A can do half of a certain piece of work in 8 days, while B can do one-third of the same work in 8 days. If A and B work together from the start at their constant rates, in how many days will they complete the entire work?

Introduction / Context:
This problem provides partial completion times for two workers, A and B, and asks for the total time required when they work together from the beginning. A completes half of the work in a given time, and B completes one-third of the work in the same time. This type of question checks your understanding of how to convert partial work into rates and then combine those rates to find the joint completion time.

Given Data / Assumptions:

• A completes 1/2 of the work in 8 days.
• B completes 1/3 of the work in 8 days.
• Both work together from the start.
• Each worker's rate is constant over time.
• We must find the number of days required to complete the full work together.

Concept / Approach:
We represent the total work as 1 unit. From the information given, we derive A's daily rate as half the job divided by 8 days, and B's daily rate as one-third of the job divided by 8 days. Adding these two rates gives the combined daily rate of A and B. The total time to complete the full job is then 1 divided by this combined rate. This method is direct and avoids unnecessary complications.

Step-by-Step Solution:
Step 1: Let the total work = 1 unit. Step 2: A completes 1/2 of the work in 8 days, so A's rate = (1/2) / 8 = 1/16 of the work per day. Step 3: B completes 1/3 of the work in 8 days, so B's rate = (1/3) / 8 = 1/24 of the work per day. Step 4: Combined daily rate of A and B = 1/16 + 1/24. Step 5: Using a common denominator of 48, 1/16 = 3/48 and 1/24 = 2/48. Step 6: Combined rate = 3/48 + 2/48 = 5/48 of the work per day. Step 7: Time to complete the full job = 1 / (5/48) = 48/5 days. Step 8: 48/5 days = 9.6 days.

Verification / Alternative check:
Approximate reasoning supports this result. A's full-time if working alone would be 16 days for the whole work, and B's alone would be 24 days. Working together must therefore take less than 16 days. Our answer 9.6 days is indeed less than 16 and closer to the value you expect when rates of 1/16 and 1/24 are added. Also, if they work for 9.6 days at 5/48 per day, total work = 9.6 × 5/48 = 48/5 × 5/48 = 1 unit, confirming correctness.

Why Other Options Are Wrong:
10.5, 11.2, 16 and 8 days all correspond to effective rates that do not match 5/48 per day. For instance, 16 days would indicate a rate of only 1/16 per day, which is A's rate alone, ignoring B's contribution. 8 days would require a higher rate than even both working together can provide. Therefore, 9.6 days is the only duration consistent with the combined rate.

Common Pitfalls:
Some students mistakenly average the partial times or assume that 1/2 and 1/3 can be simply added as if they corresponded to the same time period without adjusting for the 8 days factor. Another mistake is miscalculating the fractions when finding 1/16 and 1/24. Keeping careful track of fractions and always working in terms of daily rates prevents these errors.

Final Answer:
A and B together will complete the entire work in 9.6 days.