Question

The sum of the cubic powers of three rational numbers is -1009. The ratio between the first number and the second number is 2/3, and the ratio between the first number and the third number is 4/9. Find those numbers

Answer 1

Given:

The sum of the cubic powers of three rational numbers is -1009.

The ratio between the first number and the second number is
`(2)/(3)`.

The ratio between the first number and the third number is
`(4)/(9)`.

To find:

The three numbers.

Solution:

Let
`x,y,z` be the three numbers, then

`x^3+y^3+z^3=-1009` ...(i)

`(x)/(y)=(2)/(3)`

`3x=2y`

`(3)/(2)x=y` ...(ii)

`(x)/(z)=(4)/(9)`

`9x=4z`

`(9)/(4)x=z` ...(iii)

Using (i), (ii) and (iii), we get

`x^3+\left((3)/(2)x\right)^3+\left((9)/(4)x\right)^3=-1009`

`x^3+(27)/(8)x^3+(729)/(64)x^3=-1009`

`(64+216+729)/(64)x^3=-1009`

`(1009)/(64)x^3=-1009`

`x^3=(-1009* 64)/(1009)`

On simplification, we get

`x^3=-64`

`x^3=-4^3`

`x=-4`

Substituting
`x=-4` in (ii), we get

`(3)/(2)(-4)=y`

`-6=y`

Substituting
`x=-4` in (iii), we get

`(9)/(4)(-4)=z`

`-9=z`

Therefore, the three numbers are -4, -6 and -9 respectively.