A and B together can complete a piece of work in 40 days. A works alone for 20 days, after which B finishes the remaining work alone in 60 days. In how many days can B complete the whole work if working alone from the start?

Introduction / Context:
This question is about sharing work over time between two workers, A and B, and then deducing the time taken by B alone to finish the whole work. You are given the joint time for A and B, the time A works alone initially, and the time B takes to finish the remaining work. From this, you must determine B’s individual time to complete the entire job working alone.

Given Data / Assumptions:
A and B together can complete the work in 40 days. A works alone for 20 days and then stops. B completes the remaining work alone in 60 days. The total work is taken as one complete job. All workers operate at constant rates and there are no breaks or changes in efficiency over time.

Concept / Approach:
We first convert the given combined time into a combined daily work rate for A and B together. Let the individual rates of A and B be a and b units of work per day. From the combined rate and the information about the partial work done by A and then B, we can set up equations relating a and b. Solving these equations allows us to find B’s rate and then B’s time to complete the full job alone by taking the reciprocal of that rate.

Step-by-Step Solution:
Step 1: Let the total work be 1 unit. Step 2: Since A and B together take 40 days, their combined rate is (a + b) = 1 / 40 work per day. Step 3: A works alone for 20 days. The amount of work done by A in this period is 20a. Step 4: The remaining work is (1 - 20a). B finishes this in 60 days, so B’s rate b satisfies 60b = 1 - 20a. Step 5: We now have two equations: a + b = 1 / 40 and 60b = 1 - 20a. Step 6: From the first equation, express a in terms of b: a = 1 / 40 - b. Step 7: Substitute this expression for a into the second equation: 60b = 1 - 20(1 / 40 - b). Step 8: Simplify the term inside the parentheses: 20(1 / 40) = 1 / 2. So 60b = 1 - (1 / 2) + 20b = 1 / 2 + 20b. Step 9: Rearranging, 60b - 20b = 1 / 2, so 40b = 1 / 2. Step 10: Therefore b = (1 / 2) / 40 = 1 / 80 work per day. Step 11: Time taken by B alone to complete the entire work is the reciprocal of b: time = 1 / (1 / 80) = 80 days.

Verification / Alternative check:
We can verify by checking the total work done by A and B in the described scenario. If b = 1 / 80, then a = 1 / 40 - 1 / 80 = 2 / 80 - 1 / 80 = 1 / 80, which means A and B have equal rates. A working alone for 20 days completes 20 * (1 / 80) = 1 / 4 of the work. The remaining work is 3 / 4. B working alone at 1 / 80 completes this in 60 days, since 60 * (1 / 80) = 3 / 4. The total work done is 1 / 4 + 3 / 4 = 1, and together their rate a + b = 1 / 80 + 1 / 80 = 1 / 40, which matches the original combined time of 40 days.

Why Other Options Are Wrong:
60 days, 70 days, 90 days, and 100 days do not satisfy the equation derived from splitting the work between A and B. If B took any of these times alone, his rate would be different and the sequence of 20 days of A followed by 60 days of B would either overshoot or fail to complete the work. Only 80 days is consistent with both the joint time and the partial work distribution described in the problem.

Common Pitfalls:
One common error is to assume that B’s time is simply related in a linear way to the combined time, such as doubling or halving without using equations. Another pitfall is to forget that the work done by A in the first 20 days must be subtracted from the total work, leaving the remainder for B. Mismanaging this subtraction leads to incorrect values for B’s rate.

Final Answer:
B alone would complete the whole work in 80 days.