Worker K can build a wall in 30 days if he works alone. Worker L can completely demolish the same wall in 60 days if he works alone. If K and L work on alternate days, starting with K on the first day, after how many days will the wall be fully completed?

Introduction / Context:
This question is a classic example of alternating work with one worker building and another destroying part of the work. Worker K builds a wall while worker L demolishes it. They work on alternate days starting with K, and we must determine in how many days the wall will finally be completed.

Given Data / Assumptions:

• K alone can build the wall in 30 days.
• L alone can demolish the entire wall in 60 days.
• They work on alternate days, K on day 1, L on day 2, K on day 3, and so on.
• Both work at constant daily rates, and a full wall is considered as one unit of work.

Concept / Approach:
We treat building as positive work and demolishing as negative work. K's daily contribution is positive while L's is negative. Over a full two day cycle (one day of K followed by one day of L), there is a net amount of work completed. We compute this net work per cycle and then determine how many such cycles are needed to finish the wall exactly.

Step-by-Step Solution:
Let the total work (a fully built wall) be 1 unit. K alone finishes the wall in 30 days, so K's rate = 1 / 30 unit per day. L alone demolishes the wall in 60 days, so L's rate = -1 / 60 unit per day (negative because he removes work). In one two day cycle: day 1 K builds 1 / 30, day 2 L demolishes 1 / 60. Net work in 2 days = 1 / 30 - 1 / 60 = (2 - 1) / 60 = 1 / 60. So every 2 days, 1 / 60 of the wall is effectively completed. To complete 1 full unit of work at 1 / 60 per 2 days, we need 60 such net portions. Number of 2 day cycles = 60, so total days = 60 * 2 = 120 days.

Verification / Alternative check:
Check partial progress: After 120 days, the number of K days and L days are both 60. K's total addition = 60 * (1 / 30) = 2 units. L's total demolition = 60 * (1 / 60) = 1 unit. Net result = 2 - 1 = 1 unit of completed wall, matching the requirement. Therefore, the wall is exactly completed at the end of 120 days.

Why Other Options Are Wrong:
119, 118, and 117 days correspond to incomplete two day cycles and would leave some fraction of work unfinished or over demolished. 121 days would go beyond the exact completion point, implying extra work and demolition beyond one full wall.

Common Pitfalls:
Many learners forget to treat demolition as negative work and instead just subtract days or times directly. Another mistake is to think K's net rate is 1 / 30 - 1 / 60 per day rather than per two day cycle. Always compute rates carefully, account for sign when work is undone, and sum over a full cycle when work patterns repeat.

Final Answer:
The wall will be completed in 120 days.