Workers A and B together can finish a job in 24 days. Workers A, B and C together can finish the same job in 8 days. In how many days will worker C alone finish the job working at a constant rate?

Introduction / Context:
This is a standard type of time and work question where various groups of workers complete a job in different times. We are given the time taken by A and B together, and by A, B and C together. The goal is to determine how long C alone would take to finish the same job, which we find by subtracting known combined rates.

Given Data / Assumptions:

• A and B together finish the job in 24 days.
• A, B and C together finish the job in 8 days.
• All workers have constant individual rates of work.
• The job size is fixed and taken as one unit.

Concept / Approach:
We convert completion times to daily work rates. The rate of A and B together gives us one combined rate, and the rate of A, B and C together gives a larger combined rate. Subtracting the first from the second yields C's solo rate. Finally, we invert C's rate to obtain the number of days C alone needs to completed the job.

Step-by-Step Solution:
Let total work = 1 unit.A + B together finish in 24 days, so their combined rate rAB = 1 / 24 per day.A + B + C together finish in 8 days, so rABC = 1 / 8 per day.C's rate rC = rABC - rAB = 1 / 8 - 1 / 24.Compute: 1 / 8 = 3 / 24, so rC = 3 / 24 - 1 / 24 = 2 / 24 = 1 / 12 per day.Time taken by C alone = 1 / (1 / 12) = 12 days.

Verification / Alternative check:
We can check by recomputing group times from rates. If C's rate is 1 / 12, then A + B rate is 1 / 24. Combined A + B + C rate is 1 / 24 + 1 / 12 = 1 / 24 + 2 / 24 = 3 / 24 = 1 / 8, which matches the given 8 days for all three together. The relationships are consistent, confirming that 12 days is correct for C alone.

Why Other Options Are Wrong:
Fourteen, sixteen, and twenty four days correspond to smaller daily rates than 1 / 12. If C needed any of those longer times alone, then the combined rate A + B + C would be smaller than 1 / 8 and would not achieve completion in 8 days. Twelve days is the only duration that yields the correct rate differences.

Common Pitfalls:
Some learners incorrectly average the times instead of working with rates. Others may forget that rates add linearly while times do not. The correct process is always to convert times to rates, add or subtract the rates as appropriate, and then convert back to time at the end.

Final Answer:
Worker C alone will finish the job in 12 days.