A can do 75% of a job in 18 days and B can do 25% of the same job in 12 days. If A and B work together, in how many days will they complete 75% of the job?

Introduction / Context:
This problem provides partial work information for two workers, A and B. Instead of giving full-job times directly, the question gives how long each takes to complete certain percentages of the work. From these details we must deduce their full-job rates, then combine those rates to find the time needed to complete 75 percent of the job when they work together.

Given Data / Assumptions:
- A completes 75 percent (3/4) of the job in 18 days.
- B completes 25 percent (1/4) of the job in 12 days.
- We are asked for the time A and B together need to complete 75 percent of the job.
- Total work is taken as 1 unit.

Concept / Approach:
First, we compute A’s daily work rate from the fact that A finishes 3/4 of the work in 18 days. Then we compute B’s daily rate from the fact that B finishes 1/4 of the work in 12 days. Using these rates, we obtain the combined daily rate of A and B. Finally, we use this combined rate to determine how long it takes them to finish 3/4 of the job together.

Step-by-Step Solution:
Step 1: Let total work = 1 unit. Step 2: A completes 3 / 4 of the work in 18 days. Step 3: Rate of A = (3 / 4) / 18 = 3 / 72 = 1 / 24 work per day. Step 4: B completes 1 / 4 of the work in 12 days. Step 5: Rate of B = (1 / 4) / 12 = 1 / 48 work per day. Step 6: Combined rate of A and B = 1 / 24 + 1 / 48. Step 7: LCM of 24 and 48 is 48, so 1 / 24 = 2 / 48. Step 8: Combined rate = 2 / 48 + 1 / 48 = 3 / 48 = 1 / 16 work per day. Step 9: We need time to complete 3 / 4 of the work at a rate of 1 / 16 per day. Step 10: Time = (3 / 4) / (1 / 16) = (3 / 4) * 16 = 12 days.

Verification / Alternative check:
We can check the logic by computing individual full-job times. From rate 1 / 24, A would need 24 days for the full job, and from rate 1 / 48, B would need 48 days. Together they need 1 / (1 / 24 + 1 / 48) = 16 days for the full work. Three fourths of 16 days is 12 days, which matches the calculated answer directly from rates, confirming consistency.

Why Other Options Are Wrong:
- 16 and 20 days: These are closer to times for the full job, not three fourths of it with both working together.
- 8 and 10 days: These underestimate the required time and would only be valid if the combined rate were higher than 1 / 16 work per day, which contradicts the given data.

Common Pitfalls:
A common mistake is to misinterpret the given percentages and directly average the partial times. Another typical error is to forget that the question asks for 75 percent of the job, not the entire job. Always convert percentages into fractions of the total work, compute the exact rates, and then carefully use the fraction of work requested in the final step.

Final Answer:
A and B together will complete 75 percent of the job in 12 days.