In preparation for the family's upcoming vacation, Tony puts together five bags of jelly beans, one bag for each day of the trip, with an equal number of jelly beans in each bag.  Tony then pours all the jelly beans out of the five bags and begins making patterns with them.  One of the patterns that he makes has one jelly bean in a top row, three jelly beans in the next row, five jelly beans in the row after that, and so on:

\[\begin{array}{ccccccccc}&&&&*&&&&\\&&&*&*&*&&&\\&&*&*&*&*&*&&\\&*&*&*&*&*&*&*&\\ *&*&*&*&*&*&*&*&*\\&&&&\vdots&&&&\end{array}\]

Continuing in this way, Tony finishes a row with none left over.  For instance, if Tony had exactly $25$ jelly beans, he could finish the fifth row above with no jelly beans left over.  However, when Tony finishes, there are between $10$ and $20$ rows.  Tony then scoops all the jelly beans and puts them all back into the five bags so that each bag once again contains the same number.  How many jelly beans are in each bag?  (Assume that no marble gets put inside more than one bag.)
1. **Determine the total number of jelly beans based on the pattern:**
   - The pattern described is an arithmetic sequence where the number of jelly beans in each row increases by 2. Specifically, the number of jelly beans in the \(n\)-th row is \(2n-1\).
   - The total number of jelly beans in \(n\) rows is the sum of the first \(n\) odd numbers, which is known to be \(n^2\). This can be derived from the formula for the sum of the first \(n\) odd numbers:
     \[
     1 + 3 + 5 + \ldots + (2n-1) = n^2
     \]

2. **Identify the range for \(n\):**
   - We are given that the number of rows \(n\) is between 10 and 20. Therefore, \(10 \leq n \leq 20\).

3. **Find the total number of jelly beans:**
   - Since the total number of jelly beans is \(n^2\), we need to find a perfect square within the range \(10^2\) to \(20^2\) that is also divisible by 5.
   - The perfect squares in this range are \(100, 121, 144, 169, 196, 225, 256, 289, 324, 361\).

4. **Check for divisibility by 5:**
   - Among these perfect squares, the only one that is divisible by 5 is \(225\) (since \(225 = 15^2\)).

5. **Calculate the number of jelly beans per bag:**
   - Since there are 225 jelly beans in total and they are divided equally into 5 bags, the number of jelly beans per bag is:
     \[
     \frac{225}{5} = 45
     \]

The final answer is \(\boxed{45}\).