Question

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

Answer 1

Let the numbers be x, 2x and 3x.
Therefore

\[x^3 + \left( 2x \right)^3 + \left( 3x \right)^3 = 98784\]
\[ \Rightarrow x^3 + 8 x^3 + {27}^3 = 98784\]
\[ \Rightarrow 36 x^3 = 98784\]
\[ \Rightarrow x^3 = \frac{98784}{36} = 2744\]
\[ \Rightarrow x^3 = 2744\]
\[ \Rightarrow x = \sqrt[3]{2744} = \sqrt[3]{\left\{ 2 \times 2 \times 2 \right\} \times \left\{ 7 \times 7 \times 7 \right\}} = 2 \times 7 = 14\]

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