3. As shown in Figure $1, \triangle A B C$ is an equilateral triangle, $P M \perp A B, P N \perp A C$. Let the perimeters of quadrilateral $A M P N$ and $\triangle A B C$ be $m$ and $n$, respectively. Then ( ).
(A) $\frac{1}{2}<\frac{m}{n}<\frac{3}{5}$
(B) $\frac{2}{3}<\frac{m}{n}<\frac{3}{4}$
(C) $80 \%<\frac{m}{n}<83 \%$
(D) $78 \%<\frac{m}{n}<79 \%$
3. (D).

Let $B M=x, C N=y$. Then
$$
\begin{array}{l}
B P=2 x, P C=2 y, P M=\sqrt{3} x, P N=\sqrt{3} y, \\
A M+A N=2 \cdot B C-(B M+C N)=3(x+y) . \\
\text { Hence } \frac{m}{n}=\frac{(3+\sqrt{3})(x+y)}{3 \times 2(x+y)}=\frac{3+\sqrt{3}}{6} \approx \frac{4.7321}{6} .
\end{array}
$$