= Solution
The high-pressure <atomic basis of a crystal> has the <caesium chloride structure>. Its chemically labelled <Bravais lattice> is <simple cubic lattice>: exchanging Na and Cl is not a <translation symmetry>. Its <structure factor> is
$$
F_{hkl}=f_{\rm Na}+(-1)^{h+k+l}f_{\rm Cl}.
$$
Because <ionic bonding> gives approximately Na$^+$ and Cl$^-$, their bound-electron counts are $10$ and $18$. In the stipulated electron-count approximation, odd-index-sum reflections have $|F|=8$, while even-index-sum reflections have $|F|=28$. In particular, the first ring $\{100\}$ is weak but present; it must not be discarded using a body-centering extinction rule.
The next ring is $\{110\}$, and <Bragg's law> gives
$$
\sin\frac{\psi_{110}}2=\sqrt2\sin\frac{29.7^\circ}2,
\qquad\boxed{\psi_{110}=42.50^\circ.}
$$
The <powder diffraction multiplicities> are six and twelve. Counting all symmetry-equivalent reflections, while neglecting other angular factors, therefore gives
$$
\boxed{\frac{I_{110}}{I_{100}}\simeq\frac{12}{6}\left(\frac{28}{8}\right)^2=24.5.}
$$
The corresponding single-directed-reflection ratio would be $12.25$. Neutral-atom counts would give the wrong contrast because the material is ionic.
For measured integrated photon counts, the geometry must also be specified. More generally the ratio is $2|F_{110}/F_{100}|^2$ times the ratio of <Lorentz-polarization factors>, thermal-motion factors and detection efficiencies. For example, the common unpolarized symmetric-powder convention has
$$
LP(\theta)\propto\frac{1+\cos^2(2\theta)}{\sin^2\theta\cos\theta},
$$
and gives $24.5\,LP(21.2506^\circ)/LP(14.85^\circ)\simeq11.18$ if all other corrections are suppressed. Thus \b[24.5 is the multiplicity-and-structure-factor estimate], rather than a geometry-independent precision count ratio; actual angle-dependent ionic form factors are also not supplied.
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