Question

Three numbers are in the ratio of $2:3:4.$ The sum of their cubes is $33957.$ Find the numbers.

Answer 1

Given: Ratio of the three numbers $=2:3:4$

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

Sum of cubes $=33957$ [Given]

$\Rightarrow (2x{)}^3+(3x{)}^3+(4x{)}^3=33957$

$\Rightarrow 8x^3+27x^3+64x^3=33957$

$\Rightarrow 99x^3=33957$

$\Rightarrow x^3=\frac{33957}{99}$

$\Rightarrow x=\sqrt[3]{3343}$

$\Rightarrow x=\sqrt[3]{37\times 7\times 7}$

$\Rightarrow x=7$

Numbers are:
$2x=2\times 7=14$
$3x=3\times 7=21$
$4x=4\times 7=28$