= Solution
For independent first-order sinks acting on the same well-mixed burden, loss frequencies add:
$$
k_{\rm tot}=\frac14+\frac14+\frac18=\frac58\ \mathrm{day^{-1}},
\qquad\boxed{\tau_{\rm SO_2}=1.60\ \mathrm{days}.}
$$
The fraction converted to <sulfate> before removal by the competing direct-deposition sink is
$$
\boxed{f_{\rm oxidation}=\frac{1/4+1/8}{5/8}=0.60.}
$$
The individual event fractions are $0.40$ for direct <sulfur dioxide> deposition, $0.40$ for <cloud> <oxidation> and $0.20$ for <hydroxyl radical> <oxidation>. These are branching fractions for the <sulfur dioxide> fate; <sulfate> formed by <oxidation> can subsequently be deposited as well. The argument assumes the supplied channel lifetimes are compatible effective first-order rates, not residence times of isolated air masses that never exchange.
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