A can do a piece of work in 12 days whereas B can destroy the work in 3 days. A does the work for 10 days and then B destroys the work up to 2 days. In how many days will the work be finished?

Fraction of work accomplished W at the end of day t will be denoted W(t).

W(0)=0
W(10)=5/6
W(12)=5/6-2/3=1/6
W(22)=1/6+5/6=1 work completed.

To find out how many days it will take to finish the work, we need to calculate the remaining work after A and B have done their share.

Let's first find out the work done by A in one day. We are given that A can complete the work in 12 days, so A's one-day work is 1/12.

Similarly, B can destroy the work in 3 days, so B's one-day work is 1/3.

Now, let's calculate the work done by A in 10 days. Since A's one-day work is 1/12, the work done by A in 10 days is 10 * (1/12) = 10/12.

Next, let's calculate the work done by B in 2 days. Since B's one-day work is 1/3, the work done by B in 2 days is 2 * (1/3) = 2/3.

Now, let's calculate the remaining work after A and B have done their share. The remaining work can be found by subtracting the work done by A and B from the total work.

Total work = 1 (since we are considering the whole work as 1 unit)

Remaining work = Total work - Work done by A - Work done by B

Remaining work = 1 - (10/12) - (2/3)

To simplify this expression, we need to find a common denominator for 12 and 3, which is 12. So, let's convert 10/12 and 2/3 into fractions with the denominator 12.

10/12 = (10/12) * (1/1) = 10/12

2/3 = (2/3) * (4/4) = 8/12

Now, let's substitute these values back into the expression:

Remaining work = 1 - 10/12 - 8/12

Remaining work = 12/12 - 10/12 - 8/12

Remaining work = (12 - 10 - 8)/12

Remaining work = (12 - 18)/12

Remaining work = -6/12

Since the remaining work is negative, it means that A and B have completed more work than required. In other words, the work will be finished before the 12-day deadline.

Therefore, the work will be finished before the 12th day. However, the exact number of days cannot be determined from the given information.