18.53 If a square is cut from a rectangle, and the remaining rectangle is similar to the original rectangle, then the ratio of the longer side to the shorter side of the original rectangle is 
(A) $(1+\sqrt{5}): 2$.
(B) $3: 2$.
(C) $(1+\sqrt{3}): 2$.
(D) $(1+\sqrt{6}): 2$.
(China Jilin Province Seven Cities and Prefectures Junior High School Mathematics Competition, 1984; China Zhejiang Province Junior High School Mathematics Competition, 1992)
[Solution 1] Let the length of the rectangle be $x$, and the width be 1. According to the problem, we have

that is
$$
\begin{array}{l}
1: x=(x-1): 1, \\
x^{2}-x-1=0,
\end{array}
$$

Solving for $x$ gives $x=\frac{1 \pm \sqrt{5}}{2}$ (discard the negative value).
Thus, the ratio of the longer side to the shorter side of the rectangle is $(1+\sqrt{5}): 2$, so the answer is $(A)$.
[Solution 2] Let the length of the rectangle be $x$, and the width be $y$, with $x>y$. After cutting out the square, the remaining rectangle has a length of $y$ and a width of $x-y$. According to the problem, we have $x: y=y:(x-y)$,
which simplifies to $x^{2}-x y-y^{2}=0$.
This can be rewritten as $\left(\frac{x}{y}\right)^{2}-\frac{x}{y}-1=0$,

Solving for $\frac{x}{y}$ gives $\frac{x}{y}=\frac{1 \pm \sqrt{5}}{2}$ (discard the negative value). Therefore, the answer is $(A)$.