**Question:**

Three numbers are in the ratio 2 : 3 : 4. The sum of their cubes is 0.334125. Find the numbers.

**Solution:**

Let the numbers be $2 x, 3 x$ and $4 x$, respectively.

$\because$ Sum of their cubes $=0.334125$ [given]

According to the question,

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

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

$\Rightarrow \quad 99 x^{3}=0.334125$

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

$\Rightarrow \quad x^{3}=0.003375$

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

$\Rightarrow$ $x=\sqrt[3]{\frac{15 \times 15 \times 15}{10 \times 10 \times 10 \times 10 \times 10 \times 10}} \quad$ [taking cube root on both sides]

$\Rightarrow$ $x=\frac{15}{10 \times 10 \times 10}$

$\therefore \quad x=0.015$

Hence, the required numbers are $2 \times 0.015,3 \times 0.015$ and $4 \times 0.015$, i.e. $0.03,0.045$ and $0.06$.