The product of two positive numbers is equal to $50$ times their sum and $75$ times their difference.  Find their sum.
1. Let the two positive numbers be \( x \) and \( y \).
2. According to the problem, the product of the two numbers is equal to \( 50 \) times their sum:
   \[
   xy = 50(x + y)
   \]
3. The product of the two numbers is also equal to \( 75 \) times their difference:
   \[
   xy = 75(x - y)
   \]
4. Since both expressions are equal to \( xy \), we can set them equal to each other:
   \[
   50(x + y) = 75(x - y)
   \]
5. Rearrange the equation to isolate \( x \) and \( y \):
   \[
   50x + 50y = 75x - 75y
   \]
6. Combine like terms:
   \[
   50y + 75y = 75x - 50x
   \]
   \[
   125y = 25x
   \]
7. Simplify the equation:
   \[
   5y = x
   \]
8. Substitute \( x = 5y \) back into the original product equation \( xy = 50(x + y) \):
   \[
   (5y)y = 50(5y + y)
   \]
   \[
   5y^2 = 50(6y)
   \]
9. Simplify and solve for \( y \):
   \[
   5y^2 = 300y
   \]
   \[
   y^2 = 60y
   \]
   \[
   y(y - 60) = 0
   \]
10. Since \( y \) is a positive number, we have:
    \[
    y = 60
    \]
11. Substitute \( y = 60 \) back into \( x = 5y \):
    \[
    x = 5 \cdot 60 = 300
    \]
12. The sum of the two numbers is:
    \[
    x + y = 300 + 60 = 360
    \]

The final answer is \(\boxed{360}\)