The length of a rectangle is a two-digit number with identical digits (aa). the width is 1/10 of the length. the perimeter is 2 times the area of the rectangle. find the length and the width.

L = 10a + a
L = 11 a
W = 1/10 (11a)
W = 1.1 a

Perimeter = 2(L + W)
Perimeter = 2*(11a + 1.1a)
Perimeter = 22a + 2.2 a
Perimeter = 24.2 a

Area = L * W
Area = 11a * 1.1a
Area = 12.1 a^2
 
The perimeter is 2*area
P = 2*A
P = 2* 12.1 a^2
P = 24.2 a^2

Now put in the perimeter on the left side of the above equation.
24.2a = 24.2 a^2 Divide both sides by 24.2
1 a = a^2 Divide by a
1 = a

The length = 11*a = 11*1 = 11
The Width =  1.1 a = 1.1 * 1 = 1.1

Answers
L = 11 <<<<<<< answer
W = 1.1 <<<<<< answer