Person X can complete a task alone in 8 days, whereas person Y can complete the same task alone in 10 days. If X and Y work together on the task from start to finish, in how many days will they complete the entire task?

Introduction / Context:
This is a basic time and work problem involving two people, X and Y, each capable of completing a task alone in different amounts of time. The question checks whether you understand how to combine individual work rates to find the time required when both persons work together on the same job.

Given Data / Assumptions:

• X alone can complete the work in 8 days.
• Y alone can complete the work in 10 days.
• Both X and Y work together for the entire duration.
• Total work is considered as 1 unit.
• Work rates are constant and additive.

Concept / Approach:
We convert the time taken by each person into a daily work rate expressed as a fraction of the total work completed per day. Then we add their rates to obtain the combined rate. Finally, we take the reciprocal of that combined rate to find the total time needed for them to complete the task when working together.

Step-by-Step Solution:
Step 1: Assume the total work is 1 unit. Step 2: Rate of X = 1/8 units per day. Step 3: Rate of Y = 1/10 units per day. Step 4: Combined rate of X and Y working together = 1/8 + 1/10. Step 5: Find a common denominator for 1/8 and 1/10. The least common multiple of 8 and 10 is 40. Step 6: Convert the fractions: 1/8 = 5/40 and 1/10 = 4/40. Step 7: Add the two rates: 5/40 + 4/40 = 9/40 units per day. Step 8: Time taken when working together = total work / combined rate = 1 / (9/40) = 40/9 days.

Verification / Alternative check:
We can check the reasonableness of the answer by approximation. X alone takes 8 days and Y alone takes 10 days. When they work together, the time must be less than 8 days. 40/9 is approximately 4.44 days, which is indeed less than 8 days and greater than 0, so the answer is logical and consistent with expectations.

Why Other Options Are Wrong:

• 9/40 days: This is actually the combined rate, not the time. It is the reciprocal of the correct answer.
• 44/9 days: This is greater than 8 days, which cannot be correct because working together must be faster than either person alone.
• 9/44 days: This is a very small fraction of a day and does not match the combined rate calculation.

Common Pitfalls:
A frequent mistake is to average the two times as (8 + 10) / 2 = 9 days, which is incorrect. Time cannot be averaged directly in such problems; rates must be added instead. Another pitfall is forgetting to invert the combined rate to obtain the time. Care must also be taken when adding fractions to ensure the proper least common multiple is used.

Final Answer:
The task will be completed in 40/9 days when X and Y work together.