A piece of work is completed by three workers A, B, and C together. A and B together complete 60% of the work, while B and C together complete 70% of the work. Based on this information, who among A, B, and C is the most efficient worker (that is, who completes the greatest share of work)?

Introduction / Context:
This conceptual question focuses on comparing the efficiencies of three workers A, B, and C based on how much work various pairs of them complete together. Instead of asking for the time to complete the work, it asks which worker is the most efficient, meaning who contributes the largest share to the total work. The problem tests reasoning with percentages and systems of equations for work contributions.

Given Data / Assumptions:

• A, B, and C together complete 100% (the whole work).
• A and B together complete 60% of the work.
• B and C together complete 70% of the work.
• Work rates of A, B, and C are constant.
• Percentages refer to the same total job.

Concept / Approach:
We interpret the percentages as fractions of the total work. Let the total work be 1 unit and let a, b, and c represent the shares of work done by A, B, and C respectively when all three work together. The total contribution satisfies a + b + c = 1. The given percentages indicate that a + b = 0.6 and b + c = 0.7. Solving this system of equations allows us to find relative contributions of A, B, and C and then identify the largest contribution and thus the most efficient worker.

Step-by-Step Solution:
Step 1: Let the total work be 1 unit. Step 2: Let a, b, and c be the parts of the work that A, B, and C effectively contribute when the whole job is done. Step 3: According to the problem, A and B together complete 60% of the work, so a + b = 0.6. Step 4: Similarly, B and C together complete 70% of the work, so b + c = 0.7. Step 5: Since the entire work is finished by A, B, and C together, we have a + b + c = 1. Step 6: From a + b + c = 1 and a + b = 0.6, subtract the second equation from the first to get c = 1 - 0.6 = 0.4. Step 7: From a + b + c = 1 and b + c = 0.7, subtract the second equation to get a = 1 - 0.7 = 0.3. Step 8: Now use a + b = 0.6 to find b = 0.6 - a = 0.6 - 0.3 = 0.3. Step 9: Therefore, the contributions are a = 0.3, b = 0.3, and c = 0.4. Step 10: Since c = 0.4 is the largest share, C is the most efficient worker.

Verification / Alternative check:
We can verify consistency. A and B together contribute 0.3 + 0.3 = 0.6, which matches the 60% given. B and C together contribute 0.3 + 0.4 = 0.7, consistent with 70%. The total contribution a + b + c = 0.3 + 0.3 + 0.4 = 1, confirming that the entire work is accounted for. Therefore, our solution is correct and internally consistent.

Why Other Options Are Wrong:

• A: A contributes 0.3 of the work, which is less than C's 0.4.
• B: B also contributes 0.3, again less than C's contribution.
• None: This would be correct only if there were a tie or if the data were inconsistent, but here C clearly does the greatest share of work.

Common Pitfalls:
Some students may attempt to divide the percentages directly among workers without setting up equations, leading to confusion. Others may incorrectly assume that if two pairs involving B show high percentages, then B must be the most efficient, ignoring the actual algebraic relationships. Carefully forming and solving the equations a + b = 0.6, b + c = 0.7, and a + b + c = 1 is the best way to avoid mistakes.

Final Answer:
Among A, B, and C, the most efficient worker is C.