Workers A, B and C together can finish a task in 7.5 days. Worker C is three times as productive as worker A, and worker B alone can finish the same task in 15 days. If B goes on leave, in how many days will workers A and C together complete the task?

Introduction / Context:
This question blends group work rates with a relationship between individual efficiencies. We know the time taken by all three workers together, we know C is three times as productive as A, and we know B's solo completion time. The objective is to deduce the individual rates, then find how long A and C together will take to complete the task once B is unavailable.

Given Data / Assumptions:

• A, B and C together finish the task in 7.5 days.
• C is three times as productive as A.
• B alone finishes the task in 15 days.
• All workers have constant rates of work.
• We require the time for A and C together to finish the task.

Concept / Approach:
We convert all times into rates. B's rate is straightforward from the solo time. The combined rate of A, B and C is the reciprocal of 7.5 days. Using the relation C = 3A, we set up an equation for total rate in terms of A and B. By solving that equation we find A's rate, then C's rate from the productivity relation. Finally, we add A's and C's rates to get the combined rate and invert it to find the time required.

Step-by-Step Solution:
Let total work = 1 unit.Time for A, B and C together = 7.5 days = 15 / 2 days.So combined rate rABC = 1 / (15 / 2) = 2 / 15 per day.B alone finishes in 15 days, so B's rate rB = 1 / 15 per day.Let A's rate be a, so C's rate is 3a, because C is three times as productive as A.Then rABC = a + rB + 3a = 4a + 1 / 15.Set this equal to 2 / 15: 4a + 1 / 15 = 2 / 15.So 4a = 2 / 15 - 1 / 15 = 1 / 15, which gives a = 1 / 60.Therefore, A's rate is 1 / 60 and C's rate is 3 * 1 / 60 = 1 / 20.Combined rate of A and C together = 1 / 60 + 1 / 20 = 1 / 60 + 3 / 60 = 4 / 60 = 1 / 15.Thus A and C together take 1 / (1 / 15) = 15 days to complete the task.

Verification / Alternative check:
Check consistency: If A's rate is 1 / 60 and C's is 1 / 20, their total with B is 1 / 60 + 1 / 15 + 1 / 20. Convert to a common denominator 60: 1 / 60 + 4 / 60 + 3 / 60 = 8 / 60 = 2 / 15, which matches the combined 7.5 day time. Thus the derived rates are correct, and the resulting A + C time of 15 days is consistent with all given information.

Why Other Options Are Wrong:
Ten or twenty days would imply different combined rates for A and C that would not be compatible with the known A + B + C rate and B's solo time. Thirty days is too long and would correspond to a combined rate of 1 / 30, far smaller than the rates implied by the conditions. Only 15 days matches the precise arithmetic and relationships between the workers.

Common Pitfalls:
A common mistake is to misread C is three times as productive as A and treat it as C taking three times as long instead of having three times the rate. Another common error is to incorrectly subtract B's rate from the total combined rate. Always work with rates, not times, when combining groups of workers, and convert all quantities to consistent fractions before solving.

Final Answer:
If B goes on leave, workers A and C together will complete the task in 15 days.