Question

Three numbers are in the ratio $1:2:3$and the sum of their cubes is $4500$. Find the numbers.

Answer 1

Let us assume the three number be $n,2n,3n$

Then,

Given, sum of cube of three numbers is $4500$

i.e. $n^3+{\left(2n\right)}^3+{\left(3n\right)}^3=4500$

$\Rightarrow$ $n^3+8n^3+27n^3=4500$

$\Rightarrow$ $36n^3=4500$

$\Rightarrow$ $n^3=\frac{4500}{36}$

$\Rightarrow$ $n^3=125$

$\Rightarrow$ $n=\sqrt[3]{(5\times 5\times 5)}$

$\Rightarrow$ $n=5$

Hence, the numbers are,

$n=5$

$2n=2\times 5=10$

$3n=3\times 5=15$

Therefore, the numbers are $5,10,15$