= Solution
Assume a massless junction and common <tension> $T$. The <characteristic impedance of a stretched string> is $Z_j=\sqrt{T\mu_j}$, where $\mu_j$ is its <linear mass density>. The displacement <reflection coefficient> is $r=(Z_1-Z_2)/(Z_1+Z_2)$, so the reflected fraction of <power> is
$$
\mathcal R=|r|^2=\left(\frac{1-z}{1+z}\right)^2,\qquad z=\frac{Z_2}{Z_1}=\sqrt{\frac{\mu_2}{\mu_1}}.
$$
Set $\mathcal R=1/2$. The positive solutions are $z=3\pm2\sqrt2$, which are reciprocals. Thus the <string density ratio for prescribed reflected power> is
$$
\boxed{\frac{\mu_{\rm heavy}}{\mu_{\rm light}}=(3+2\sqrt2)^2=17+12\sqrt2\simeq34.0.}
$$
The impedance ratio is about $5.83$, rather than the required density ratio. Reversing which string is on the incident side changes the reflected displacement sign but not the reflected <power> fraction.
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