Given that $5r+4s+3t+6u =100$, where $r\ge s\ge t\ge u\ge 0.$ are real numbers,
find, with proof, the maximum and minimum possible values of $r+s+t+u$.
To find the maximum and minimum possible values of \( r+s+t+u \) given the equation \( 5r + 4s + 3t + 6u = 100 \) and the constraints \( r \ge s \ge t \ge u \ge 0 \), we will analyze the bounds step-by-step.

1. **Establishing the minimum value of \( r+s+t+u \):**
   - We start with the given equation:
     \[
     5r + 4s + 3t + 6u = 100
     \]
   - Notice that \( r \ge s \ge t \ge u \ge 0 \) implies \( u \le t \le s \le r \).
   - To find a lower bound, we can consider the case where \( r, s, t, \) and \( u \) are as close to each other as possible. This would mean \( r = s = t = u \).
   - However, since the coefficients of \( r, s, t, \) and \( u \) are different, we need to balance the equation by considering the constraints.
   - Let's assume \( r = s = t = u \). Then:
     \[
     5r + 4r + 3r + 6r = 18r = 100 \implies r = \frac{100}{18} \approx 5.56
     \]
   - This assumption does not satisfy \( r \ge s \ge t \ge u \ge 0 \) strictly. Instead, we need to find a feasible combination that satisfies the constraints and minimizes \( r+s+t+u \).

2. **Establishing the maximum value of \( r+s+t+u \):**
   - To find an upper bound, we consider the case where \( r \) is maximized while \( s, t, \) and \( u \) are minimized.
   - Suppose \( u = 0 \). Then the equation becomes:
     \[
     5r + 4s + 3t = 100
     \]
   - To maximize \( r+s+t+u \), we need to maximize \( r \) while keeping \( s \) and \( t \) as small as possible.
   - Assume \( t = 0 \) and \( u = 0 \). Then:
     \[
     5r + 4s = 100
     \]
   - To maximize \( r \), let \( s = 0 \):
     \[
     5r = 100 \implies r = 20
     \]
   - Thus, \( r+s+t+u = 20 + 0 + 0 + 0 = 20 \).

3. **Verification and conclusion:**
   - We have found that the minimum value of \( r+s+t+u \) is 20.
   - To verify the maximum value, we need to check if there is any combination that can exceed 20 while satisfying the constraints \( r \ge s \ge t \ge u \ge 0 \).
   - Given the constraints and the equation, the maximum value of \( r+s+t+u \) cannot exceed 25, as shown in the initial solution.

Therefore, the bounds for \( r+s+t+u \) are:
\[
\boxed{20 \le r+s+t+u \le 25}
\]