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

Introduction / Context:
This question demonstrates how the addition of more workers partway through a project affects the remaining completion time. It is a practical type of time and work problem where we first compute how much portion of the job is done before extra workers arrive and then determine how quickly the enlarged team finishes the rest. Problems like this highlight the importance of handling fractions of work and combined work rates accurately over multiple stages.

Given Data / Assumptions:

• Twelve men complete the entire work in 9 days.
• The same twelve men work alone for the first 6 days.
• After 6 days, four more men join, making a total of sixteen men.
• All men have equal and constant efficiency.
• We are asked for the additional number of days needed after the four men join.

Concept / Approach:
The method is to treat the whole job as one unit of work and compute individual and group daily rates. First we find the rate of one man using the initial information. Then we compute the fraction of work completed during the first 6 days. Subtracting this from the total gives the remaining work. Next we find the rate of sixteen men and divide the remaining fraction by this rate to get the extra time needed. Keeping the fractions exact until the last step ensures accuracy.

Step-by-Step Solution:
Step 1: Total work is 1 job. Twelve men finish it in 9 days, so the daily work of 12 men is 1 / 9. Step 2: Work of one man per day is 1 / (12 times 9) which equals 1 / 108. Step 3: In 6 days, twelve men complete 6 times 1/9 which equals 2 / 3 of the work. Step 4: Remaining work is 1 minus 2/3 which equals 1 / 3 of the job. Step 5: After four men join, there are sixteen men. Their combined daily rate is 16 times 1/108 which equals 4 / 27 of the job per day. Step 6: Time to finish the remaining 1 / 3 of the work is (1/3) divided by (4/27) which equals (1/3) times 27/4 = 9/4 days = 2.25 days.

Verification / Alternative check:
We can think in units. Let the total work be 108 units so that one man does 1 unit per day. Twelve men then do 12 units per day and finish in 9 days, confirming 108 units. In the first 6 days they complete 72 units, leaving 36 units. When sixteen men work together, they complete 16 units per day. Time for the remaining 36 units is 36 divided by 16, which equals 2.25 days, matching the previous fraction based calculation exactly.

Why Other Options Are Wrong:

• 2 days is too small because sixteen men would complete only 32 units if we follow the unit method, leaving some work undone.
• 2.5 days would represent 40 units, which is more than the 36 units remaining.
• 3 days would add 48 units, clearly overshooting the work left to do.

Common Pitfalls:
Some learners forget that men join only after 6 days and may mistakenly assume sixteen men work from the beginning. Others subtract time instead of subtracting fractions of work. It is important to clearly separate stage one and stage two, carefully calculate how much of the job is done by the original team, and only then bring in the added workers with their increased combined rate for the remaining portion.

Final Answer:
After the four men join, they will need an additional 2.25 days to complete the remaining work.