= Solution
In the electron-count estimate $f_{\rm Ga}\simeq31$, $f_{\rm As}\simeq33$, the <structure factors> are $F_{111}=4(31-33i)$ and $F_{200}=4(31-33)$. Thus a single-directed-reflection estimate is
$$
\frac{|F_{111}|^2}{|F_{200}|^2}=\frac{31^2+33^2}{(31-33)^2}=512.5.
$$
A <powder X-ray diffraction> peak collects all equivalent reflections. The <powder diffraction multiplicities> are eight for $\{111\}$ and six for $\{200\}$, so the multiplicity-and-electron-count estimate is
$$
\boxed{\frac{I_{\{111\}}}{I_{\{200\}}}\simeq\frac86\frac{31^2+33^2}{(31-33)^2}=6.83\times10^2.}
$$
At $(200)$ the two sublattice amplitudes are opposite in phase and nearly cancel; at $(111)$ their relative phase is a quarter cycle, so this cancellation does not occur. In <silicon>, the two <atomic scattering factors> are identical and the <diamond cubic structure> has \b[an exactly forbidden $(200)$ reflection], while $(111)$ remains allowed in the <kinematic approximation>.
The numerical ratio neglects the angular dependence of the <atomic scattering factors>, the <Lorentz-polarization factor> and thermal-motion corrections. No wavelength or measurement geometry is supplied to refine those factors. This is therefore a stated structural estimate, rather than a unique precision ratio of measured peak areas.
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