Question

please respond it soon it is a question based on chapter ratio and proportion ::​

Answer 1

Question -

Given $\dfrac{ {x}^3 + 12x }{ 6 {x}^2 + 8 } = \dfrac{ { y }^ 3 + 27y }{ 9 { y }^ 2 + 27 }$ .

Using componendo and dividendo, find x : y .

Solution -

The given Information is -

$\dfrac{ {x}^3 + 12x }{ 6 {x}^2 + 8 } = \dfrac{ { y }^ 3 + 27y }{ 9 { y }^ 2 + 27 } \\ \\ \sf{ Using \: componendo \: and \: dividendo \: - } \\ \\ => \dfrac{ {x}^3 + 6 {x}^2 + 12x + 8 }{ {x}^ 3 - 6 { x } ^ 2 + 12x - 8 } = \dfrac{ { y } ^ 3 + 9 { y } ^ 2 + 27y + 27 }{ { y } ^ 3 - 9 { y } ^ 2 + 27y - 27 } \\ \\ =>{ \dfrac{ (x + 3) }{ ( x - 3 ) } } ^ 3 = { \dfrac{ ( y + 3 ) }{ ( y - 3 ) } } ^ 3 \\ \sf{ Taking \: Cube \: root \: both \: sides \::- } \\ \\ => \dfrac{ x + 3 }{ x - 3 } = \dfrac{ y + 3 }{ y - 3 } \\ \\ \sf{ Using \: componendo \: and \: dividendo \: - } \\ \\ => \dfrac{ 2x }{ 6 } = \dfrac{ 2y }{ 6 } \\ \\ => \dfrac{ 2x }{ \cancel{6} } = \dfrac{ 2y }{ \cancel{6} } \\ => 2x = 2y \\ \\ => x = y \\ \\ \sf{ Hence \: - } \\ \\ x : y = 1 : 1 \: ............ [ A ]$