= Solution
An <H II region> is photoionized interstellar gas around young massive stars. Its optical spectrum contains hydrogen and helium recombination lines, collisionally excited forbidden metal lines, and weaker free-free, free-bound and two-<photon> continua. Typical hot main-sequence ionizing stars have effective <temperatures> of a few $10^4\,\mathrm K$; their ultraviolet output, age and <metallicity> determine the ionization balance. Gas and dust can also emit strongly outside the visible band.
A <planetary nebula> is expelled evolved-star material illuminated by a much hotter exposed stellar core. Its central star can reach <temperatures> of order $10^5\,\mathrm K$, allowing higher ionization stages and prominent He II emission. The gas may carry nucleosynthetic enrichment from its progenitor, while expansion lowers density and changes line strengths with age. Both object classes produce nebular spectra, but their ionizing sources and evolutionary contexts differ; neither spectrum is simply the blackbody spectrum of the exciting star.
For an optically thin recombination line under <Case B recombination>, the line emissivity is
$$
j_\lambda=n_en(X^{i+1})\alpha_{\lambda}^{\rm eff}(T_e,n_e)\frac{hc}{\lambda}.
$$
The effective coefficient includes recombination cascades feeding the line. For approximately common conditions and matching sampled volumes,
$$
\boxed{\frac{n(X^{i+1})}{n({\rm H}^+)}
=\frac{F_\lambda}{F_{\rm H\beta}}\frac{\alpha_{\rm H\beta}^{\rm eff}}{\alpha_\lambda^{\rm eff}}\frac{\lambda}{\lambda_{\rm H\beta}}.}
$$
The wavelength factor is essential because the tabulated <energy> flux counts <photons> weighted by their <energies>. More generally the result is an emissivity-weighted ionic ratio, not an unweighted pointwise abundance.
Assume corrected extinction and underlying stellar absorption, suitable <electron> <temperature> and density, negligible additional collisional or optical-depth effects on the chosen lines, and essentially ionized hydrogen. The total helium abundance is then
$$
\frac{\mathrm{He}}{\mathrm H}\simeq\frac{n({\rm He}^+)+n({\rm He}^{++})}{n({\rm H}^+)},
$$
provided neutral helium is negligible or an ionization correction is applied. A line emitted by neutral He after recombination measures the recombining He$^+$ population; He II $4686$ measures recombining He$^{++}$.
Using the supplied line coefficients and fluxes gives
$$
\frac{n({\rm He}^+)}{n({\rm H}^+)}=\frac5{100}\frac{3.02}{1.35}\frac{0.4471}{0.4861}=0.1029,
$$
$$
\frac{n({\rm He}^{++})}{n({\rm H}^+)}=\frac{50}{100}\frac{3.02}{29.4}\frac{0.4686}{0.4861}=0.04951.
$$
Therefore
$$
\boxed{\mathrm{He/H}\simeq0.1524\quad\hbox{by number}.}
$$
If hydrogen and helium dominate the <mass>, this corresponds to helium <mass> fraction $Y\simeq4y/(1+4y)\simeq0.379$, with $y=\mathrm{He/H}$. It is an approximate elemental estimate under the ionization assumptions, not merely a ratio of two observed line fluxes.
An ordinary H II region would typically have $\mathrm{He/H}\sim0.08$–$0.10$ by number, with <metallicity>-dependent enrichment, rather than necessarily the larger value inferred here for an evolved-star nebula. Hydrogen ionization requires $13.6\,\mathrm{eV}$, neutral helium $24.6\,\mathrm{eV}$ and He$^+$ $54.4\,\mathrm{eV}$. Typical H II region stars often produce little He$^{++}$, so He II $4686$ can be extremely weak or absent despite substantial helium. He I $4471$ relative to H$\beta$ can still be useful, but neutral helium inside the H$^+$ zone may require an ionization correction. A very hot planetary-nebula core is more favorable for observing both ionic stages, although ionization structure must still be checked.
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