Question

A school library charges a fine of 5 cents on the first day a book is overdue, then doubles the amount owed each day until payment is made. Jada and Juan each had a book that was overdue. Jada paid a fine that was $2.40 more than Juan’s fine. Which statement about the overdue books is true?
Jada’s book was overdue by 1 more day than Juan’s book.
Jada’s book was overdue by 2 more days than Juan’s book.
Jada’s book was overdue by 24 more days than Juan’s book.
Jada’s book was overdue by 48 more days than Juan’s book.

Answer 1

Answer: So the first day the fine is 5 cents, the "doubles the amount owed each day" so the second day the fine is 10 cents, the third day is 20 cents, and so on.

The function that describes it is: F(d) = $0.05*$2^{d-1}$

where d are the number of days and F is the fine.

so if the function of jada is $2.40 bigger than the one of juan, we want to know the difference in days, we can do this, call d1 the number of days of jada, and d2 the number of days for juan.

So F1 = $0.05*$2^{d1-1}$ and F2 = $0.05*$2^{d2-1}$

and we also know that:

F1 - F2 = $2.40 so:

 $0.05( $2^{d1-1} - 2^{d2-1}$) = $2.40

( $2^{d1-1} - 2^{d2-1}$) = 2.40/0.05 = 48

 ( $2^{d1-1} - 2^{d2-1}$) = 48

In both our exponentials we have the "-1"

we can write this as

$2^{d1-1} - 2^{d2-1} = \frac{2^{d1} - 2^{d2}}{2}$ = 48

${2^{d1} - 2^{d2}$ = 48*2 = 96

now we want to know the value of d1-d2 = x. Then d2 = d1 - x.

then:

${2^{d1} - 2^{d1 -x}$ = 96

${2^{d1}$ = $\frac{96}{1 - \frac{1}{2^{x} } }$

if x = 1, we get:

${2^{d1}$ = 96/(1/2) = 180, which isn't a potency of 2.

if x = 2 we get:

${2^{d1}$  = 96/( 3/4) = 128, which is equal the seventh potency of 2, so "jada's book was overdue by 2 more days than Juan's book" is a solution.

is easy to see that for x =24 and x =48 you don't have an integer solution for d1.

so the only solution is x = 2.

Answer 2

Jada's book was overdue by 2 more days than Juan's book