sides of a rectangle given a ratio and a surface

Question

I am trying to find the sides of a rectangle given a ratio and a surface area.

Here is where i am:

Given the ratio formula where m:n

height * (m / n) = width

Given the surface is width * height = surface

I get:

height * (m / n) * height = surface

2 * height * (m/ n) = surface

(2 * n) * (height * n) * ((m / n) * n) = surface * n

(2 * n) * (height * n) * (m * n) = surface * n

Am i going in the right direction? Can anyone correct and/or complete this?

thanks in advance.

[edit]

So:

height * (m / n) * height = surface

height ^ 2 * (m /n) = surface

Do i still eliminate the fraction by multiplication with 'n'?

Answer 1

Suppose $$ wh = A $$ and $$ w/h = r $$ are both known. Then $w = rh$, so $$ wh = (rh)h = A \\ r h^2 = A \\ h^2 = \frac{A}{r} $$ hence $$ h = \sqrt{\frac{A}{r}}\\ w = r\sqrt{\frac{A}{r}} = \sqrt{Ar} $$ which expresses the width and height in terms of the area and the ratio of width to height. $$