If A and B together can complete a job in 7.5 days and A alone can complete the same job in 10 days, how many days will B alone take to complete the job?

Introduction / Context:
This question provides the combined time of A and B working together and the individual time of A alone. We are required to determine how long B alone would take to finish the job. This is a typical time and work problem that uses the relationship between time, work, and rate to isolate the unknown worker’s efficiency.

Given Data / Assumptions:
- A and B working together complete the job in 7.5 days.
- A alone completes the job in 10 days.
- We need the number of days B alone will take to complete the job.
- Total work is taken as 1 unit.

Concept / Approach:
We convert 7.5 days into a fraction (15 / 2 days) and then compute the combined daily rate of A and B as 1 divided by 7.5. A’s rate is 1 / 10 per day. B’s rate is obtained by subtracting A’s rate from the combined rate. Finally, we find B’s time by taking the reciprocal of B’s rate. Handling the decimal 7.5 as a fraction simplifies the calculations.

Step-by-Step Solution:
Step 1: Let total work = 1 unit. Step 2: Combined time of A and B = 7.5 days = 15 / 2 days. Step 3: Combined rate of A and B = 1 / (15 / 2) = 2 / 15 work per day. Step 4: Rate of A alone = 1 / 10 work per day. Step 5: Rate of B = combined rate - rate of A = 2 / 15 - 1 / 10. Step 6: LCM of 15 and 10 is 30, so 2 / 15 = 4 / 30 and 1 / 10 = 3 / 30. Step 7: Rate of B = 4 / 30 - 3 / 30 = 1 / 30 work per day. Step 8: Time taken by B alone = 1 / (1 / 30) = 30 days.

Verification / Alternative check:
Using the rates, A does 1 / 10 per day and B does 1 / 30 per day. Together they do 1 / 10 + 1 / 30 = 3 / 30 + 1 / 30 = 4 / 30 = 2 / 15 work per day. Time required at this rate is 1 / (2 / 15) = 15 / 2 = 7.5 days, which matches the given combined time, confirming that B’s individual time of 30 days is correct.

Why Other Options Are Wrong:
- 40, 25, 50, 20: Each of these values yields an incorrect individual rate for B. When combined with A’s rate, none of them reproduces the correct combined time of 7.5 days.

Common Pitfalls:
A frequent mistake is misinterpreting 7.5 days, either leaving it in decimal form and miscalculating, or failing to convert properly to a fraction. Another common error is to average the times of A and B instead of using rate subtraction. Always work with rates, especially when combining or separating worker contributions.

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