If x is the mean proportional between 12.8 and 64.8 and y is the third proportional to 38.4 and 57.6, then 2x : y is equal to:

x is the mean proportional between 12.8 and 64.8

$$\Rightarrow$$  $$\frac{12.8}{\text{x}}=\frac{\text{x}}{64.8}$$

$$\Rightarrow$$  x$$^2$$ = $$12.8\times64.8$$

$$\Rightarrow$$  x$$^2$$ = $$25.6\times32.4$$

$$\Rightarrow$$  x$$^2$$ = $$2.56\times324$$

$$\Rightarrow$$  x = $$1.6\times18$$

$$\Rightarrow$$  x = 28.8

y is the third proportional to 38.4 and 57.6

$$\Rightarrow$$  $$\frac{38.4}{57.6}=\frac{57.6}{\text{y}}$$

$$\Rightarrow$$  y = $$\frac{57.6\times57.6}{38.4}$$

$$\Rightarrow$$  y = $$\frac{57.6\times28.8}{19.2}$$

$$\Rightarrow$$  y = $$3\times28.8$$

$$\therefore\ $$2x : y = $$(2\times28.8)$$ : $$(3\times28.8)$$ = 2 : 3

Hence, the correct answer is Option A