We are interested in the dimensions of a certain square. a rectangle has a length of 5 units more than the side of this square and width half of the side of this square. if two areas are equal, what are the squares dimensions

Lets consider the side of square be 'x' units
So as per data,
Length of rectangle is x+5
Breadth of rectangle is x/2

and also as per data, the areas of rectangle and square are equal.
Area of rectangle = Length * Breadth = (x+5)*x/2 = ( x^{2} +5x)/2 <- equation1
Area of square = Side* Side= x*x =  x^{2}  <- equation2

As per given data, Equation1 and equation 2 are equal
so
( x^{2} +5x)/2 = tex] x^{2} [/tex]  
x^{2} +5x = 2 x^{2}
2 x^{2} - x^{2} = 5x
x^{2} = 5x
x = 5

So the side of square = 5 units  

For Square, both dimensions are equal

Yea they are simplyfiy the ratios