A takes twice as much time as B or thrice as much time as C to finish a piece of work. Working together, they can finish the work in 2 days. B can do the work alone in:4 days6 days8 days12 days

Let's denote the amount of work as W.

1. According to the problem, A takes twice as much time as B to finish the work, so we can say that the work rate of A is 1/2 of B. We denote this as A = 1/2B.

2. Similarly, A takes thrice as much time as C to finish the work, so the work rate of A is 1/3 of C. We denote this as A = 1/3C.

3. Working together, A, B, and C can finish the work in 2 days. This means that 2(A + B + C) = W.

4. We can substitute A from the first and second equations into the third equation to get: 2((1/2B + B + 1/3C) = W. Simplifying this gives us B + C = W/2.

5. From the problem, we know that B can do the work alone in x days. This means that B = W/x.

6. Substituting B = W/x into the equation B + C = W/2 gives us W/x + C = W/2. Solving for C gives us C = W/2 - W/x.

7. Substituting C = W/2 - W/x into the equation A = 1/3C gives us A = 1/3(W/2 - W/x). Simplifying this gives us A = W/6 - W/3x.

8. Substituting A = W/6 - W/3x and B = W/x into the equation 2(A + B + C) = W gives us 2((W/6 - W/3x) + W/x + (W/2 - W/x)) = W. Simplifying this gives us W/3 + W/2 = W.

9. Solving for x gives us x = 6.

Therefore, B can do the work alone in 6 days.