In patty's room, 1/2 of her award ribbons are for horse riding and 1/6 are for ballet. what fraction of the awards are for another activity?

1/2 = 3/6
So, 
(1/2) + (1/6) = (3/6) + (1/6)
(1/2) + (1/6) = (3+1)/6
(1/2) + (1/6) = 4/6
(1/2) + (1/6) = 2/3

2/3 of the awards are for horse riding or ballet
So 1/3 is for the rest since 1 - (2/3) = (3/3) - (2/3) = (3-2)/3 = 1/3

1/3