A contractor undertook to complete a certain work in 124 days and employed 120 men. After 64 days, he found that two thirds of the work was already completed. How many men can he discharge so that the remaining work is finished in exactly the originally planned 124 days?

Introduction / Context:
This question deals with manpower planning in a project. The contractor initially estimates a certain workforce to complete the job in a fixed time. Later, he discovers that the work is progressing faster than expected, which allows him to reduce the workforce and still meet the deadline. Problems like this are common in aptitude tests and project management contexts and require careful handling of work rate, remaining work, and revised workforce calculations.

Given Data / Assumptions:

• The planned completion time is 124 days with 120 men.
• After 64 days of work with 120 men, two thirds of the work is completed.
• The remaining one third of the work must be completed in the remaining time to still finish in 124 days.
• We assume each man works at a constant rate and all men have equal efficiency.
• We want to know how many men the contractor can discharge, meaning how many can be removed, while still finishing on time.

Concept / Approach:
We treat the total work as one unit and calculate each man's daily work rate from the information that 120 men complete two thirds of the work in 64 days. Then we compute how many man days are needed to finish the remaining one third and how many days are left. Dividing the remaining required man days by the remaining time gives the number of men needed for the rest of the project. Subtracting this from the current 120 men gives the number of men who can be safely discharged.

Step-by-Step Solution:
Step 1: Total work is 1 job. In 64 days, 120 men complete 2/3 of the job. Step 2: Let each man do r units per day. Then 120 men working for 64 days complete 120 times 64 times r = 2/3. Hence r = (2/3) divided by (120 times 64) = 1 / 11520. Step 3: Remaining work is 1 - 2/3 = 1/3 of the job. Step 4: Remaining time is 124 - 64 = 60 days. Step 5: Let N be the number of men required. Then N times 60 times r must equal 1/3. So N times 60 times 1/11520 = 1/3. Step 6: Simplifying, N times 60 / 11520 = 1/3 which gives N = (1/3) times 11520 / 60 = 64 men. Step 7: Initially there were 120 men, so men to be discharged equals 120 - 64 = 56 men.

Verification / Alternative check:
We can confirm using man days. Each man does 1/11520 of the job per day, so 64 men do 64 / 11520 = 1/180 of the job per day. Over 60 days, they complete 60 times 1/180 which equals 1/3 of the job, exactly the remaining part. Adding this to the initial two thirds yields one full job completed in 124 days. This confirms that 64 men are sufficient for the last stage and that discharging 56 men is correct.

Why Other Options Are Wrong:

• 64 men as an answer would be correct only if the question asked for the remaining workforce, not for the men to be discharged.
• 62 men or 58 men would not be enough to finish the remaining one third within 60 days.
• Only discharging 56 men leaves exactly 64 men, which matches the requirement computed from the remaining work and time.

Common Pitfalls:
Some learners confuse men required with men discharged and therefore stop after finding 64 men. Others may incorrectly assume that the original estimate of 120 men for 124 days implies constant usage of 120 men, ignoring the updated progress data. It is important to use the actual completed work to infer the true work rate and then recompute manpower requirements for the remaining part.

Final Answer:
The contractor can safely discharge 56 men and still finish the work in the originally planned 124 days.