= Solution
If the quoted coupling is $k=\varepsilon_y/\varepsilon_x$, then $\varepsilon_y=k\varepsilon_x$. Combining the horizontal factor of eight with $k$ falling from $0.1$ to $0.02$ gives
$$
\boxed{\frac{\varepsilon_{y,\rm new}}{\varepsilon_{y,\rm old}}
\simeq\frac{0.02}{0.1}\frac18=\frac1{40}.}
$$
This is the usual leading estimate treating the <horizontal emittance> in part <S19/1/a> as unchanged by small coupling.
If one instead accounts for exact redistribution of a fixed natural-emittance sum, $\varepsilon_x=\varepsilon_0/(1+k)$ and $\varepsilon_y=k\varepsilon_0/(1+k)$. Then the horizontal ratio is $(1/8)(1.1/1.02)\simeq0.1348$ and the vertical ratio is $0.02696$, an improvement by about $37$. These are consistent refinements of the stated approximate factor forty, not a second independent reduction to apply afterward.
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