The ratio of the total surface areas of two cubes is 49 : 81. What is the ratio of their volumes?

The ratio of the total surface areas of two cubes is 49 : 81.

$$\frac{total\ surface\ area\ of\ first\ cube}{total\ surface\ area\ of\ \sec ond\ cube}\ =\ \frac{6\times\left(side\ of\ first\ cube\right)^2}{6\times\left(side\ of\ \sec ond\ cube\right)^2}$$

$$\frac{49}{81} = \frac{6\times(side\ of\ first\ cube)^2}{6\times(side\ of\ second\ cube)^2}$$

$$\frac{49}{81} = \frac{(side\ of\ first\ cube)^2}{(side\ of\ second\ cube)^2}$$

$$\frac{7}{9} = \frac{(side\ of\ first\ cube)}{(side\ of\ second\ cube)}$$    Eq.(i)

The ratio of their volumes = $$\frac{(side\ of\ first\ cube)^3}{(side\ of\ second\ cube)^3}$$

Put Eq.(i) in the above formula.

= $$\frac{(7)^3}{(9)^3}$$

= $$\frac{343}{729}$$

= 343 : 729