First observe that the area of the quarter circle with centre B is made up of area X + the area of the two semi circles - area Y (The area Y needs to be taken away because it has been counted twice, once in each semi circle.)
Since we are looking for a ratio we can choose any scale we like to measure the problem. To ease the calculations suppose the square has a side of 2 units.
Then the big quarter circle has area 1/4*Pi*2^2 = Pi
Each semi circle has area 1/2 Pi 1^2 = Pi/2
So by the above observation Pi = X + Pi - Y
Or X = Y So ratio is 1.
