Area perimeter

Let A be the total area  of the largest square

The smallest  square occupies [ (2 ) / (2 +9)]A  =   2/11  of the total area  =  (2/11)A

The shaded area occupies  [ (3) / (3 + 7)]A =   3/10 of the total area  =  (3/10)A

Shaded area                                  (3/10)A      (3/10)

____________________   =        ______  =    ____   =   (3/10) (11/2)  =   33/20 =  33 : 20

Area of smallest square                (2/11)A       (2/11)

Let the area of small square he S

Shaded area be x.

Unshaded area be y.

as the largest square is made up of shaded and unshaded area So area of largest square = x + y.

Now given ratio of area of smallest square to largest square is 2:9.

i.e.                   S/(x + y) = 2/9 - 1

ratio of shaded area & unshaded area is 3:7.

(Shaded area)/(Unshaded area) = 3/7

x/y = 3/7

7x = 3y

y = 7/3x

put value of y in 1

S/(x + 7/3x) = 2/9

9s = 2(x + 7/3x)

9s = 20x/3

27/20 = x/s

27/20 = (Shaded area)/(Smallest Square)

Ratio of shaded area to the area of smallest square is 27:20

I got the same as slimesewer, good work!

Sorry Chris, I believe you made a mistake somewhere.  

I draw the scale pic (just for fun)

I actually used a different, although probably not easier method to Slimesewer

I said that if the ratio fo areas is 2 to 9 

then the ratio of the sides is   sqrt2  to  sqrt9

hence on my pic the first sides I could label were   sqrt(2) *x     and    3x

Then I took it from there.