Explanation:

Let A take ‘a’ days to complete the task and B take ‘b’ days to complete the task.
Thus in one day, A will complete (1/a)th of the task.
Similarly in one day, B will complete (1/b)th of the task.
So in one day, if A and B work together they will complete (1/a + 1/b)^th of the task.
Given that A and B together take 12 days to complete the task, then in one day A and B together complete (1/12)th of the task.
Thus, 1/a + 1/b = 1/12 ……Eqn. 1
If A worked half as efficiently as he usually does, then A will take twice the time as he usually takes, i.e., 2a days.
Thus in one day, A completes (1/2a)^th of the task.
Similarly if B worked thrice as efficiently as he usually does, then B will take one-third the time he usually takes, i.e., b/3 days.
Thus in one day, B completed (1/(b/?/3))^th or (3/b)^th of the task.
Thus when both of them work together, they will complete (1/2a + 3/b)^th of the task, given that A and B take 9 days to complete the task.
Thus, 1/a + 1/b = 1/12 ……Eqn. 1
1/2a + 3/b = 1/9 …… Eqn. 2
1/2a + 3/b = 1/9 …… Eqn. 2
Solving Equations 1 and 2 for ‘a’ we should get the answer,
From equation (1) we get 12(a + b) = ab
From equation (2), we get 9(b + 6a) = 2ab
Substituting ab as 12(a + b) in equation (2) we get 9b + 54a = 2 x 12 x ( a + b)
9b + 54a = 24a + 24b;
Or, 30a = 15b,
Or, b = 2a
Now, 12(a + b) = ab, or 12 x 3a = 2a²
a = 18 days.