If john can drink one barrel of water in 6 days, and mary can drink one barrel of water in 12 days, how long would it take them to drink one barrel of water together?

Hello,

A good first step to take would be to calculate how much of a barrel John and Mary can drink together in 1 day.

If John can drink 1 barrel in 6 days, then every day he can drink 1/6 of a barrel.
If Mary can drink 1 barrel in 12 days, then every day she can drink 1/12 of a barrel. 

Every day, the total that John and Mary can drink will be (1/6) + (1/12) = (1/4) of a barrel.

If we want to know how many days it will take for them to drink 1 barrel of water together, and they drink 1/4 of a gallon every day, we do 

(1) / (1/4) = 1 * (4/1) = 4 days

It will take 4 days for John and Mary to drink 1 barrel together.

Hope this helps!