No of sides of the two polygons is 4x, 5x since ratio is 4:5

Interior angle of regular polygon of n sides is (n-2)180/n = 180 - 360/n

thus for 4x sides it is 180 - 360/4x = 180 - 90/x

and for 5x sides it is 180- 360/5x = 180 - 72/x

ratio of the interior angles of 4x to 5x side is 15/16

so

(180 - 90/x)/(180 - 72/x) = 15/16

15(180 - 72/x) = 16( 180 - 90/x)

15*180 - 15*72/x = 16*180 - 16*90/x

16*180 - 15*180 = 16*90/x - 15*72/x

180 = 360/x

x = 360/180 = 2

so no of sides are 8 and 10