Three numbers are in the ratio 1 : 2 : 3. The sum of their cubes is 98784.
Question:

Three numbers are in the ratio 1 : 2 : 3. The sum of their cubes is 98784. Find the numbers.

Solution:

Let the numbers be x, 2x and 3x.

Therefore

$x^{3}+(2 x)^{3}+(3 x)^{3}=98784$

$\Rightarrow x^{3}+8 x^{3}+27^{3}=98784$

$\Rightarrow 36 x^{3}=98784$

$\Rightarrow x^{3}=\frac{98784}{36}=2744$

$\Rightarrow x^{3}=2744$

$\Rightarrow x=\sqrt[3]{2744}=\sqrt[3]{\{2 \times 2 \times 2\} \times\{7 \times 7 \times 7\}}=2 \times 7=14$

Hence, the numbers are $14,(2 \times 14=28)$ and $(3 \times 14=42)$.