What percent of the numbers $1, 2, 3, ... 1000$ are divisible by exactly one of the numbers $4$ and $5?$
To determine the percentage of numbers from $1$ to $1000$ that are divisible by exactly one of the numbers $4$ and $5$, we need to follow these steps:

1. **Count the multiples of $4$ within $1$ to $1000$:**
   \[
   \left\lfloor \frac{1000}{4} \right\rfloor = 250
   \]
   So, there are $250$ multiples of $4$.

2. **Count the multiples of $5$ within $1$ to $1000$:**
   \[
   \left\lfloor \frac{1000}{5} \right\rfloor = 200
   \]
   So, there are $200$ multiples of $5$.

3. **Count the multiples of both $4$ and $5$ (i.e., multiples of $20$) within $1$ to $1000$:**
   \[
   \left\lfloor \frac{1000}{20} \right\rfloor = 50
   \]
   So, there are $50$ multiples of both $4$ and $5$.

4. **Apply the principle of inclusion-exclusion to find the number of integers divisible by either $4$ or $5$:**
   \[
   250 + 200 - 50 = 400
   \]
   So, there are $400$ numbers that are divisible by either $4$ or $5$.

5. **Count the numbers that are divisible by exactly one of $4$ or $5$:**
   \[
   400 - 50 = 350
   \]
   This is because we need to exclude the $50$ numbers that are divisible by both $4$ and $5$.

6. **Calculate the percentage of numbers from $1$ to $1000$ that are divisible by exactly one of $4$ or $5$:**
   \[
   \frac{350}{1000} \times 100\% = 35\%
   \]

The final answer is $\boxed{35\%}$.