A and B together can complete a job in 6.75 days, and A alone can complete the same job in 9 days. If B works alone, in how many days will B complete the job?

Introduction / Context:
This time and work problem asks us to find the time taken by B alone, given the time taken by A alone and the time taken by A and B together. Such questions are common in aptitude tests and rely on the principle that work rates are additive: the combined rate is the sum of individual rates. We must first find the combined daily rate, subtract A's rate, and then invert to get B's time.

Given Data / Assumptions:
- A and B together complete the job in 6.75 days. - A alone completes the job in 9 days. - Work rates are constant for both A and B. - Total work is one complete job. - We need to find the number of days B alone would take to complete the job.

Concept / Approach:
If a job is finished in T days, the daily work rate is 1 / T. For two people working together, the combined rate is the sum of their individual rates. We first compute the combined rate of A and B using the given 6.75 days. We then subtract A's rate (1 / 9) to get B's rate. Finally, we take the reciprocal of B's rate to find how many days B alone would need to complete the job.

Step-by-Step Solution:
Step 1: Convert 6.75 days to a fraction: 6.75 = 6 and 3 / 4 = 27 / 4 days. Step 2: Combined rate of A and B = 1 / (27 / 4) = 4 / 27 job per day. Step 3: A's rate alone = 1 / 9 job per day. Step 4: Express A's rate with denominator 27: 1 / 9 = 3 / 27 job per day. Step 5: B's rate = combined rate - A's rate = 4 / 27 - 3 / 27 = 1 / 27 job per day. Step 6: Time taken by B alone = 1 / (1 / 27) = 27 days.

Verification / Alternative check:
We can verify by adding the rates back. A's rate is 1 / 9 job per day, B's rate is 1 / 27 job per day, so their combined rate is 1 / 9 + 1 / 27. With a common denominator 27, we get 3 / 27 + 1 / 27 = 4 / 27 job per day. The time needed at this rate to complete one job is 1 / (4 / 27) = 27 / 4 = 6.75 days, which matches the given value. This confirms that B alone takes 27 days.

Why Other Options Are Wrong:
18 or 21 days: These imply a faster rate for B than 1 / 27 job per day and do not satisfy the given combined time of 6.75 days.
24 days: This would lead to an inconsistent combined rate that does not reproduce 6.75 days when combined with A's rate.
30 days: This implies a slower rate than our computed 1 / 27 job per day, again failing to match the given combined time.

Common Pitfalls:
Students sometimes forget to correctly convert 6.75 into a fraction, or they treat 6.75 as 6 + 75 / 100 without simplifying. Another frequent mistake is to subtract times instead of subtracting rates. Always remember: you subtract and add rates (fractions of work per day), not the number of days. Carefully converting mixed numbers to improper fractions helps avoid calculation errors.

Final Answer:
B alone will complete the job in 27 days.