A can do a certain work in the same time as B and C together can do it. A and B together can complete the work in 20 days, and C alone can do the work in 60 days. In how many days can B alone complete the work?

Introduction / Context:
This time and work question uses a relationship between the time taken by A alone and the time taken by B and C together. You must translate that relationship into equations, use the given combined time for A and B, and the individual time for C, and then solve for B's time alone.

Given Data / Assumptions:

• A alone takes the same time as B and C working together.
• A and B together finish the work in 20 days.
• C alone finishes the work in 60 days.
• Work rate of each person is constant.

Concept / Approach:
Let T_A, T_B and T_C denote the days taken by A, B and C alone. Using work rate form, 1 / T_A is A's daily work. The condition that A alone takes as long as B and C together implies that 1 / T_A = 1 / T_B + 1 / T_C. Also, 1 / T_A + 1 / T_B = 1 / 20 and T_C = 60. Solve these equations to get T_B.

Step-by-Step Solution:
Let T_A, T_B be days for A and B alone, and T_C = 60 days. Condition 1: A alone takes same time as B and C together. So 1 / T_A = 1 / T_B + 1 / 60. Condition 2: A and B together complete in 20 days. So 1 / T_A + 1 / T_B = 1 / 20. From condition 1, 1 / T_A = 1 / T_B + 1 / 60. Substitute into condition 2: (1 / T_B + 1 / 60) + 1 / T_B = 1 / 20. So 2 / T_B + 1 / 60 = 1 / 20. 1 / 20 - 1 / 60 = (3 - 1) / 60 = 2 / 60 = 1 / 30. Hence 2 / T_B = 1 / 30, so T_B = 60 days.
Verification / Alternative check:
Now find T_A: 1 / T_A = 1 / T_B + 1 / 60 = 1 / 60 + 1 / 60 = 1 / 30, so T_A = 30 days. Check A and B together: their daily work = 1/30 + 1/60 = 1/20, so they finish in 20 days, which matches given data.
Why Other Options Are Wrong:
20 or 40 days for B would give different combined rates and would not satisfy both given conditions. 50 days does not satisfy the precise equations either; substituting would give inconsistent times.
Common Pitfalls:
One common error is misreading the phrase "same time in which B and C together can do it" and writing 1 / T_A = 1 / T_B + 1 / T_C incorrectly. Some students also manipulate fractions incorrectly when subtracting 1/60 from 1/20. Careful equation setup and fraction arithmetic are essential in such problems.
Final Answer:
B alone can complete the work in 60 days.