Contractor K can lay a highway road between two cities in 16 days, and contractor L can lay the same road in 12 days. With the help of contractor M, all three together complete the job in 4 days. In how many days would M alone complete the work?

Introduction / Context:
This is a standard time and work problem involving three workers. We know the individual times for two workers and the combined time when all three work together. The goal is to find the individual time for the third worker. This requires translating times into daily work rates, using the total work equation, and solving for the unknown rate of worker M.

Given Data / Assumptions:

• K alone can complete the work in 16 days.
• L alone can complete the work in 12 days.
• K, L and M together complete the work in 4 days.
• All workers have constant efficiencies over time.
• The work is identical in all scenarios.

Concept / Approach:
Let the total work be 1 unit. The daily work rates of K and L can be written as the reciprocals of their times. The combined rate of K, L and M is also the reciprocal of 4 days. Using the fact that combined rate equals the sum of individual rates, we can solve for M's rate and then take the reciprocal to find his individual time.

Step-by-Step Solution:
Let total work = 1 unit. K's daily rate = 1 / 16 of the work. L's daily rate = 1 / 12 of the work. Combined rate of K, L and M = 1 / 4 of the work per day. Let M's daily rate be m units of work per day. Then 1 / 16 + 1 / 12 + m = 1 / 4. First, compute 1 / 16 + 1 / 12. LCM of 16 and 12 is 48, so 1 / 16 = 3 / 48 and 1 / 12 = 4 / 48. Sum = 3 / 48 + 4 / 48 = 7 / 48. So 7 / 48 + m = 1 / 4. Convert 1 / 4 to denominator 48: 1 / 4 = 12 / 48. Thus m = 12 / 48 − 7 / 48 = 5 / 48. Therefore M's daily rate is 5 / 48 of the work per day. Time taken by M alone = 1 / m = 1 / (5 / 48) = 48 / 5 days.

Verification / Alternative check:
Check the sum of the three rates: 1 / 16 + 1 / 12 + 5 / 48. Using denominator 48, this is 3 / 48 + 4 / 48 + 5 / 48 = 12 / 48 = 1 / 4, which corresponds exactly to 4 days for all three together. This confirms that the computed rate of M is correct.

Why Other Options Are Wrong:
47/7 days, 59/6 days or 57/5 days are simply other fractions that do not satisfy the rate equation. Using any of those times for M would give a different value for m and the combined rate would no longer be 1 / 4 of the work per day.

Common Pitfalls:
Common mistakes include forgetting to convert all fractions to a common denominator when adding rates or misinterpreting “4 days together” as meaning something other than the reciprocal rate. It is crucial to do all fraction operations carefully and to remember that the total work has been assumed to be 1 unit for simplicity.

Final Answer:
M alone will complete the work in 48/5 days (which is 9.6 days).