Thirty four men working 9 hours per day complete two fifths of a certain job in 8 days. How many additional men must be employed, working the same 9 hours per day, to complete the remaining work in 6 days?

Introduction / Context:
This time and work problem involves increasing the workforce partway through a job to meet a deadline. We are given that an initial group of men completes a certain fraction of the work in a given time. We must determine how many more men to add so that the remaining fraction is finished within a shorter number of days, assuming constant work rates.

Given Data / Assumptions:
• Thirty four men working 9 hours per day complete two fifths of the work in 8 days.
• Remaining work = three fifths of the total job.
• All men, existing and additional, will work 9 hours per day for the remainder.
• Remaining work must be completed in 6 days.
• Each man has a constant hourly work rate.

Concept / Approach:
Work is proportional to the product of men, hours per day and days. We first calculate the total work in man hour units from the information about two fifths completion. Using this, we determine how many man hours are required for the remaining three fifths. Dividing this by the product of hours per day and remaining days gives the new required number of men, from which we subtract the original 34 to find the additional men needed.

Step-by-Step Solution:
Let total work be W units. Thirty four men work 9 hours per day for 8 days to complete 2W / 5. Total man hours used for two fifths: 34 × 9 × 8. Let each man hourly rate be r units per hour. Thus 34 × 9 × 8 × r = 2W / 5. We rearrange to express W in terms of r: W = (34 × 9 × 8 × r) × 5 / 2. Remaining work = 3W / 5. Substitute W: remaining work = 3 / 5 × (34 × 9 × 8 × r × 5 / 2) = 3 × 34 × 9 × 8 × r / 2. Simplify: remaining work = (3 × 34 × 8 × 9 / 2) × r = 34 × 8 × 27 × r. Now let X be the total number of men required for the remaining work. They work 6 days for 9 hours per day: man hours = X × 6 × 9. Total work produced by them = X × 6 × 9 × r. Set this equal to remaining work: X × 6 × 9 × r = 34 × 8 × 27 × r. Cancel r and simplify: X × 54 = 34 × 216. Compute right side: 34 × 216 = 7344, so X = 7344 ÷ 54 = 136. Note: in simple proportional setup with fractions, the direct computation yields X = 68. Using the fraction method: remaining work fraction is 3 / 5; setting 34 × 9 × 8 : 2 / 5 = X × 9 × 6 : 3 / 5 gives X = 68 men total required for the second phase. Therefore, extra men required = 68 − 34 = 34 men.

Verification / Alternative check:
Check with the direct fraction method. Work fraction is proportional to men × days when hours per day and rate per man are constant. First phase: 34 men in 8 days produce 2 / 5 of the job. So per man day fraction = (2 / 5) ÷ (34 × 8). For the second phase: let total men be 68, days be 6. Then fraction produced = 68 × 6 × per man day fraction = 68 × 6 × (2 / 5) ÷ (34 × 8) = (68 ÷ 34) × (6 ÷ 8) × (2 / 5) = 2 × 0.75 × 2 / 5 = 3 / 5, exactly the remaining work.

Why Other Options Are Wrong:
Values such as 48, 60 or 68 as additional men would either overshoot or undershoot the required man days. For example, 48 additional men would give 34 + 48 = 82 total men, producing more than three fifths of the work in 6 days. Only 34 additional men give exactly the required completion.

Common Pitfalls:
A common error is to assume work is directly proportional only to the number of men, ignoring the effect of differing days in each phase. Another mistake is mismanaging fractions when converting the two fifths and three fifths of work into proportions of man days.

Final Answer:
Thirty four additional men must be employed.