A and B can do a piece of work separately in 24 days and 30 days respectively. They work together for 6 days, then B leaves and C joins A. A and C together complete the remaining work in 11 days. In how many days can C alone complete the entire work?

Introduction / Context:
This is a multi-stage time and work problem. First A and B work together for a few days, after which B leaves and C joins A to finish the remaining work. We are asked to find C's individual time to complete the entire work alone. The problem tests your ability to track fractions of work across stages and to use the resulting information to determine an unknown worker's rate.

Given Data / Assumptions:

• A alone can complete the work in 24 days.
• B alone can complete the work in 30 days.
• A and B work together for 6 days.
• Then B leaves and C works with A for 11 more days to finish the work.
• All workers have constant rates during the time they work.
• We must find the number of days C alone would need to complete the entire work.

Concept / Approach:
We take the total work as 1 unit. First we compute the combined rate of A and B, then calculate how much work they complete in 6 days. Subtracting this from 1 gives the remaining work. Next, using the time taken by A and C together to finish this remaining work, we find their combined rate and subtract A's individual rate to derive C's rate. Finally, we take the reciprocal of C's rate to get C's individual completion time.

Step-by-Step Solution:
Step 1: Let total work = 1 unit. Step 2: A's rate = 1/24 of the work per day. Step 3: B's rate = 1/30 of the work per day. Step 4: Combined rate of A and B = 1/24 + 1/30. Step 5: Using common denominator 120, 1/24 = 5/120 and 1/30 = 4/120, so combined rate = 9/120 = 3/40. Step 6: Work done by A and B together in 6 days = 6 × (3/40) = 18/40 = 9/20 of the total work. Step 7: Remaining work after 6 days = 1 − 9/20 = 11/20 of the job. Step 8: A and C together finish this remaining 11/20 of the job in 11 days, so their combined rate = (11/20) / 11 = 1/20 of the work per day. Step 9: A's rate is known as 1/24 per day, so C's rate = (A + C) rate − A's rate = 1/20 − 1/24. Step 10: Using common denominator 120, 1/20 = 6/120 and 1/24 = 5/120, so C's rate = 1/120 of the work per day. Step 11: Time taken by C alone to complete the whole job = 1 / (1/120) = 120 days.

Verification / Alternative check:
We can verify by recomputing the work distribution. In 6 days, A and B do 9/20 of the job. The remaining 11/20 is done in 11 days by A and C at 1/20 per day, giving exactly 11/20 more work. Also, checking rates: A + C = 1/24 + 1/120 = 5/120 + 1/120 = 6/120 = 1/20 as required. All stages match the information given, confirming that C's time of 120 days is correct.

Why Other Options Are Wrong:
80, 90, 100, and 130 days correspond to different daily work rates for C that would disrupt the precise balance of rates and times. They would not reproduce the 11 days required for A and C to complete the remaining 11/20 of the work. Only 120 days gives C a rate that fits all parts of the problem exactly.

Common Pitfalls:
Common mistakes include miscalculating the fraction of work completed in the first 6 days, or forgetting to subtract this correctly from 1. Another frequent error is mishandling the fractions when computing the rate of A and C together. Always proceed step by step: find A + B work, find remaining work, use A + C time, then isolate C's rate. This structured method prevents most errors.

Final Answer:
C alone can complete the entire work in 120 days.