Working together, A and B can complete a certain job in 40 days, B and C can complete the same job in 36 days, and A, B and C together can complete it in 24 days. Based on this time and work data, in how many days can B alone finish the entire job?

Introduction / Context:
This is a classic time and work problem where pairwise and group working times are given for three workers A, B and C. The task is to compute the time taken by one particular worker, B, when working alone. Such questions check understanding of simultaneous equations using work rates rather than directly using days.

Given Data / Assumptions:
- A and B together complete the job in 40 days.
- B and C together complete the job in 36 days.
- A, B and C together complete the job in 24 days.
- Each worker's efficiency remains constant over time.
- Total work is assumed to be 1 complete job.

Concept / Approach:
Let the individual daily work rates of A, B and C be a, b and c respectively (in units of job per day). The given times can be converted into equations involving these rates. We then solve step by step to isolate B's rate. Once we find b, we can compute the time required for B alone to finish the job as the reciprocal 1/b.

Step-by-Step Solution:
Step 1: Let the total job be 1 unit. Step 2: A and B together finish in 40 days, so a + b = 1/40. Step 3: B and C together finish in 36 days, so b + c = 1/36. Step 4: A, B and C together finish in 24 days, so a + b + c = 1/24. Step 5: From the third equation, subtract the first: (a + b + c) − (a + b) = 1/24 − 1/40. Step 6: This gives c = 1/24 − 1/40. Compute with common denominator 120: c = (5 − 3) / 120 = 2/120 = 1/60. Step 7: Use b + c = 1/36, so b = 1/36 − 1/60. Step 8: With denominator 180, 1/36 = 5/180 and 1/60 = 3/180, so b = (5 − 3) / 180 = 2/180 = 1/90. Step 9: Time taken by B alone = 1 / (1/90) = 90 days.

Verification / Alternative check:
Check by reconstructing other rates. From a + b = 1/40 and b = 1/90, we get a = 1/40 − 1/90. With denominator 360, that is 9/360 − 4/360 = 5/360 = 1/72. Now test the group rate: a + b + c = 1/72 + 1/90 + 1/60. Finding a common denominator 360, we get 5/360 + 4/360 + 6/360 = 15/360 = 1/24, which matches the given 24 days for all three together. Thus the value of b is consistent.

Why Other Options Are Wrong:
Option 60 days: This corresponds to b = 1/60, which would distort the equations and not match the given combined times.
Option 72 days: Would give b = 1/72, leading to inconsistent pairwise and three person working times.
Option 120 days: Implies a rate slower than required, again failing consistency with the provided data.

Common Pitfalls:
A common error is to attempt solving directly with days instead of converting to rates, which leads to incorrect algebra. Another mistake is mismanaging fractions when subtracting or adding. Always convert all times to rates 1/time, form clear equations, and use a common denominator when combining fractions.

Final Answer:
B alone can complete the job in 90 days.