r. \( -4 \frac{1}{8}+5 \frac{1}{16} \div \frac{1}{8} \times \frac{1}{32}-\frac{1}{8} \)

To solve the expression, let's break it down step by step while keeping it engaging! First, we start with \( 5 \frac{1}{16} \div \frac{1}{8} \). Dividing by a fraction is the same as multiplying by its reciprocal, so this becomes \( 5 \frac{1}{16} \times 8 \). First, let's convert \( 5 \frac{1}{16} \) into an improper fraction: \[ 5 \frac{1}{16} = \frac{5 \times 16 + 1}{16} = \frac{80 + 1}{16} = \frac{81}{16}. \] Now we can compute: \[ \frac{81}{16} \times 8 = \frac{81 \times 8}{16} = \frac{81 \times 1}{2} = \frac{81}{2}. \] Next, we multiply \( \frac{81}{2} \) by \( \frac{1}{32} \): \[ \frac{81}{2} \times \frac{1}{32} = \frac{81}{64}. \] Now we can substitute this back into the expression: \[ -4 \frac{1}{8} + \frac{81}{64} - \frac{1}{8}. \] Converting \( -4 \frac{1}{8} \) into an improper fraction gives us: \[ -4 \frac{1}{8} = -\frac{32 + 1}{8} = -\frac{33}{8}. \] We also convert \( \frac{1}{8} \) into a fraction with a denominator of 64: \[ \frac{1}{8} = \frac{8}{64}. \] Combining these, we have: \[ -\frac{33}{8} + \frac{81}{64} - \frac{8}{64}. \] Now, changing \( -\frac{33}{8} \) to a fraction with a denominator of 64: \[ -\frac{33}{8} = -\frac{264}{64}. \] Finally, we can combine everything together: \[ -\frac{264}{64} + \frac{81}{64} - \frac{8}{64} = \frac{-264 + 81 - 8}{64} = \frac{-191}{64}. \] So, the final result of the expression is: \[ -\frac{191}{64}. \]