Given:
QUANTITY A: \(\frac{1}{6}- (\frac{1}{2})^2 + (\frac{-1}{4})^2\)
QUANTITY B: \(\frac{1}{6}\)

Expand the two powers in Quantity A to get:
QUANTITY A: \(\frac{1}{6}- \frac{1}{4} + \frac{1}{16}\)
QUANTITY B: \(\frac{1}{6}\)

Rewrite the fractions in Quantity A with a common denominator of 48 to get:
QUANTITY A: \(\frac{8}{48}- \frac{12}{48} + \frac{3}{48}\)
QUANTITY B: \(\frac{1}{6}\)

Combine the fractions in Quantity A to get:
QUANTITY A: \(-\frac{1}{48}\)
QUANTITY B: \(\frac{1}{6}\)

Answer: B