Example 2. Find a three-digit number such that the ratio of the number to the sum of its digits is minimized.

 untranslated text remains unchanged:
例2. 求一个三位数, 便它与它的各位数字之和的比为最小. 

However, for a proper translation, it should be:
Example 2. Find a three-digit number such that the ratio of the number to the sum of its digits is minimized.
Solve: Arrange the ratios of all three-digit numbers to the sum of their respective digits in a $9 \times 100$ grid (Table 1).

We examine the numbers in each row and select the smaller ones. By the inequality $\frac{a}{b}>\frac{a+1}{b+1}(a>b)$, from left to right, every 10 numbers form a group, and the last number in each group is the smallest. Extract these numbers and arrange them in a $9 \times 10$ grid.
\begin{tabular}{|c|c|c|c|c|c|c|c|c|}
\hline \begin{tabular}{l}
$\frac{100}{1}$
\end{tabular} & \begin{tabular}{l}
$\frac{101}{2}$
\end{tabular} & \begin{tabular}{l}
$\frac{102}{3}$
\end{tabular} & $\cdots$ & \begin{tabular}{l}
$\frac{109}{10}$
\end{tabular} & \begin{tabular}{l}
$\frac{110}{2}$
\end{tabular} & \begin{tabular}{l}
$\frac{111}{3}$
\end{tabular} & $\cdots$ & \begin{tabular}{l}
$\frac{199}{19}$
\end{tabular} \\
\hline \begin{tabular}{l}
$\frac{200}{2}$
\end{tabular} & \begin{tabular}{l}
$\frac{201}{3}$
\end{tabular} & \begin{tabular}{l}
$\frac{202}{4}$
\end{tabular} & $\cdots$ & \begin{tabular}{l}
$\frac{209}{11}$
\end{tabular} & \begin{tabular}{l}
$\frac{210}{3}$
\end{tabular} & \begin{tabular}{l}
$\frac{211}{4}$
\end{tabular} & $\cdots$ & \begin{tabular}{l}
$\frac{299}{20}$
\end{tabular} \\
\hline $\mathbf{\vdots}$ & $\vdots$ & $\mathbf{\vdots}$ & & $\mathbf{\vdots}$ & : & $\vdots$ & & $\mathbf{\vdots}$ \\
\hline \begin{tabular}{l}
$\frac{900}{9}$
\end{tabular} & \begin{tabular}{l}
$\frac{901}{10}$
\end{tabular} & \begin{tabular}{l}
$\frac{902}{11}$
\end{tabular} & $\cdots$ & \begin{tabular}{l}
$\frac{909}{18}$
\end{tabular} & \begin{tabular}{l}
$\frac{910}{10}$
\end{tabular} & \begin{tabular}{l}
$\frac{911}{11}$
\end{tabular} & $\cdots$ & $\frac{999}{27}$ \\
\hline
\end{tabular}

Table 1
From Table 2, it is easy to see that the numbers in each row from left to right are
\begin{tabular}{|c|c|c|c|c|}
\hline \begin{tabular}{l}
$\frac{109}{10}$
\end{tabular} & \begin{tabular}{l}
$\frac{119}{11}$
\end{tabular} & \begin{tabular}{l}
$\frac{129}{12}$
\end{tabular} & $\cdots$ & \begin{tabular}{l}
$\frac{199}{19}$
\end{tabular} \\
\hline \begin{tabular}{l}
$\frac{209}{11}$
\end{tabular} & \begin{tabular}{l}
$\frac{219}{12}$
\end{tabular} & \begin{tabular}{l}
$\frac{229}{13}$
\end{tabular} & $\cdots$ & \begin{tabular}{l}
$\frac{299}{20}$
\end{tabular} \\
\hline$\cdots$ & $\cdots$ & $\cdots$ & $\cdots$ & $\cdots$ \\
\hline \begin{tabular}{l}
$\frac{909}{18}$
\end{tabular} & \begin{tabular}{l}
$\frac{919}{19}$
\end{tabular} & \begin{tabular}{l}
$\frac{929}{20}$
\end{tabular} & $\cdots$ & \begin{tabular}{l}
$\frac{999}{27}$
\end{tabular} \\
\hline
\end{tabular}

Table 2
gradually decreasing, while the rightmost column increases from top to bottom. Therefore, the smallest number is $\frac{190}{19}$, and the required three-digit number is 159.