Solution:

Let $n$ be a four-digit number, so $1000 \leq n \leq 9999$. Let $s_n$ denote the sum of the digits of $n$.

We are to find the smallest value of $\frac{n}{s_n}$.

Let $n = 1000a + 100b + 10c + d$, where $a, b, c, d$ are digits and $a \geq 1$.
Then $s_n = a + b + c + d$.

We want to minimize $\frac{n}{s_n} = \frac{1000a + 100b + 10c + d}{a + b + c + d}$.

To minimize this, for a fixed $n$, we want $s_n$ to be as large as possible, i.e., the digits as large as possible.

Let us try $a = 1$ (the smallest possible value for a four-digit number):

If $a = 1$, $n = 1000 + 100b + 10c + d$, $s_n = 1 + b + c + d$.

To maximize $s_n$, set $b = c = d = 9$:

Then $n = 1000 + 100 \times 9 + 10 \times 9 + 9 = 1000 + 900 + 90 + 9 = 1999$.
$s_n = 1 + 9 + 9 + 9 = 28$.
$\frac{n}{s_n} = \frac{1999}{28} \approx 71.39$.

Try $a = 2$, $b = c = d = 9$:
$n = 2000 + 900 + 90 + 9 = 2999$.
$s_n = 2 + 9 + 9 + 9 = 29$.
$\frac{2999}{29} \approx 103.41$.

So increasing $a$ increases the ratio. So $a = 1$ is best.

Try $b = 9, c = 9, d = 8$:
$n = 1000 + 900 + 90 + 8 = 1998$.
$s_n = 1 + 9 + 9 + 8 = 27$.
$\frac{1998}{27} = 74$.

Try $b = 9, c = 8, d = 9$:
$n = 1000 + 900 + 80 + 9 = 1989$.
$s_n = 1 + 9 + 8 + 9 = 27$.
$\frac{1989}{27} = 73.666...$

Try $b = 8, c = 9, d = 9$:
$n = 1000 + 800 + 90 + 9 = 1899$.
$s_n = 1 + 8 + 9 + 9 = 27$.
$\frac{1899}{27} = 70.333...$

Try $b = 9, c = 9, d = 7$:
$n = 1000 + 900 + 90 + 7 = 1997$.
$s_n = 1 + 9 + 9 + 7 = 26$.
$\frac{1997}{26} \approx 76.81$

Try $b = 9, c = 8, d = 8$:
$n = 1000 + 900 + 80 + 8 = 1988$.
$s_n = 1 + 9 + 8 + 8 = 26$.
$\frac{1988}{26} \approx 76.46$

Try $b = 8, c = 9, d = 8$:
$n = 1000 + 800 + 90 + 8 = 1898$.
$s_n = 1 + 8 + 9 + 8 = 26$.
$\frac{1898}{26} \approx 73$.

Try $b = 8, c = 8, d = 9$:
$n = 1000 + 800 + 80 + 9 = 1889$.
$s_n = 1 + 8 + 8 + 9 = 26$.
$\frac{1889}{26} \approx 72.65$

Try $b = 9, c = 7, d = 9$:
$n = 1000 + 900 + 70 + 9 = 1979$.
$s_n = 1 + 9 + 7 + 9 = 26$.
$\frac{1979}{26} \approx 76.12$

Try $b = 7, c = 9, d = 9$:
$n = 1000 + 700 + 90 + 9 = 1799$.
$s_n = 1 + 7 + 9 + 9 = 26$.
$\frac{1799}{26} \approx 69.19$

Try $b = 8, c = 8, d = 8$:
$n = 1000 + 800 + 80 + 8 = 1888$.
$s_n = 1 + 8 + 8 + 8 = 25$.
$\frac{1888}{25} = 75.52$

Try $b = 7, c = 9, d = 8$:
$n = 1000 + 700 + 90 + 8 = 1798$.
$s_n = 1 + 7 + 9 + 8 = 25$.
$\frac{1798}{25} = 71.92$

Try $b = 7, c = 8, d = 9$:
$n = 1000 + 700 + 80 + 9 = 1789$.
$s_n = 1 + 7 + 8 + 9 = 25$.
$\frac{1789}{25} = 71.56$

Try $b = 6, c = 9, d = 9$:
$n = 1000 + 600 + 90 + 9 = 1699$.
$s_n = 1 + 6 + 9 + 9 = 25$.
$\frac{1699}{25} = 67.96$

Try $b = 7, c = 8, d = 8$:
$n = 1000 + 700 + 80 + 8 = 1788$.
$s_n = 1 + 7 + 8 + 8 = 24$.
$\frac{1788}{24} = 74.5$

Try $b = 6, c = 9, d = 8$:
$n = 1000 + 600 + 90 + 8 = 1698$.
$s_n = 1 + 6 + 9 + 8 = 24$.
$\frac{1698}{24} = 70.75$

Try $b = 6, c = 8, d = 9$:
$n = 1000 + 600 + 80 + 9 = 1689$.
$s_n = 1 + 6 + 8 + 9 = 24$.
$\frac{1689}{24} = 70.375$

Try $b = 5, c = 9, d = 9$:
$n = 1000 + 500 + 90 + 9 = 1599$.
$s_n = 1 + 5 + 9 + 9 = 24$.
$\frac{1599}{24} = 66.625$

Try $b = 6, c = 8, d = 8$:
$n = 1000 + 600 + 80 + 8 = 1688$.
$s_n = 1 + 6 + 8 + 8 = 23$.
$\frac{1688}{23} \approx 73.39$

Try $b = 5, c = 9, d = 8$:
$n = 1000 + 500 + 90 + 8 = 1598$.
$s_n = 1 + 5 + 9 + 8 = 23$.
$\frac{1598}{23} \approx 69.48$

Try $b = 5, c = 8, d = 9$:
$n = 1000 + 500 + 80 + 9 = 1589$.
$s_n = 1 + 5 + 8 + 9 = 23$.
$\frac{1589}{23} \approx 69.09$

Try $b = 4, c = 9, d = 9$:
$n = 1000 + 400 + 90 + 9 = 1499$.
$s_n = 1 + 4 + 9 + 9 = 23$.
$\frac{1499}{23} \approx 65.17$

Try $b = 5, c = 8, d = 8$:
$n = 1000 + 500 + 80 + 8 = 1588$.
$s_n = 1 + 5 + 8 + 8 = 22$.
$\frac{1588}{22} = 72.18$

Try $b = 4, c = 9, d = 8$:
$n = 1000 + 400 + 90 + 8 = 1498$.
$s_n = 1 + 4 + 9 + 8 = 22$.
$\frac{1498}{22} \approx 68.09$

Try $b = 4, c = 8, d = 9$:
$n = 1000 + 400 + 80 + 9 = 1489$.
$s_n = 1 + 4 + 8 + 9 = 22$.
$\frac{1489}{22} \approx 67.68$

Try $b = 3, c = 9, d = 9$:
$n = 1000 + 300 + 90 + 9 = 1399$.
$s_n = 1 + 3 + 9 + 9 = 22$.
$\frac{1399}{22} \approx 63.59$

Try $b = 2, c = 9, d = 9$:
$n = 1000 + 200 + 90 + 9 = 1299$.
$s_n = 1 + 2 + 9 + 9 = 21$.
$\frac{1299}{21} = 61.857...$

Try $b = 1, c = 9, d = 9$:
$n = 1000 + 100 + 90 + 9 = 1199$.
$s_n = 1 + 1 + 9 + 9 = 20$.
$\frac{1199}{20} = 59.95$

Try $b = 0, c = 9, d = 9$:
$n = 1000 + 0 + 90 + 9 = 1099$.
$s_n = 1 + 0 + 9 + 9 = 19$.
$\frac{1099}{19} \approx 57.84$

Try $b = 0, c = 8, d = 9$:
$n = 1000 + 0 + 80 + 9 = 1089$.
$s_n = 1 + 0 + 8 + 9 = 18$.
$\frac{1089}{18} = 60.5$

Try $b = 0, c = 7, d = 9$:
$n = 1000 + 0 + 70 + 9 = 1079$.
$s_n = 1 + 0 + 7 + 9 = 17$.
$\frac{1079}{17} \approx 63.47$

Try $b = 0, c = 6, d = 9$:
$n = 1000 + 0 + 60 + 9 = 1069$.
$s_n = 1 + 0 + 6 + 9 = 16$.
$\frac{1069}{16} = 66.8125$

Try $b = 0, c = 5, d = 9$:
$n = 1000 + 0 + 50 + 9 = 1059$.
$s_n = 1 + 0 + 5 + 9 = 15$.
$\frac{1059}{15} = 70.6$

Try $b = 0, c = 4, d = 9$:
$n = 1000 + 0 + 40 + 9 = 1049$.
$s_n = 1 + 0 + 4 + 9 = 14$.
$\frac{1049}{14} \approx 74.93$

Try $b = 0, c = 3, d = 9$:
$n = 1000 + 0 + 30 + 9 = 1039$.
$s_n = 1 + 0 + 3 + 9 = 13$.
$\frac{1039}{13} \approx 79.92$

Try $b = 0, c = 2, d = 9$:
$n = 1000 + 0 + 20 + 9 = 1029$.
$s_n = 1 + 0 + 2 + 9 = 12$.
$\frac{1029}{12} = 85.75$

Try $b = 0, c = 1, d = 9$:
$n = 1000 + 0 + 10 + 9 = 1019$.
$s_n = 1 + 0 + 1 + 9 = 11$.
$\frac{1019}{11} \approx 92.64$

Try $b = 0, c = 0, d = 9$:
$n = 1000 + 0 + 0 + 9 = 1009$.
$s_n = 1 + 0 + 0 + 9 = 10$.
$\frac{1009}{10} = 100.9$

Try $b = 0, c = 0, d = 8$:
$n = 1000 + 0 + 0 + 8 = 1008$.
$s_n = 1 + 0 + 0 + 8 = 9$.
$\frac{1008}{9} = 112$

Try $b = 0, c = 0, d = 1$:
$n = 1000 + 0 + 0 + 1 = 1001$.
$s_n = 1 + 0 + 0 + 1 = 2$.
$\frac{1001}{2} = 500.5$

So, the minimum value occurs when $b = 0, c = 9, d = 9$:
$n = 1099$, $s_n = 19$, $\frac{1099}{19} \approx 57.84$.

But let's try $b = 0, c = 8, d = 9$:
$n = 1089$, $s_n = 18$, $\frac{1089}{18} = 60.5$ (larger).

Try $b = 0, c = 9, d = 8$:
$n = 1000 + 0 + 90 + 8 = 1098$, $s_n = 1 + 0 + 9 + 8 = 18$, $\frac{1098}{18} = 61$.

Try $b = 0, c = 9, d = 7$:
$n = 1000 + 0 + 90 + 7 = 1097$, $s_n = 1 + 0 + 9 + 7 = 17$, $\frac{1097}{17} \approx 64.53$.

So, $n = 1099$, $s_n = 19$, $\frac{1099}{19} \approx 57.84$ is the minimum so far.

But let's try $b = 0, c = 9, d = 9$ for $a = 1$.

Now, try $a = 1, b = 0, c = 9, d = 9$:
$n = 1000 + 0 + 90 + 9 = 1099$, $s_n = 1 + 0 + 9 + 9 = 19$, $\frac{1099}{19} \approx 57.84$.

Now, try $a = 1, b = 0, c = 8, d = 9$:
$n = 1000 + 0 + 80 + 9 = 1089$, $s_n = 1 + 0 + 8 + 9 = 18$, $\frac{1089}{18} = 60.5$.

Try $a = 1, b = 0, c = 7, d = 9$:
$n = 1000 + 0 + 70 + 9 = 1079$, $s_n = 1 + 0 + 7 + 9 = 17$, $\frac{1079}{17} \approx 63.47$.

Try $a = 1, b = 0, c = 6, d = 9$:
$n = 1000 + 0 + 60 + 9 = 1069$, $s_n = 1 + 0 + 6 + 9 = 16$, $\frac{1069}{16} = 66.8125$.

Try $a = 1, b = 0, c = 5, d = 9$:
$n = 1000 + 0 + 50 + 9 = 1059$, $s_n = 1 + 0 + 5 + 9 = 15$, $\frac{1059}{15} = 70.6$.

So, the minimum is $\boxed{\frac{1099}{19}}$.

Therefore, the smallest value of $\frac{n}{s_n}$ for four-digit $n$ is $\boxed{\frac{1099}{19}} \approx 57.84$.