Ganesh, Ram, and Sohan together can complete a piece of work in 16 days. Ganesh and Ram together can complete the same work in 24 days. How many days will Sohan alone take to finish the entire work?

Introduction / Context:
This question asks us to find the time taken by one worker, Sohan, given the time taken by a group that includes him and the time taken by a subset of that group without him. Such problems are classic applications of time and work concepts and highlight how partial group times can be used to isolate the efficiency of one individual worker.

Given Data / Assumptions:

• Ganesh, Ram, and Sohan together can complete the work in 16 days.
• Ganesh and Ram together can complete the work in 24 days.
• We must find how many days Sohan alone would take to complete the entire work.
• Total work is considered as 1 complete job.
• All workers have constant daily work rates.

Concept / Approach:
Let the daily work rates of Ganesh, Ram, and Sohan be G, R, and S respectively. From the given times, we can write equations for G + R + S and for G + R. Subtracting the second equation from the first isolates S, the daily rate of Sohan. Once we know S rate, we can obtain the time for Sohan alone by taking the reciprocal of that rate. This method is simple and very effective for combined work problems.

Step-by-Step Solution:
Step 1: Let total work be 1 job. Step 2: Since Ganesh, Ram, and Sohan together complete the job in 16 days, G + R + S = 1/16 job per day. Step 3: Since Ganesh and Ram together complete the job in 24 days, G + R = 1/24 job per day. Step 4: Subtract the second equation from the first to isolate Sohan rate. Step 5: S = (G + R + S) - (G + R) = 1/16 - 1/24. Step 6: Take LCM of 16 and 24, which is 48. Then 1/16 = 3/48 and 1/24 = 2/48. Step 7: S = (3/48 - 2/48) = 1/48 job per day. Step 8: Time taken by Sohan alone = 1 / (1/48) = 48 days.

Verification / Alternative check:
We can check consistency by reconstructing the combined rates. If Sohan rate is 1/48 and Ganesh plus Ram together have rate 1/24, then G + R + S = 1/24 + 1/48 = (2/48 + 1/48) = 3/48 = 1/16 job per day, which matches the given information. Thus, Sohan taking 48 days alone is consistent with both statements about the combined work times.

Why Other Options Are Wrong:
If Sohan took 32, 30, 40, or 24 days, his rate would be higher, leading to G + R + S being more than 1/16 job per day, which contradicts the given data. Only 48 days produces the correct combined rate and stays consistent with both given completion times for the two groups.

Common Pitfalls:
One mistake is to average the two time values instead of working with rates. Another is to miscalculate the subtraction of fractions, especially when not taking the right LCM. Always remember that adding or subtracting work times directly is incorrect; we must always add or subtract work rates instead.

Final Answer:
Sohan alone will take 48 days to complete the work.