To solve this expression, we first need to simplify the terms inside the parentheses.

2/15 - 3 5/6 = 2/15 - (3+(5/6))
= 2/15 - (18/6 + 5/6)
= 2/15 - 23/6
= (4/30) - (115/30)
= (-111/30) or (-37/10)

Now we plug this result back into the original expression and simplify further:

2 1/2 (-37/10) + 1/4
= (5/2) (-37/10) + 1/4
= -37/4 + 1/4
= -36/4
= -9

Therefore, the final result of the expression 2 1/2×(2/15-3 5/6)+1/4 is -9.