A can complete a piece of work in 10 days and B can complete the same work in 12 days. They work together for 3 days, after which B leaves and A continues alone for 2 more days. Two days later, C joins A and together they finish the remaining work in 2 additional days. If C works alone from start to finish, in how many days would C be able to complete the entire work?

Introduction / Context:

This time and work question tests how to convert different time durations into work rates, add and subtract those rates in stages, and then back calculate the individual capacity of one worker. Such questions are common in quantitative aptitude tests and competitive exams because they combine ratio, fraction, and basic algebra concepts in a single scenario.

Given Data / Assumptions:

• A alone finishes the work in 10 days.
• B alone finishes the same work in 12 days.
• A and B work together for 3 days.
• After that B stops, and A works alone for 2 more days.
• Then A and C work together for 2 days and complete the remaining work.
• Work is assumed to be uniform and divisible, and total work is taken as one unit.

Concept / Approach:

The main idea is to treat the whole work as one unit and express daily work done by each person as a fraction of this unit. Rates add when people work together. We compute the total work already done by A and B, then by A alone, and finally use the last phase with A and C together to deduce C's daily work rate. Once C's rate is known, its reciprocal gives the total number of days C alone would take to finish the entire work.

Step-by-Step Solution:

Verification / Alternative Check:

We can check by reconstructing the entire sequence. In 3 days A and B complete 11/20 of the work, then A adds 1/5 more, for a total of 11/20 + 4/20 = 15/20 = 3/4. The remaining 1/4 is exactly completed by two days of work at rate 1/8 per day. Substituting C's rate of 1/40 into 1/10 + 1/40 gives 1/8, which matches the combined rate used, so the calculation is consistent.

Why Other Options Are Wrong:

30 days or 50 days would correspond to C being faster or slower than 40 days, but they do not satisfy the equation for the combined rate in the last phase. Similarly, 60 days gives an even smaller rate for C and makes the total work done by A and C in the final 2 days less than 1/4, so the work would not be completed on time.

Common Pitfalls:

A common mistake is to treat days as simple numbers and not as reciprocals of work rates, or to forget that A continues working during the last phase together with C. Another frequent error is to compute C's time from only the final 2 days without correctly subtracting the earlier completed portions from the total work.

Final Answer:

The worker C would take 40 days to complete the entire work if C works alone from start to finish.