Example 1 Find the minimum value of the ratio of a three-digit number (in decimal) to the sum of its digits.



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Let the three-digit number be $100x + 10y + z$, where $x, y, z$ are integers, and $1 \leqslant x \leqslant 9$, $0 \leqslant y, z \leqslant 9$. Thus, the ratio is
$$f(x, y, z) = \frac{100x + 10y + z}{x + y + z} = 1 + \frac{99x + 9y}{x + y + z}.$$

In the expression on the right-hand side of (1), only the denominator contains $z$, so when $x$ and $y$ are fixed, $f(x, y, z)$ attains its minimum value if and only if $z = 9$. Similarly, we have
$$\begin{aligned}
f(x, y, z) & \geqslant 1 + \frac{99x + 9y}{9 + x + y} = 10 + \frac{90x - 81}{9 + x + y} \\
& \geqslant 10 + \frac{90x - 81}{18 + x} = 100 - \frac{1701}{18 + x} \\
& \geqslant 100 - \frac{1701}{18 + 1} = \frac{199}{19}.
\end{aligned}$$

Therefore,
$$f_{\min}(x, y, z) = f(1, 9, 9) = \frac{199}{19}.$$