Prabhat completes one-half of a job in 12 days. Santosh completes the remaining half of the same job in 6 days. If both Prabhat and Santosh work together from the beginning on the entire job, in how many days will they complete it?

Introduction / Context:
This problem involves two workers, Prabhat and Santosh, each completing half of a job in different times when working alone. The question asks for the total time needed if they work together from the start. It tests your ability to interpret partial work information and transform it into full-work rates for each person before combining them.

Given Data / Assumptions:

• Prabhat completes 1/2 of the job in 12 days.
• Santosh completes the remaining 1/2 of the job in 6 days.
• We assume total work is 1 unit.
• Both workers have constant work rates.
• We need the time when both work together on the full job from the beginning.

Concept / Approach:
From the time taken to complete half the work, we can infer the time each worker would need for the full job. This leads to daily work rates for both Prabhat and Santosh. Once we know their individual rates, we add them to find the combined rate and then invert that rate to obtain the total time required when they work together on the complete job. This approach uses the idea that work rates add for simultaneous workers.

Step-by-Step Solution:
Step 1: Let total work be 1 unit. Step 2: Prabhat completes 1/2 of the work in 12 days. So his rate = (1/2) / 12 = 1/24 units per day. Hence, he would finish the full job in 24 days. Step 3: Santosh completes 1/2 of the work in 6 days. So his rate = (1/2) / 6 = 1/12 units per day. Hence, he would finish the full job in 12 days. Step 4: When they work together, combined rate = 1/24 + 1/12. Step 5: Express 1/12 with denominator 24: 1/12 = 2/24. Step 6: Combined rate = 1/24 + 2/24 = 3/24 = 1/8 units per day. Step 7: Time taken to complete the full job together = 1 / (1/8) = 8 days.

Verification / Alternative check:
Check the work done in 8 days. At a combined rate of 1/8 units per day, in 8 days they do 8 * (1/8) = 1 unit of work, which exactly equals the full job. The time is less than both 24 days (Prabhat alone) and 12 days (Santosh alone), which is consistent because combining workers should always reduce the total time.

Why Other Options Are Wrong:

• 12 days: This is the time Santosh would need alone, and it ignores the contribution of Prabhat.
• 4 days: This would require a combined rate of 1/4 units per day, which is much larger than the actual rate of 1/8 units per day derived from the given data.
• 16 days: This is longer than Santosh's individual time and therefore cannot be correct when both are working together.

Common Pitfalls:
Students may incorrectly assume that if half is done in 12 days and half in 6 days, then the total time together is some direct average of 12 and 6. However, time cannot be added in such a way here; instead, we must work with rates. Another pitfall is forgetting that Prabhat and Santosh are working on the entire job together, not only on their original halves.

Final Answer:
Prabhat and Santosh together will complete the entire job in 8 days.