The excess pressure inside the first soap bubble is three times that inside the second soap bubble. The ratio of the volume of the first soap bubble to the volume of the second soap bubble is

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The excess pressure inside a soap bubble of radius $R$ and surface tension $T$ is given by $P = \frac{4 T}{R}$ .

Given $P_{1} = 3 P_{2}$ .

$\frac{4 T}{R _{1}} = 3 (\frac{4 T}{R _{2}})$

$\frac{R _{1}}{R _{2}} = \frac{1}{3}$

The volume of a soap bubble is $V = \frac{4}{3} π R^{3}$ .

The ratio of their volumes is $\frac{V _{1}}{V _{2}} = (\frac{R _{1}}{R _{2}})^{3} = (\frac{1}{3})^{3} = \frac{1}{27}$ .

Thus, the ratio of the volumes is $1 : 27$ .

Answer: $1 : 27$

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