This is probably a remedial question for all you experts out there, but I am having some trouble with it.

A certain library assesses fines for overdue books as follows. On the first day that a book is overdue, the total fine is $0.10. For each addtional day that the book is overdue, the total fine is either increased by $0.30 or doubled, whichever results in the lesser amount. What is the total fine for a book on the fourth day it is overdue?

-.60
-.70 - correct
-.80
-.90
-1.00

The way I solved this was
Days 1 2 3 4
2X .10 .20 .40 .80
+.30 .10 .40 .70 1.00

Im sure it is probably more complicated then the way I laid it out, can someone break it down into simple terms..thanks.

you're making the faulty assumption that you're locked into whichever of the two options - doubling or adding thirty cents - you choose on the first day. but it's not so:
each day, you make the decision again, based on whichever would render the fine smaller the next day.

it works like this, then:
1st day = 10 cents
2nd day = 20 cents (doubling gives lower fine)
3rd day = 40 cents (doubling gives lower fine)
4th day = 70 cents (this time, adding thirty cents gives the lower fine)

--

you don't have to go any further than that, of course, but it should be clear that, from this point onward, adding thirty cents would always be the correct procedure.