Given a number $\overline{abcd}$, where $a$, $b$, $c$, and $d$, represent the digits of $\overline{abcd}$, find the minimum value of
\[\frac{\overline{abcd}}{a+b+c+d}\]
where $a$, $b$, $c$, and $d$ are distinct
[hide=Answer]$\overline{abcd}=1089$, minimum value of $\dfrac{\overline{abcd}}{a+b+c+d}=60.5$[/hide]
To find the minimum value of \(\frac{\overline{abcd}}{a+b+c+d}\) where \(a, b, c,\) and \(d\) are distinct digits, we need to consider the following steps:

1. **Express the number \(\overline{abcd}\) in terms of its digits:**
   \[
   \overline{abcd} = 1000a + 100b + 10c + d
   \]

2. **Formulate the expression to be minimized:**
   \[
   \frac{\overline{abcd}}{a+b+c+d} = \frac{1000a + 100b + 10c + d}{a + b + c + d}
   \]

3. **Analyze the expression:**
   We need to find the combination of distinct digits \(a, b, c,\) and \(d\) that minimizes this ratio. Since \(a, b, c,\) and \(d\) are distinct digits, they range from 0 to 9, but \(a\) cannot be 0 because \(\overline{abcd}\) is a four-digit number.

4. **Consider the constraints and simplify:**
   To minimize the ratio, we need to maximize the denominator \(a + b + c + d\) while keeping the numerator \(1000a + 100b + 10c + d\) as small as possible.

5. **Check possible combinations:**
   Let's start with the smallest possible value for \(a\), which is 1 (since \(a\) cannot be 0).

   - If \(a = 1\), we need to find \(b, c,\) and \(d\) such that \(b, c,\) and \(d\) are distinct and maximize \(b + c + d\).

6. **Evaluate specific cases:**
   - If \(a = 1\), let's try \(b = 0\), \(c = 8\), and \(d = 9\):
     \[
     \overline{abcd} = 1000 \cdot 1 + 100 \cdot 0 + 10 \cdot 8 + 9 = 1000 + 0 + 80 + 9 = 1089
     \]
     \[
     a + b + c + d = 1 + 0 + 8 + 9 = 18
     \]
     \[
     \frac{1089}{18} = 60.5
     \]

7. **Verify if this is the minimum value:**
   We need to check if there are any other combinations of distinct digits that yield a smaller ratio. However, given the constraints and the need to maximize the sum of the digits in the denominator, it is unlikely that a smaller ratio can be achieved with distinct digits.

Therefore, the minimum value of \(\frac{\overline{abcd}}{a+b+c+d}\) is achieved with \(\overline{abcd} = 1089\).

The final answer is \(\boxed{60.5}\).