Question

If (4x²+ xy) : (3xy – y²) = 12 : 5, find (2x + y) : (x + 2y).

Answer 1

`(4x^2 + xy)/(3xy - y^2) = 12/5`

5(4x² + xy) = 12(3xy − y²)

20x² + 5xy = 36xy − 12y²

20x² + 5xy = 36xy − 12y² = 0

20x² − 31xy + 12y² = 0

Divide the entire equation by y² to get a quadratic equation in terms of the ratio `x/y`

Let p = `x/y`

`20(x^2/y^2) - 31((xy)/y^2) + 12(y^2/y^2) = 0`

`20(x/y)^2 - 31(x/y) + 12` = 0

20p² − 31p + 12 = 0

Factor the quadratic equation:

20p² − 16p − 15p + 12 = 0

4p(5p − 4) − 3(5p − 4) = 0

(4p − 3) (5p − 4) = 0

This gives two possible values for p:

4p − 3 = 0

p = `3/4`

`x/y = 3/4`

5p − 4 = 0

p = `4/5`

`x/y = 4/5`

Substitute the two values of `x/y` into the expression for the required ratio, `(2x + y)/(x + 2y)`.

Divide the numerator and the denominator by y

`(2x + y)/(x + 2y) = (2(x/y) + 1)/((x/y) + 2)`

Case 1: If `x/y = 3/4`

`(2(3/4) + 1)/((3/4) + 2)`

= `(3/2 + 2/2)/(3/4 + 8/4)`

= `(5/2)/(11/4)`

= `5/4 xx 4/11`

= `20/22`

= `10/11`

The ratio is 10 : 11.

Case 2: If `x/y = 4/5`

= `(2(4/5) + 1)/((4/5) + 2)`

= `(8/5 + 5/5)/(4/5 + 10/5)`

= `(13/5)/(14/5)`

= `13/5 xx 5/14`

= `13/14`

The ratio is 13 : 14.

The possible values for the ratio (2x + y) : (x + 2y) are 10 : 11 or 13 : 14.