(1) Triangles XOY and XOM have the same height if measured from vertex x. (The height is the distance from x to the line MY.) The bases are in the ratio of 2:5, so the areas are in the ratio 2:5 or 4:10. ((If two triangles have equal heights, then the ratio of their areas equals the ratio of their bases.))
(2) I believe that there is some piece of information missing -- probably that XY||MN. We need triangles XOY and NOM to be similar. (I'm not sure what allowed you to write in OY=2 and OM=5 -- whatever information that came from could be used to show the triangles are similar.)
With the triangles similar, the ratio of their areas is the square of the ratios of the scale factor. Since XY corresponds to MN the scale factor is 2:5 and the ratio of the areas is 4:25.
Thus the ratio of the areas is 4:10:25