If John can drink one barrel of water in 6 days, the rate at which he drinks is 1/6 of a barrel per day. If Mary can drink one barrel of water in 12 days, her rate is 1/12 of a barrel per day. To find how long it would take them to drink one barrel together, you add their rates:

Combined rate=16+112=212+112=312=14\text{Combined rate}=\frac{1}{6}+\frac{1}{12}=\frac{2}{12}+\frac{1}{12}=\frac{3}{12}=\frac{1}{4}Combined rate=61​+121​=122​+121​=123​=41​

This means together they drink 1/4 of a barrel per day. Therefore, it would take them 4 days to drink one barrel together. This solution is confirmed by a cognitive reflection test example and a detailed explanation showing how their drinking rates add up to finish the barrel in 4 days