SOLUTION: A library charges a fixed amount for the first day that a book is overdue and an additional charge for each day thereafter. Raoul paid $0.75 for one book that was 7 days overdue an

Question 61800: A library charges a fixed amount for the first day that a book is overdue and an additional charge for each day thereafter. Raoul paid $0.75 for one book that was 7 days overdue and $1.95 for a book that was 19 days overdue. Find the fixed charge and the charge for each additional day.
Answer by joyofmath(189) (Show Source): You can put this solution on YOUR website!
Let n = the number of days that a book is overdue.
Let f = fixed amount for the first day a book is overdue.
Let a = additional charge for each day after the first day.
Then, the fee for a book that's n days overdue is f+(n-1)*a.
This is because of the fixed rate, f, plus a charge of a per additional day, but not for the first day. That's why it's (n-1) and not n.
Raoul paid 75 cents for a 7 day overdue book. So, f+(7-1)*a = 75, or f+6a = 75.
Raoul also paid 195 cents for a book 19 days overdue. So, f+(19-1)*a = 195 or f+18a = 195.
So, we have two equations to solve together: f+6a=75 and f+18a=195.
Subtract the first equation from the second and you get 12a=120 so a=10.
Since f+6a = 75 and a=10 we know that f+60=75 so f=15.
So, the first day late fee is 15 cents and additional days cost 10 cents.
To verify:
The 7 day overdue book cost 15 cents for the first day and 10 cents for each of the other six days (60) so the total was 15+60 = 75 which is what the problem states.
The 19 day overdue book cost 15 cents for the first day and 10 cents for each of the other 18 days (180 cents) so the total was 15+180 = 195.
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