A and B can complete a piece of work in 12 days. whereas the work will take 3 more days to finish the work if A do not work for last 7 days. Find in how many days A can alone finish the work.

Let A's work rate = 1/a per day and B's work rate = 1/b per day. Working together: 1/a + 1/b = 1/12. When A doesn't work for the last 7 days, B works alone for those 7 days and they work together for (15-7) = 8 days. So 8(1/a + 1/b) + 7(1/b) = 1. Substituting 1/a + 1/b = 1/12: 8/12 + 7/b = 1, which gives 7/b = 4/12 = 1/3, so b = 21. Then 1/a = 1/12 - 1/21 = 7/84 - 4/84 = 3/84 = 1/28, so a = 28 days.