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

Given:

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

To do:

We have to find the numbers.

Solution:

Let the first number be $x$.

This implies,

The second number is $2x$.

The third number is $3x$.

According to the question,

The sum of their cubes is $98784$.

$\Rightarrow x^3+( 2x)^3+( 3x)^3=98784$

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

$\Rightarrow 36x^3=98784$

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

$\Rightarrow x^3=2744$

$\Rightarrow x=\sqrt[3]{2744}$

$\Rightarrow x=\sqrt[3]{2\times2\times2\times7\times7\times7}$

$\Rightarrow x=2\times7=14$

Therefore,

The first number $=x=14$

The second number$=2x=2\times14=28$

The third number$=3x=3\times14=42$

Hence, the three numbers are $14,\ 28$ and $42$.