I have a rectangle whose sides are in the proportion b:a, where b is the longer side. I draw a line parallel to a inside the rectangle in order to break it up into a square with sides of length a and a second rectangle. Amazingly, the sides of the new smaller rectangle are in the exact same proportion as my original rectangle, except that a is the longer side.

Question

I have a rectangle whose sides are in the proportion b:a, where b is the longer side. I draw a line parallel to a inside the rectangle in order to break it up into a square with sides of length a and a second rectangle.  Amazingly, the sides of the new smaller rectangle are in the exact same proportion as my original rectangle, except that a is the longer side.

Draw a picture and solve for the proportion b:a.  (Note that if you set a equal to 1, b automatically equals the proportion)  You will need to use the quadratic formula.  The formula will give you two answere, but only the positive answer is relevant.