Three number are in the ratio $2:3:4$. The sum of their cube is $0.334125$, find the numbers.

Given: Three number are in the ratio $2:3:4$. The sum of their cube is $0.334125$.

To do: To find the numbers.

Suppose the numbers $= 2x,\ 3x,\ 4x$

Given, the sum of their cubes $=0.334125$

As given, The sum of their cube is $0.334125$.

$( 2x)^3+( 3x)^3+( 4x)^3=0.334125$

$\Rightarrow  8x^3+27x^3+64x^3=0.334125$

$\Rightarrow  99x^3=0.334125$

$\Rightarrow x^3=0.334125\div99$

$\Rightarrow  x^3=\frac{0.334125}{99}=0.003375$

$\Rightarrow x^3=\frac{3375}{1000000}$

$\Rightarrow  x=\sqrt[3]{\frac{15\times15\times15}{10\times10\times10\times10\times10\times10}}=\frac{15}{10\times10\times10}=0.015$

Therefore, the numbers are:

$2x=2\times0.015=0.03$

$3x=3\times0.015=0.045$

$4x=4\times0.015=0.06$