A can complete one-third of a piece of work in 5 days, while B can complete two-fifths of the same work in 10 days. If both A and B work together from the beginning, in how many days will they complete the entire work?

Introduction / Context:
This question explores partial work information for two workers and asks for the total time when they work together. Instead of directly giving complete-time values for A and B, it provides fractions of work completed in specific times. This is a common technique in aptitude tests to check if you can convert fractional work into individual work rates and then combine them correctly.

Given Data / Assumptions:

• A completes one-third (1/3) of the work in 5 days.
• B completes two-fifths (2/5) of the work in 10 days.
• They both start together and work simultaneously.
• Work rate of each person is constant over time.
• We need the total time required to complete the full work when they work together.

Concept / Approach:
We use the concept of work rate, defined as fraction of work done per day. From the partial work information, we find A's daily rate and B's daily rate. Then we add the two rates to get the combined daily rate. Finally, we compute total time as the total work (taken as 1 unit) divided by this combined rate. The concept of fractional work is central here.

Step-by-Step Solution:
Step 1: Let total work = 1 unit. Step 2: A completes 1/3 of the work in 5 days, so A's rate = (1/3) / 5 = 1/15 of the work per day. Step 3: B completes 2/5 of the work in 10 days, so B's rate = (2/5) / 10 = 1/25 of the work per day. Step 4: Combined rate of A and B = 1/15 + 1/25. Step 5: Compute the sum: 1/15 + 1/25 = (5/75) + (3/75) = 8/75. Step 6: Time to complete 1 unit of work together = 1 / (8/75) = 75/8 days. Step 7: 75/8 days = 9.375 days, or 9 3/8 days.

Verification / Alternative check:
To verify, we can check how much work they would complete in 9.375 days. Combined rate is 8/75 per day, so work done = (8/75) × (75/8) = 1 unit, which matches the total job. Also, A alone would take 15 days to do the whole work and B alone would take 25 days. Since they work together, the combined time should be less than 15 days, and 9.375 days fits this expectation well.

Why Other Options Are Wrong:
7.375 days and 8.5 days are too small and would imply a combined rate larger than what is actually possible from the individual rates. 10 and 12.5 days are larger than 9.375; 12.5 is even greater than the time that B would need alone, which is illogical for combined work. Only 9.375 days satisfies the exact calculation based on their rates.

Common Pitfalls:
A typical mistake is to directly average the partial times or to convert fractions incorrectly, such as treating 2/5 in 10 days as equivalent to 1/5 in 5 days without proper calculation. Another frequent error is not reducing the fractions to a common denominator when adding the rates. Carefully computing each person's rate and then summing them avoids these issues.

Final Answer:
A and B together will complete the entire work in 9.375 days (that is, 9 3/8 days).