$2 \cdot 73$ an item is worth $C$ dollars, and it is sold for 100 dollars at a loss of $x \%$ of the selling price, then it is sold again at a profit of $x \%$ of the new selling price $S$. If the difference between $S$ and $C$ is $1 \frac{1}{9}$ dollars, then $x$ is
(A) undetermined.
(B) $\frac{80}{9}$.
(C) 10.
(D) $\frac{95}{9}$.
(E) $\frac{100}{9}$
(16th American High School Mathematics Examination, 1965)
[Solution] Let $C=100+100 \cdot x \% =100+x$, and $S=C+1 \frac{1}{9}=(100+x)+1 \frac{1}{9}=101 \frac{1}{9}+x$, also
$$
S=100+\frac{x}{100}\left(101 \frac{1}{9}+x\right) \text {. }
$$

From the above two equations, we have $\quad \frac{x^{2}}{100}+\frac{x}{90}-\frac{10}{9}=0$, solving this gives $x=10$ (the negative value has been discarded).
Therefore, the answer is $(C)$.