Six men can complete a piece of work in 12 days, eight women can complete the same work in 18 days, and eighteen children can complete it in 10 days. Four men, twelve women and twenty children work together for 2 days. If only men are to complete the remaining work in 1 more day, how many men in total are required on that last day?

Introduction / Context:
This mixed work rate question involves three categories of workers with different efficiencies: men, women, and children. The problem first gives separate completion times for each group and then asks for the number of men required at the end to finish the remaining portion of the work in one day. Such scenarios are frequently used in aptitude exams to test comfort with fractional work, rate comparisons, and combining contributions from different groups over different time intervals.

Given Data / Assumptions:

• Six men complete the work in 12 days.
• Eight women complete the same work in 18 days.
• Eighteen children complete the work in 10 days.
• Four men, twelve women, and twenty children work together for 2 days.
• After this, only men will work and must complete the remaining work in 1 day.
• All workers of the same type have equal and constant efficiency.

Concept / Approach:
We use the idea of daily work rates for each type of worker. From the total time required by each group we calculate the daily work done by one man, one woman, and one child. Then we find how much work is completed in the first two days by the mixed group. Subtracting that from the total leaves the remaining work. Finally, we compute how many men are needed so that, at the known rate of one man, the remaining work is finished in exactly one day.

Step-by-Step Solution:
Step 1: Total work can be taken as 1 job. Rate of one man is 1 divided by (6 times 12) which is 1 / 72 per day. Step 2: Rate of one woman is 1 divided by (8 times 18) which is 1 / 144 per day. Step 3: Rate of one child is 1 divided by (18 times 10) which is 1 / 180 per day. Step 4: Daily work of 4 men, 12 women and 20 children is 4(1/72) + 12(1/144) + 20(1/180) which simplifies to 1 / 4 of the job per day. Step 5: Work done in 2 days by this group is 2 times 1/4 which equals 1 / 2 of the job. Step 6: Remaining work is 1 minus 1/2 which equals 1 / 2 of the job. Step 7: Let N be the number of men required on the last day. Then N times 1/72 times 1 day equals 1/2, so N / 72 = 1/2, giving N = 36.

Verification / Alternative check:
Check that 36 men at the rate of 1 / 72 each would complete exactly half the job in one day. Total output is 36 times 1/72 which equals 1/2. Adding the first half completed by the mixed group, the total equals 1, meaning the entire work is done. All calculations are consistent with the data given, and the number of required men is therefore correct.

Why Other Options Are Wrong:

• 38 men would complete slightly more than half the job on the last day and are more than necessary.
• 72 men would complete the entire job in one day, which is not needed because half the work has already been done.
• 76 men are even more excessive and do not match the requirement to finish exactly the remaining half in one day.

Common Pitfalls:
One common mistake is to confuse total group times with individual rates or to assume that each category has equal efficiency. Another pitfall is to forget that men, women, and children all contribute simultaneously in the first two days, so their rates must be added. Some learners also misinterpret the phrase men required totally and attempt to include earlier men, but the question clearly asks for the number of men needed on the final day to finish the remaining work.

Final Answer:
The total number of men required on the last day to finish the remaining work in one day is 36 men.