When A, B, and C are deployed for a task, A and B together complete 70% of the work, and B and C together complete 50% of the work. Based on this information alone, who is the most efficient worker?

Introduction / Context:
This conceptual time and work question focuses on comparing individual efficiencies using partial information about combined work. We are told how much of the job is completed by certain pairs of workers but are not given direct data about each individual. The aim is to determine who is the most efficient worker based only on the given pairwise completion percentages.

Given Data / Assumptions:
When A and B work together, they complete 70% of the work in the same reference time period. When B and C work together, they complete 50% of the work in that same period. No additional information about individual times or total time is given. We are asked to compare individual efficiencies (rates of work) of A, B, and C and to identify the most efficient worker if possible.

Concept / Approach:
Let the work done in the reference time be 1 unit. The information that A and B do 70% of the work means their combined rate is 0.7 units per reference time. Similarly, B and C together doing 50% of the work means their combined rate is 0.5 units per reference time. We then express these relationships in terms of variables representing individual efficiencies and analyze whether the given system of equations is enough to uniquely rank A, B, and C by efficiency.

Step-by-Step Solution:
Step 1: Let the efficiencies (work rates) of A, B, and C in the given time frame be a, b, and c units of work respectively. Step 2: A and B together complete 70% of the work in that time, so a + b = 0.7. Step 3: B and C together complete 50% of the work in the same time, so b + c = 0.5. Step 4: Subtract the second equation from the first: (a + b) - (b + c) = 0.7 - 0.5. Step 5: This simplifies to a - c = 0.2, so a = c + 0.2. This tells us that A is more efficient than C by 0.2 units per reference time. Step 6: To compare B with A and C, rewrite b from either equation. From a + b = 0.7, b = 0.7 - a. From b + c = 0.5, b = 0.5 - c. Step 7: Substituting a = c + 0.2 into b = 0.7 - a gives b = 0.7 - (c + 0.2) = 0.5 - c. Step 8: This expression for b shows that its value depends on c. Different choices of c consistent with the equations will give different relative positions of B compared to A and C. Step 9: For example, if c is relatively small, B may be more efficient than both A and C. If c is relatively large (but still consistent with the equations), B may be less efficient than one or both of them. Step 10: Because many sets of (a, b, c) satisfy the two equations, we cannot uniquely determine which worker is the most efficient overall.

Verification / Alternative check:
Consider one possible scenario. Let c = 0.1. Then a = c + 0.2 = 0.3. From b + c = 0.5, we get b = 0.4. In this case, b (0.4) is the most efficient, then a (0.3), then c (0.1). Now consider another scenario: let c = 0.24. Then a = 0.44 and b = 0.26. In this scenario, a is the most efficient, b is in the middle, and c is the least efficient. Both sets of values satisfy the original equations, but they produce different conclusions about who is the most efficient worker. This confirms that the data are insufficient to decide uniquely.

Why Other Options Are Wrong:
Options A, B, and C each claim that a specific worker is definitely the most efficient, which is not justified by the given information. We have shown that multiple configurations of efficiencies satisfy the conditions with different workers being the most efficient in different scenarios. The option stating that all are equally efficient contradicts the equation a - c = 0.2, which clearly indicates that A and C do not have the same efficiency. Hence, only the answer that says the result cannot be determined is correct.

Common Pitfalls:
A common mistake is to assume that A must be the most efficient simply because A appears in the larger combined percentage (with B). Another error is to ignore the fact that B appears in both pairs and thus introduce unintended symmetry. Always check whether the number of independent equations matches the number of unknowns; if there are fewer equations, there may be infinitely many solutions and no unique ranking of efficiencies.

Final Answer:
Based on the given information alone, the most efficient worker cannot be determined.