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?
Yes
To solve this problem, we need to set up an equation and calculate the fines for both Jada and Juan. Let's denote the number of days the book is overdue as 'd.'
According to the given information, the fine for the first day is 5 cents, and it doubles each subsequent day. So, we can form the equation for Jada's fine as:
Jada's fine = 5 cents + 2 * 5 cents + 2^2 * 5 cents + ... + 2^(d-1) * 5 cents
Similarly, we can form the equation for Juan's fine:
Juan's fine = 5 cents + 2 * 5 cents + 2^2 * 5 cents + ... + 2^(d-1) * 5 cents
We are also given that Jada's fine is $2.40 more than Juan's fine. Thus, we can write the equation as:
Jada's fine = Juan's fine + $2.40
Now, let's simplify the equations. The sum of a geometric series is given by the formula:
Sum = a * (r^n - 1) / (r - 1)
Applying this formula to Jada's fine, we have:
Jada's fine = 5 cents * (2^d - 1) / (2 - 1) = 5 cents * (2^d - 1)
Similarly, for Juan's fine:
Juan's fine = 5 cents * (2^d - 1) / (2 - 1) = 5 cents * (2^d - 1)
Now, let's substitute these values into the equation Jada's fine = Juan's fine + $2.40:
5 cents * (2^d - 1) = 5 cents * (2^d - 1) + $2.40
Simplifying this equation, we find:
0 = $2.40
However, this equation has no solution because $2.40 cannot be equal to zero. Therefore, there is no possible scenario where Jada's fine is $2.40 more than Juan's fine. Hence, the statement "Jada paid a fine that was $2.40 more than Juan’s fine" is false.
Don't know, but we know that
5*2^r2 = 5*2^r1 + 240
2^r2 = 2^r1 + 24
2^r1 (r2-r1+1) = 24 = 2^3(2+1)
So, r2-r1 = 2 and r1=3
Juan's book was 4 days overdue
Jada's book was 6 days overdue
Juan's fine: 5+10+20+40 = 75
Jada's fine: 75+80+160 = 315