A can do a piece of work in 24 days and B can do the same work in 40 days. If they work together on the job for 10 days, what fraction of the work is still left to be completed?

Introduction / Context:
This question asks us to determine the fraction of work remaining after two workers, A and B, have jointly worked for a fixed period. We are given the individual times required by A and B to complete the work alone and must find how much of the job remains undone after they work together for 10 days. It tests fundamental understanding of work rates and partial completion of tasks.

Given Data / Assumptions:
- A can complete the work alone in 24 days. - B can complete the same work alone in 40 days. - A and B work together for 10 days. - Work rates remain constant. - Total work is treated as one unit.

Concept / Approach:
The daily work rate of someone who finishes a job in T days is 1 / T of the job per day. When two people work together, their combined rate is the sum of their individual rates. We compute the combined rate of A and B, multiply it by 10 to get the fraction completed in 10 days and then subtract this from 1 to find the remaining fraction. Finally, we simplify the fraction to one of the given options.

Step-by-Step Solution:
Step 1: A's daily work rate = 1 / 24 of the job per day. Step 2: B's daily work rate = 1 / 40 of the job per day. Step 3: Combined daily rate = 1 / 24 + 1 / 40. Step 4: Take common denominator 120: 1 / 24 = 5 / 120, 1 / 40 = 3 / 120, so sum = 8 / 120 = 1 / 15. Step 5: Work done in 10 days = 10 * (1 / 15) = 10 / 15 = 2 / 3 of the job. Step 6: Remaining work = 1 - 2 / 3 = 1 / 3 of the job.

Verification / Alternative check:
We can check if 1 / 3 remaining is reasonable. If 2 / 3 of the job is done in 10 days, then the full job at this rate would take 10 / (2 / 3) = 15 days. This matches the combined time we would get by solving for 1 / (1 / 24 + 1 / 40) = 15 days, which is consistent. So the fraction remaining, 1 / 3, is correct.

Why Other Options Are Wrong:
1 / 2 or 2 / 3: These would suggest that 1 / 2 or 1 / 3 of the work is completed, not 2 / 3, and conflict with the actual combined rate and time.
3 / 4: This implies only 1 / 4 is done, which is far too small for 10 days of joint work at the calculated rate.
1 / 4: This would mean 3 / 4 of the work is done, which is more than the calculated 2 / 3.

Common Pitfalls:
Students may forget to convert to a common denominator when adding fractions or may try to average the times (24 and 40 days) instead of summing rates. Another error is to miscalculate the fraction done versus remaining. Always carefully calculate the combined rate, multiply by the actual working time, and subtract from 1 to find the fraction of work left.

Final Answer:
The fraction of work left after 10 days is 1/3.