Hello everyone. I am needing some help with a word problem that has to do with rational equations. Something about this problem is messing with me. Here it is:

A painter worked on a job for 10 days and is then joined by an associate. Together they finish the job in 6 more days. The associate could have done the job alone in 30 days. How long would it have taken the painter to do the job alone?

I usually do not have a problem with these type of word problems, but the fact that the associate joined the painter after the painter had already worked at some rate for 10 days messes with me. I was thinking along the lines of (rate of painter alone) * (rate of associate alone) = rate of them together. The problem is I do not know how to describe their rate together. It would not simply be: (1/x) * (1/30) = (1/16) because they did not work on the project together for the entire 16 days.

Sorry if this problem is too easy or whatever, but I just can't see it. I guess we all hit a wall sometimes. Any help would be greatly appreciated. Thanks so much!!