Hi, there. I'm happy to help with this.
This one has confusing wording, but the question itself is getting at a very simple idea.
Prompt:
A health club charges an annual $10 renewal fee for all memberships that are not renewed at least 30 days before they would expire. If Eileen renewed her membership 7 days ago, was she charged the renewal fee?
So, when someone is a member, they can renew without a fee up until a certain day (that day is 30 days before the membership would expire). Before that day, no fee. After that day, pay a fee. We want to know, where did the day Eileen paid land in with respect to this line?
Statement #1:
If Eileen had renewed her membership 14 days later, she would have paid the renewal fee.
In the diagrams I have drawn, the E represents the day Eileen paid, the F represents the day the fee began, green days have no fee, and red days have a fee.
scenario #1: 14 days
after[/i ]Eileen paid there was a fee, but Eileen paid no fee
E_ _ _ _ _ _ F _ _ _ _ _ _
scenario #2: Eileen paid a fee, and 14 days after her was still a fee
_ _ _ F _ _ E _ _ _ _ _ _ _ _ _ _ _ _ _ _
Because we can construct both these scenarios, this statement doesn't allow us to answer the question of whether Eileen paid a fee or not. Statement #1, by itself, is insufficient.
Statement #2: If Eileen had renewed her membership 3 days later, she would not have paid the renewal fee.
Here, the day 3 day after Eileen renews has no fee, so it must be before the beginning of the fee period. So:
(Eileen renews) is before (3 days later) is before (fees begin)
Therefore, by "transitivity of before-ness" , Eileen must have renewed before the fee began, so she paid no renewal fee. Statements #2, by itself, is sufficient to answer the question.
Answer = B
Does this make sense? Please let me know if you have any further questions on this.
Mike 
