Several ways to go about it. One way would be to determine a least common multiple.
We have 5 min, 10 min, and 15 min. It appears that 30 min is a common multiple.
Converting
Steffon can mix 120 drinks in 30 minutes, Dwayne can mix 60 drinks in 30 minutes, and Jacob can mix 40 drinks in 30 minutes
Adding 220 drinks in 30 minutes
Scaling
Adding 220/11 drinks in 30/11 minutes This step is of course correct, but its rationale may be obscure. Where does the 11 come from? The combined rate of drink making is 220 drinks / 30 minutes (see above) but we want the minutes m for making 20 drinks. Well the rate is assumed to be independent of the time so 20 drinks / m minutes = 220 drinks /30 minutes. Now one way to solve this is by cross-multiplying m = 20 * 30 / 220 = 2.73 minutes approximately.
If however you see that 220 / 11 = 20, then 220/ 30 = (220/11) / (30/11) = 20 / (11/30) = 20/2.73 approximately. Both ways are right and so give the same answer, but, at least for me, the cross-multiplication method requires a little less intuition than tkhunny's scaling method. So now you have two methods for solving this class of problem