It takes $3$ rabbits $5$ hours to dig $9$ holes. It takes $5$ beavers $36$ minutes to build $2$ dams. At this rate, how many more minutes does it take $1$ rabbit to dig $1$ hole than it takes $1$ beaver to build $1$ dam?

[i]2016 CCA Math Bonanza Team #1[/i]
1. **Determine the rate at which one rabbit digs holes:**
   - Given that 3 rabbits take 5 hours to dig 9 holes, we can find the rate of one rabbit.
   - Let \( r \) be the rate of one rabbit in holes per hour.
   - The total work done by 3 rabbits in 5 hours is \( 3 \times 5 \times r = 9 \).
   - Solving for \( r \):
     \[
     15r = 9 \implies r = \frac{9}{15} = \frac{3}{5} \text{ holes per hour}
     \]

2. **Convert the rate of one rabbit to holes per minute:**
   - Since there are 60 minutes in an hour, the rate in holes per minute is:
     \[
     \frac{3}{5} \text{ holes per hour} \times \frac{1 \text{ hour}}{60 \text{ minutes}} = \frac{3}{5} \times \frac{1}{60} = \frac{3}{300} = \frac{1}{100} \text{ holes per minute}
     \]

3. **Determine the time it takes for one rabbit to dig one hole:**
   - The time \( t \) in minutes for one rabbit to dig one hole is the reciprocal of the rate:
     \[
     t = \frac{1}{\frac{1}{100}} = 100 \text{ minutes}
     \]

4. **Determine the rate at which one beaver builds dams:**
   - Given that 5 beavers take 36 minutes to build 2 dams, we can find the rate of one beaver.
   - Let \( b \) be the rate of one beaver in dams per minute.
   - The total work done by 5 beavers in 36 minutes is \( 5 \times 36 \times b = 2 \).
   - Solving for \( b \):
     \[
     180b = 2 \implies b = \frac{2}{180} = \frac{1}{90} \text{ dams per minute}
     \]

5. **Determine the time it takes for one beaver to build one dam:**
   - The time \( t \) in minutes for one beaver to build one dam is the reciprocal of the rate:
     \[
     t = \frac{1}{\frac{1}{90}} = 90 \text{ minutes}
     \]

6. **Calculate the difference in time between one rabbit digging one hole and one beaver building one dam:**
   - The difference in time is:
     \[
     100 \text{ minutes} - 90 \text{ minutes} = 10 \text{ minutes}
     \]

The final answer is \(\boxed{10}\) minutes.