A can complete 50 percent (one half) of a job in 16 days, while B can complete one fourth of the same job in 24 days. If A and B work together from the start, in how many days can they complete three fourths of the job?

Introduction / Context:
This time and work question involves fractions of a job completed by different workers in different times. We are given partial completion information for A and B and asked to find the time required for them together to complete three fourths of the same job. These types of questions highlight the importance of converting fractional work into daily work rates and then combining rates.

Given Data / Assumptions:
- A completes 50 percent (1/2) of the job in 16 days.
- B completes 25 percent (1/4) of the job in 24 days.
- Total work is considered as 1 complete job.
- Work rates of both A and B are constant over time.
- We are asked for the time to complete 3/4 of the job when they work together from the beginning.

Concept / Approach:
We use the concept that work rate equals fraction of work done divided by time. First we find A's rate from his fraction and days, and B's rate from his fraction and days. Then we add these rates to get the combined rate. Finally we compute how long it takes to finish three fourths of the job at the combined rate.

Step-by-Step Solution:
Step 1: Let the total job be 1 unit. Step 2: A does 1/2 of the job in 16 days, so A's daily rate = (1/2) / 16 = 1/32 of the job per day. Step 3: B does 1/4 of the job in 24 days, so B's daily rate = (1/4) / 24 = 1/96 of the job per day. Step 4: Combined daily rate of A and B = 1/32 + 1/96. Step 5: With denominator 96, 1/32 becomes 3/96, so combined rate = 3/96 + 1/96 = 4/96 = 1/24 of the job per day. Step 6: Required work to be done = 3/4 of the job. Step 7: Time required = (3/4) / (1/24) = (3/4) * 24 = 18 days.

Verification / Alternative check:
If A and B together do 1/24 of the job per day, then in 18 days they complete 18 * 1/24 = 18/24 = 3/4 of the job, which is exactly the required fraction. Also note that A alone in 18 days would complete 18 * 1/32 = 9/16, and B alone in 18 days would complete 18 * 1/96 = 3/16. Combined, that makes 12/16 = 3/4, confirming consistency of the rates and the time computed.

Why Other Options Are Wrong:
Option 24 days: At a rate of 1/24 per day, 24 days would complete the whole job, which is more than 3/4.
Option 9 days: At that duration, they would complete only 9 * 1/24 = 3/8 of the job, which is less than 3/4.
Option 21 days: This would lead to 21 * 1/24 = 7/8 of the job, which is more than 3/4.

Common Pitfalls:
Some learners mistakenly average the days rather than converting fractions to rates. Another mistake is to treat 3/4 of the job as if it required 3/4 of the total time at the combined rate without recomputing correctly. Always use the formula: time = required work / rate, using proper fractions and careful arithmetic.

Final Answer:
A and B together can complete three fourths of the job in 18 days.