Twelve men can complete a certain work in 9 days. After they have worked together for 6 days, 6 more men join the group. How many additional days are required to complete the remaining work after the new men join?

Introduction / Context:
This time and work question presents a situation where additional workers join partway through a job. We must calculate how much work remains at that point and then determine how quickly the larger team can finish it. The focus is on correctly handling total work and work done in different phases of the project.

Given Data / Assumptions:

• Twelve men can complete the entire work in 9 days.
• These 12 men work together for the first 6 days.
• Then 6 more men join, giving a total of 18 men.
• All men have equal and constant efficiency.
• We need the number of extra days needed after the new men join.

Concept / Approach:
First, we compute the total amount of work in man days. Then we see how much work is done in the first 6 days by the initial 12 men. The difference between the total work and the completed portion is the remaining workload. Next, we compute the daily capacity of the larger group of 18 men and divide the remaining work by this daily capacity to find the extra days required.

Step-by-Step Solution:
Total work in man days = 12 men * 9 days = 108 man days.Work done in first 6 days by 12 men = 12 * 6 = 72 man days.Remaining work = 108 - 72 = 36 man days.After 6 days, 6 more men join, so total men = 12 + 6 = 18.Daily work capacity of 18 men = 18 man days per day.Time to finish remaining work = 36 man days / 18 man days per day = 2 days.

Verification / Alternative check:
We can also express the problem in terms of fractions of the job. Twelve men complete the job in 9 days, so the rate of 12 men is 1 / 9 job per day. In 6 days, they complete 6 * 1 / 9 = 6 / 9 = 2 / 3 of the job. Thus, 1 / 3 remains. Eighteen men are 1.5 times as many as 12 men, so their rate is 1.5 * 1 / 9 = 1 / 6 job per day. Time for 1 / 3 job at rate 1 / 6 is (1 / 3) / (1 / 6) = 2 days, confirming our answer.

Why Other Options Are Wrong:
Three, four, or five days would overestimate the time for the remaining 36 man days when 18 men are available. At 18 man days per day, even three days would represent 54 man days, which is more than the remaining work. Hence, only 2 days match both the man day and fractional analyses.

Common Pitfalls:
A common error is to forget that extra men join after 6 days, and to compute time based only on the original team. Another mistake is miscounting total man days or mixing up which portion of the work was completed before extra men joined. Writing down the full man day equation usually helps to avoid such confusion.

Final Answer:
The work will be completed in an additional 2 days after the six new men join.