Days do not add; rates do. If A takes 10 days, A does one tenth of the job a day, and rates can be summed, subtracted and scaled freely. Converting to rates in the first line is the whole technique.
The faster version is to work in units: take the LCM of the given days as the total work, and everybody’s rate becomes a whole number. No fractions for the rest of the question.
Try each one before opening the answer. A solution you have read is not a solution you can produce under time pressure, which is the only kind that counts.
Q1 A can do a job in 12 days and B in 18. Working together, how long do they take?
7.2 days.
Take the total work as LCM(12, 18) = 36 units. A does 3 units a day, B does 2, so together 5. 36/5 = 7.2 days. The formula ab/(a + b) gives 216/30, which is the same.
Q2 A and B together finish a job in 8 days. A alone takes 12. How long does B alone take?
24 days.
Work in rates. Together they do 1/8 a day and A does 1/12, so B does 1/8 − 1/12 = 3/24 − 2/24 = 1/24. B alone takes 24 days. Subtracting the days - 12 from 8 - is meaningless, and is the error here.
Q3 Two pipes fill a tank in 20 and 30 minutes. A third empties it in 40. With all three open, how long does the tank take to fill?
17 and 1/7 minutes, about 17.14.
Take 120 units as the tank. The fillers add 6 and 4 units a minute; the drain removes 3. Net 7 units a minute, so 120/7 = 17.14 minutes. The drain is a negative rate, which is why it subtracts rather than adding to the time.