An architect designs a rectangular flower garden such that the width is exactly two-thirds of the length. if 410 feet of antique picket fencing are to be used to enclose the garden, find the dimensions of the garden.

width = 82 feet, length = 123 feet
       
First, write down equations that express what you've been given.
   
w = 2/3 l
   
p = 410
   
p = 2(w+l)
       
Since we have an expression that defines w in terms of l, substitute for w, the expression. Also substitute the known value for p in the expression.
   
p = 2(w + l)
   
410 = 2( 2/3 l + l)
       
Combine the l terms
   
410 = 2( 5/3 l)
       
Multiply
   
410 = 10/3 l
       
Multiple both sides by 3
   
1230 = 10 l
       
Divide both sides by 10
   
123 = l
       
So we now know the length is 123 feet. So plug in the known value of l into the formula w = 2/3 l to get the width w = 2/3 123 = 82