A alone can complete a piece of work in 20 days. Worker B is 25% more efficient than A, that is, B's efficiency is 125% of A's efficiency. A and B work together for 4 days, after which the remaining work is completed by C alone in 22 days. How many days would C alone take to complete the entire work from the beginning?

Introduction / Context: This is a multi-step time and work problem that involves comparing efficiencies of workers A and B and then adding a third worker C who finishes the remaining work. The question checks your understanding of percentage-based efficiency, conversion of efficiency into time, combined work, and then determining the individual time of a third worker based on partial work information.

Given Data / Assumptions:

• A alone finishes the work in 20 days.
• B is 25% more efficient than A (that is, B's efficiency is 125% of A's efficiency).
• A and B work together for 4 days.
• The remaining work is completed by C alone in 22 days.
• The work rate of each worker is constant throughout.
• Total work is assumed to be 1 unit.

Concept / Approach: We first convert efficiency information into work rates. Since B is 25% more efficient than A, B's rate is 1.25 times A's rate. Using A's time, we find A's work rate and then B's work rate. We compute how much work A and B complete in 4 days, subtract that from the total work, and then use the remaining work and time taken by C to find C's rate. Finally, we invert C's rate to find C's total time for the entire job.

Step-by-Step Solution: Step 1: Let the total work be 1 unit. Step 2: A's time = 20 days, so A's rate = 1/20 units per day. Step 3: B is 25% more efficient than A, so B's rate = 1.25 * (1/20) = 1/16 units per day (which corresponds to B taking 16 days for the whole work). Step 4: Combined rate of A and B = 1/20 + 1/16 = (4/80 + 5/80) = 9/80 units per day. Step 5: In 4 days, work done by A and B together = 4 * 9/80 = 36/80 = 9/20 of the work. Step 6: Remaining work after 4 days = 1 - 9/20 = 11/20. Step 7: C alone completes this remaining 11/20 of the work in 22 days. Step 8: Therefore, C's daily work rate = (11/20) / 22 = 11 / 440 = 1/40 units per day. Step 9: Time taken by C to do the entire work alone = 1 / (1/40) = 40 days.

Verification / Alternative check: To verify, calculate actual work done. A and B together do 9/20 of the work. The remaining 11/20 is done by C. At rate 1/40, in 22 days C does 22 * (1/40) = 22/40 = 11/20, which matches the remaining work. This confirms the correctness of C's rate and the final time of 40 days for C alone.

Why Other Options Are Wrong:

• 30 days: This would make C's rate 1/30, in which case 22 days of work would complete 22/30 of the job, which is not equal to 11/20.
• 35 days: This gives C a rate of 1/35, which does not match the requirement that 11/20 of the work is done in 22 days.
• 42 days: This again leads to a mismatched rate and incorrect remaining work calculation.

Common Pitfalls: One common mistake is to misinterpret "125% more efficient" as 225% of A's efficiency instead of 125% of A's efficiency. Another error is incorrectly adding the work done or forgetting to subtract the work already completed by A and B. It is crucial to keep track of fractions carefully and ensure that each step builds accurately on the previous calculations.

Final Answer: C alone would take 40 days to complete the entire work from the beginning.