P can complete a work in 15 days and Q can complete the same work in 24 days. They begin the work together, but Q leaves the job 2 days before it is finished. In how many days in total was the work completed?

Introduction / Context:
This problem involves two workers, P and Q, who start a job together but one of them stops working before the work is completed. You need to determine the total number of days taken to finish the work. This combines ideas of work rates and careful time accounting when one worker leaves early.

Given Data / Assumptions:

P alone can complete the work in 15 days.
Q alone can complete the work in 24 days.
They start working together from the beginning.
Q leaves the work 2 days before completion, while P continues until the end.
Work rates are constant and the total work is one complete job.

Concept / Approach:
We let the total time taken to finish the work be t days. P works for all t days, but Q works only for t - 2 days. We express their contributions in terms of their daily rates and set the sum equal to 1 (the whole work). This gives a linear equation in t, which we solve to get the total time required.

Step-by-Step Solution:
Let total work = 1 unit. P's daily rate = 1/15 of the work per day. Q's daily rate = 1/24 of the work per day. Let total time to complete the work be t days. P works for t days, so P's contribution = t * (1/15). Q leaves 2 days before completion, so Q works for (t - 2) days, giving contribution = (t - 2) * (1/24). Total work done by both = t/15 + (t - 2)/24 = 1. Find a common denominator; LCM of 15 and 24 is 120. Rewrite as (8t)/120 + (5(t - 2))/120 = 1. Simplify numerator: 8t + 5t - 10 = 13t - 10. So (13t - 10)/120 = 1, which implies 13t - 10 = 120. Solve 13t = 130, so t = 10 days.

Verification / Alternative check:
If t = 10 days, then P works for all 10 days, doing 10/15 = 2/3 of the work. Q works for 8 days (since t - 2 = 8), doing 8/24 = 1/3 of the work. Together they complete 2/3 + 1/3 = 1 whole unit of work, which matches the requirement. This confirms that the total time to finish the job is indeed 10 days.

Why Other Options Are Wrong:
Option 12 days leads to P doing 12/15 = 4/5 and Q doing 10/24 = 5/12, which does not sum to 1.
Option 9 days gives P's work as 9/15 and Q's as 7/24, which also does not equal 1 when added.
Option 11 days yields P's contribution of 11/15 and Q's of 9/24, again not equal to the full job.
Option 8 days is clearly too short given the individual times of 15 and 24 days and results in less than one complete job when contributions are summed.

Common Pitfalls:
A typical mistake is to assume that Q works for exactly half the time or to forget that Q leaves 2 days before the end, not halfway. Another pitfall is mismanaging fractions when combining the work expressions. Carefully set up the equation using t, and always check the final contributions of each worker to ensure they add up to 1.

Final Answer:
Thus, the total work is completed in 10 days, so the correct option is 10 days.