b and c is 3.5.
Discussion & Comments