A can do a piece of work in 8 days and B can do the same work in 10 days when each works alone. If A and B work together at their usual constant rates from start to finish, in how many days will they complete the work?

Introduction / Context:
Here, you are given the number of days two workers A and B take individually to complete a job. The question asks how long they will take to finish the job when they work together. This is a standard time and work problem that uses basic rate addition.

Given Data / Assumptions:

A alone can complete the work in 8 days.
B alone can complete the same work in 10 days.
They work together at constant rates from the start to completion.
The total work is considered as one complete unit of job.

Concept / Approach:
If a worker can finish a job in T days, the worker's daily rate is 1/T of the job per day. When multiple workers cooperate, their individual rates add up. We thus find the combined rate of A and B and then take the reciprocal of that combined rate to find the number of days required to finish the job.

Step-by-Step Solution:
Let total work = 1 unit. Daily rate of A = 1/8 of the job per day. Daily rate of B = 1/10 of the job per day. Combined daily rate = 1/8 + 1/10. Take the LCM of 8 and 10, which is 40. Convert fractions: 1/8 = 5/40 and 1/10 = 4/40. Combined rate = 5/40 + 4/40 = 9/40 of the job per day. Time taken together = total work / combined rate = 1 / (9/40) = 40/9 days.

Verification / Alternative check:
If they complete 9/40 of the job per day, then over 40/9 days they would do (9/40)*(40/9) = 1 job, which is correct. Also, note that A alone needs 8 days and B alone needs 10 days, so the combined time must be less than 8 days but reasonably close. 40/9 is approximately 4.44 days, which fits within this expectation and confirms the plausibility of the result.

Why Other Options Are Wrong:
Option 33/8 days is approximately 4.125 days, which implies a slightly higher combined rate than 9/40 and does not match the correct sum of rates.
Option 41/10 days is 4.1 days, which similarly conflicts with the combined rate computed by proper fraction addition.
Option 42/11 days is around 3.82 days and is too small given the individual times of 8 and 10 days.
Option 4 days would require a combined rate of 1/4 per day, which is larger than 9/40 and not equal to 1/8 + 1/10.

Common Pitfalls:
A common mistake is to average the times 8 and 10 to get 9 days, which is incorrect. Another pitfall is miscalculating the LCM or addition of fractions, leading to a wrong combined rate. Remember to always work with rates 1/T, add them carefully, and then invert the sum to find the total time.

Final Answer:
Thus, A and B working together can complete the work in 40/9 days, so the correct option is 40/9 days.