Worker K can finish a certain piece of work alone in 18 days, and worker L can finish the same work alone in 15 days. L works on the job for 10 days and then leaves. How many more days will K, working alone, take to complete the remaining work?

Introduction / Context:
This is a standard time and work question where one worker starts the job and then another finishes it. We know the individual capacities of both workers and the time for which the first worker contributes. The task is to compute how long the second worker needs to finish the remaining portion.

Given Data / Assumptions:
• K can complete the work alone in 18 days.
• L can complete the work alone in 15 days.
• L works alone for 10 days and then leaves.
• K then works alone to finish the remaining work.
• Work rates of K and L are constant.

Concept / Approach:
We treat the total work as 1 job. First, we compute how much of the work L completes in 10 days using his daily rate. The difference between 1 and that amount is the remaining work, which K must finish. We use K daily rate to find the time required to complete the remaining portion.

Step-by-Step Solution:
Let total work = 1 job. Daily rate of K = 1 / 18 job per day. Daily rate of L = 1 / 15 job per day. Work done by L in 10 days = 10 × 1 / 15 = 10 / 15 = 2 / 3 of the job. Remaining work = 1 − 2 / 3 = 1 / 3 of the job. K must complete this remaining 1 / 3 of the job. Time taken by K = remaining work ÷ K daily rate = (1 / 3) ÷ (1 / 18). Compute: (1 / 3) ÷ (1 / 18) = (1 / 3) × 18 = 6 days.

Verification / Alternative check:
We can think of the entire job as made of 90 equal units (least common multiple of 15 and 18). L finishes 90 ÷ 15 = 6 units per day and in 10 days completes 60 units. Remaining units = 90 − 60 = 30 units. K works at 90 ÷ 18 = 5 units per day, so he needs 30 ÷ 5 = 6 days, confirming the answer.

Why Other Options Are Wrong:
Options like 5 or 4.2 days give too short a time for K to complete one third of the work, given his rate of 1 / 18 job per day. Option 5.5 is arbitrary and does not correspond to a neat fraction of the work. Only 6 days is consistent with both fractional and unit based calculations.

Common Pitfalls:
A common error is to add or average the times 18 and 15 rather than using reciprocal rates. Another mistake is to think K and L worked together initially, which is not the case. Carefully reading that L works first and then K finishes is vital to forming the correct calculation.

Final Answer:
K will take 6 more days to finish the remaining work.