An artificial lake is in the shape of a rectangle and has an area of 9/20 square mile the width of the lake is 1/5 the length of the lake what are the dimensions of the lake?

 The dimensions are:   " 1.5 mi.  ×  ³⁄₁₀  mi. " .

             { length = 1.5 mi. ;  width =  ³⁄₁₀  mi. } .

Explanation:

Area of a rectangle:

A = L * w ; 

in which:  A = Area = (9/20) mi.² ,
                L = Length = ?
                w = width = (1/5)*L = (L/5) = ?

  A = L * w ;  we want to find the dimensions; that is, the values for
                         "Length (L)"  and "width (w)" ; 

Plug in our given values:

 (9/20) mi.² = L * (L/5) ;  in which: "w = L/5" ; 
 
     → (9/20) = (L/1) * (L/5) = (L*L)/(1*5) = L² / 5 ;
   
          ↔  L² / 5  = 9/20 ;
 
            →  (L² * ? / 5 * ?) = 9/20 ?    

                →     20÷5 = 4 ;  so; L² *4 = 9 ;
 
                   ↔    4 L² = 9 ; 
 
                   →  Divide EACH side of the equation by "4" ;
           
                   →   (4 L²) / 4 = 9/4 ;

           to get:  →  L² = 9/4 ; 
 Take the POSITIVE square root of each side of the equation; to isolate "L" on one side of the equation; and to solve for "L" ;
         
 
     →   ⁺√(L²)   =   ⁺√(9/4) ;

    →   L  =  (√9) / (√4) ; 

    →  L = 3/2 ; 

    → w = L/5 = (3/2) ÷ 5 = 3/2 ÷ (5/1) = (3/2) * (1/5) = (3*1)/(2*5) = 3/10;

Let us check our answers:

(3/2 mi.) * (3/10 mi.) =? (9/20) mi.² ??

→ (3/2)mi. * (3/10)mi.  =  (3*3)/(2*10) mi.² = 9/20 mi.² ! Yes!

So the dimensions are: 

Length = (3/2) mi. ;  write as: 1.5 mi.

width = ³⁄₁₀ mi.

or; write as:  " 1.5 mi.  ×  ³⁄₁₀ mi. " .