Pankaj has completed half of a job in 12 days. Sainath then completes the remaining half of the same job in 6 days. If Pankaj and Sainath work together on the entire job from the beginning, in how many days will they finish it?

Introduction / Context:
This question involves two workers, Pankaj and Sainath, each completing half of a job in different times. We are asked to find how long it would take if they started together and worked jointly on the entire job. This is a straightforward application of converting partial-completion times into work rates and then combining those rates.

Given Data / Assumptions:

• Pankaj completes half (1/2) of the job in 12 days.
• Sainath completes the remaining half (1/2) in 6 days.
• Each works at a constant rate whenever working.
• We are asked for the time for the whole job when both work together from the beginning.

Concept / Approach:
We consider the full job as 1 unit. From the statement, we derive Pankaj's daily work rate as half the job divided by 12 days. Similarly, Sainath's rate is half the job divided by 6 days. Adding the two rates gives the combined rate when both work simultaneously. The total time to complete the job is then 1 divided by this combined rate. This is a direct and clean application of basic work-rate arithmetic.

Step-by-Step Solution:
Step 1: Let total job = 1 unit. Step 2: Pankaj completes 1/2 of the job in 12 days, so his rate = (1/2) / 12 = 1/24 of the job per day. Step 3: Sainath completes 1/2 of the job in 6 days, so his rate = (1/2) / 6 = 1/12 of the job per day. Step 4: Combined rate when both work together = 1/24 + 1/12. Step 5: Convert 1/12 to 2/24, so combined rate = 1/24 + 2/24 = 3/24 = 1/8 of the job per day. Step 6: Time to complete the entire job together = 1 / (1/8) = 8 days.

Verification / Alternative check:
To verify, consider their work in 8 days at the combined rate of 1/8 per day. Total work done = 8 × (1/8) = 1 unit, which is the full job. Check the relative speeds: Sainath is twice as fast as Pankaj since 1/12 is double 1/24, so together they should finish in less time than either alone. Pankaj alone would need 24 days for the full job, Sainath alone 12 days, so 8 days together is reasonable and consistent.

Why Other Options Are Wrong:
4 days is too small; at 1/8 per day, only half the job would be done. 12 and 16 days are too large, implying a slower combined rate than either individual, which is impossible. 6 days is also too small because at 1/8 per day only 6/8 = 3/4 of the job would be completed. Therefore, 8 days is the only option that matches the combined rate correctly.

Common Pitfalls:
A frequent error is to average the given times 12 and 6 to get 9 days, which would be incorrect as work rates do not average linearly in this way. Another mistake is to treat half-jobs as full jobs, leading to wrong rates. Carefully converting "half of the job in N days" into a rate of 1/(2N) job per day is the key to solving this correctly.

Final Answer:
Pankaj and Sainath together will complete the job in 8 days.