The shaded region in the figure above represents a
rectangular frame with length 18 inches and width 15
inches. The frame encloses a rectangular picture that
has the same area as the frame itself. If the length and
width of the picture have the same ratio as the length
and width of the frame, what is the length of the
picture, in inches?
answ: 9*SQROOT OF 2

Area of the picture = (18*15)/2

Assume height be 'h' then length of the picture would be (18/15)*h (Because length and height are in same proportion as the dimensions of the frame)

Hence (6/5)h^2 = (18*15)/2

h = 15/(2)^1/2, length = (6/5)*h = 18/(2)^1/2 = 9(2)^1/2

Why is the area of the picture equal to (18*15)/2? If the area of picture is equal to the area of the frame, shouldn't the area of the picture equal 18*15? Thank you.

The area enclosed by the figure of dimensions (18 by 15) = Area of the frame + Area of the picture

and Area of the frame = Area of the picture

Area enclosed by the figure of dimensions (18 by 15) = 2(Area of the picture)

Area of the pic = (Area enclosed by the figure of dimensions (18 by 15)) / 2 or (18*15)/2

Hope it helps