Workers A and B together can finish a job in 12 days, while A alone can finish the same job in 20 days. If B works alone on the job, how many days will B take to complete it?

Introduction / Context:
This time and work question asks us to determine the time required by a single worker when we know how long it takes for that worker together with another person, and also how long it takes for one of them individually. This is a standard use of combined work rates to isolate the unknown worker's rate.

Given Data / Assumptions:

• A and B together complete the job in 12 days.
• A alone completes the same job in 20 days.
• B alone works at a constant rate throughout.
• We need to find the number of days B alone takes to complete the job.

Concept / Approach:
The central concept is that a worker who can complete a job in T days has a work rate of 1 / T of the job per day. When two workers work together, their combined rate is the sum of their individual rates. We are given the combined rate of A and B and the individual rate of A. Subtracting A's rate from the combined rate yields B's rate. The reciprocal of this rate provides the time needed by B alone.

Step-by-Step Solution:
Step 1: Let the total work be 1 unit.Step 2: Rate of A alone = 1 / 20 of the work per day.Step 3: Rate of A and B together = 1 / 12 of the work per day.Step 4: Rate of B alone = rate of (A and B together) minus rate of A alone.Step 5: Compute rate of B = 1 / 12 - 1 / 20.Step 6: To subtract, take the common denominator: LCM of 12 and 20 is 60. So 1 / 12 = 5 / 60 and 1 / 20 = 3 / 60.Step 7: Rate of B = 5 / 60 - 3 / 60 = 2 / 60 = 1 / 30 of the work per day.Step 8: Time required by B alone to complete the job = 1 divided by (1 / 30) = 30 days.

Verification / Alternative check:
If B alone takes 30 days, his rate is 1 / 30 of the work per day. Combined with A's rate of 1 / 20, their total rate is 1 / 20 + 1 / 30. With common denominator 60, we get 3 / 60 + 2 / 60 = 5 / 60 = 1 / 12. That is exactly the given combined rate, confirming that B taking 30 days alone is consistent with the data.

Why Other Options Are Wrong:

• 25 days: This corresponds to a rate of 1 / 25 per day, which would give a combined rate of 1 / 20 + 1 / 25 = 9 / 100, not equal to 1 / 12.
• 24 days: This rate of 1 / 24 per day would result in a combined rate of 1 / 20 + 1 / 24 which is different from 1 / 12.
• 15 days: This is much faster and leads to a combined rate that is too high, implying a total time less than 12 days, which contradicts the information given.

Common Pitfalls:
Sometimes students try to use averages such as (20 + 12) / 2 instead of working with rates, which is incorrect. Another error is subtracting times directly rather than converting to rates. Always convert days into rates of work, perform additions or subtractions on those rates, and then convert back to time using reciprocals.

Final Answer:
B alone will take 30 days to complete the job.