Middle man

Note that $48 \%$ of the area of the smaller square is equivalent to $27 \%$ of the area of the larger square. Thus, the total areas must be in a $\frac{100}{48}:\frac{100}{27}=27:48$ ratio. The ratio of the side lengths is thus $\sqrt{\frac{27}{48}}=\boxed{\frac{3}{4}}$ .

We note that if smaller one has an area of a^2 and the area of larger square is b^2 then 48 a^2 is equal to 27 b^2 then the rest solution can be proceeded. After that it's easy. This problem was easy. No need to see the solution.

S = small square side length

B = Big square side length

The overlapping surface area is the following :

( 1 - 0.52 ) S^2 = ( 1 - 0.73 ) B^2

Solving for S / B :

( S^2 ) / (B^2) = ( 0.27 / 0.48 )

S / B = sqrt ( 0.27 / 0.48 )

S / B = 3 / 4

each side of the yellow square is 3/4 of each the yellow square