In a room, each person is an painter and/or a musician. $2$ percent of the painters are musicians, and $5$ percent of the musicians are painters. Only one person is both an painter and a musician. How many people are in the room?

[i]Proposed by Evan Chang[/i]
1. Let \( P \) be the number of painters and \( M \) be the number of musicians. We are given that 2% of the painters are musicians and 5% of the musicians are painters. Additionally, we know that there is exactly one person who is both a painter and a musician.

2. Since 2% of the painters are musicians, we can write:
   \[
   0.02P = 1
   \]
   Solving for \( P \):
   \[
   P = \frac{1}{0.02} = 50
   \]

3. Similarly, since 5% of the musicians are painters, we can write:
   \[
   0.05M = 1
   \]
   Solving for \( M \):
   \[
   M = \frac{1}{0.05} = 20
   \]

4. To find the total number of people in the room, we need to account for the overlap (the one person who is both a painter and a musician). Therefore, the total number of people is:
   \[
   P + M - 1 = 50 + 20 - 1 = 69
   \]

The final answer is \(\boxed{69}\).