Robin can complete a job in 10 hours when working alone. Robin and Pradeep together can complete the same job in 4 hours, and Pradeep and Nitin together can complete it in 5 hours. In how many hours can Nitin alone complete the job?

Introduction / Context:
This time and work problem involves three workers, Robin, Pradeep, and Nitin. We are given the time taken by Robin alone and by pairs of workers to complete the same job. The task is to find how long Nitin alone would take to finish the job. Such questions test the ability to work with multiple equations representing combined work rates and to isolate the rate for one individual worker.

Given Data / Assumptions:

• Robin alone can complete the job in 10 hours.
• Robin and Pradeep together can complete the job in 4 hours.
• Pradeep and Nitin together can complete the job in 5 hours.
• We must find the time taken by Nitin alone to complete the job.
• Total work is taken as 1 job.

Concept / Approach:
Let the hourly work rates of Robin, Pradeep, and Nitin be R, P, and N jobs per hour. If someone completes a job in T hours, their rate is 1/T job per hour. We can write equations for Robin alone, Robin plus Pradeep, and Pradeep plus Nitin. Solving these equations gives the rate for Nitin, and the time Nitin alone needs is the reciprocal of this rate. This algebraic approach is standard for combined work rate problems.

Step-by-Step Solution:
Step 1: Let total work be 1 job. Step 2: Rate of Robin, R = 1/10 job per hour. Step 3: Rate of Robin and Pradeep together = 1/4 job per hour, so R + P = 1/4. Step 4: Substitute R = 1/10 into R + P = 1/4 to get 1/10 + P = 1/4. Step 5: P = 1/4 - 1/10. Using LCM 20, 1/4 = 5/20 and 1/10 = 2/20, so P = (5/20 - 2/20) = 3/20 job per hour. Step 6: Rate of Pradeep and Nitin together = 1/5 job per hour, so P + N = 1/5. Step 7: Substitute P = 3/20 into P + N = 1/5 to get 3/20 + N = 1/5. Step 8: Convert 1/5 to twentieths: 1/5 = 4/20. So N = 4/20 - 3/20 = 1/20 job per hour. Step 9: Time taken by Nitin alone = 1 / (1/20) = 20 hours.

Verification / Alternative check:
We can verify our result by checking the combined rates. With N = 1/20 and P = 3/20, P + N = (3/20 + 1/20) = 4/20 = 1/5, which matches the given time of 5 hours for Pradeep and Nitin together. With R = 1/10 and P = 3/20, R + P = 1/10 + 3/20 = 2/20 + 3/20 = 5/20 = 1/4, consistent with the given 4 hours for Robin and Pradeep. Therefore, Nitin taking 20 hours alone is correct.

Why Other Options Are Wrong:
12 hours, 15 hours, and 18 hours are all too short and would imply a higher rate for Nitin, which would disrupt the consistency of the pairwise work rates and given times. 24 hours is too long and would reduce Nitin rate too much, again violating the equations set by the data. Only 20 hours keeps both pairwise conditions valid.

Common Pitfalls:
Learners often mismanage fraction arithmetic or skip forming proper equations. Some try to guess the answer based on averages, which does not work reliably for these problems. It is crucial to clearly define variables for each worker rate, write equations from the given times, and solve them step by step without skipping algebraic details.

Final Answer:
Nitin alone can complete the job in 20 hours.