Worker A can do 75% of a job in 9 days, while worker B can complete 50% of the same job in 8 days. If A and B work together at their constant rates, how many days will they take to complete half of the job?

Introduction / Context:
This problem involves two workers, A and B, whose rates are given in terms of fractions of a job completed in given times. We are asked how long they will take together to complete half the job. This requires translating each worker's partial completion information into daily rates and then combining those rates to find the required time for a specified fraction of the job.

Given Data / Assumptions:

• A completes 75% (three quarters) of the job in 9 days.
• B completes 50% (one half) of the job in 8 days.
• We assume a fixed job size equal to one complete unit.
• A and B work simultaneously at their constant rates.
• We need the time taken to complete 50% of the job together.

Concept / Approach:
First, we find A's daily rate using the relation fraction of work divided by days. We do the same for B. The combined daily rate of A and B is the sum of their individual rates. Finally, we use that combined rate to calculate the time needed to accomplish 1/2 of the job by rearranging time = work / rate.

Step-by-Step Solution:
Take total work = 1 job.A does 75% = 3 / 4 of the job in 9 days, so A's rate = (3 / 4) / 9 = 3 / 36 = 1 / 12 per day.B does 50% = 1 / 2 of the job in 8 days, so B's rate = (1 / 2) / 8 = 1 / 16 per day.Combined rate of A and B = 1 / 12 + 1 / 16.Compute: 1 / 12 = 4 / 48, 1 / 16 = 3 / 48, so combined rate = 7 / 48 per day.We want time to complete 1 / 2 of the job at rate 7 / 48 per day.Time = (1 / 2) / (7 / 48) = (1 / 2) * (48 / 7) = 24 / 7 days.

Verification / Alternative check:
We can approximate 24 / 7 as about 3.4286 days. In that time, A would do 3.4286 * 1 / 12 ≈ 0.2857 of the job, and B would do 3.4286 * 1 / 16 ≈ 0.2143 of the job. Their combined contribution is about 0.2857 + 0.2143 ≈ 0.5, or exactly one half when calculated symbolically, confirming that 24 / 7 days is correct.

Why Other Options Are Wrong:
Forty over seven days would correspond to a much longer time and therefore more than half of the job. Seven over two and nine over two days both give smaller or larger contributions that do not total exactly 1 / 2 when multiplied by the combined rate of 7 / 48. Only 24 / 7 days provides the correct fraction of the job completed.

Common Pitfalls:
Some learners misinterpret 75% and 50% or convert them incorrectly into fractions. Others incorrectly average the times instead of working with rates. There can also be errors in adding the fractions 1 / 12 and 1 / 16 without using a common denominator. Careful attention to fraction operations is essential in such problems.

Final Answer:
A and B together will take 24/7 days to complete half of the job.