Simplify \( \frac{16 \frac{3}{4} \times 4^{1 / 2}}{85 / 2} \)

To simplify \( \frac{16 \frac{3}{4} \times 4^{1 / 2}}{85 / 2} \), we start by converting the mixed number \( 16 \frac{3}{4} \) into an improper fraction. \[ 16 \frac{3}{4} = 16 + \frac{3}{4} = \frac{16 \times 4}{4} + \frac{3}{4} = \frac{64}{4} + \frac{3}{4} = \frac{67}{4} \] Next, we evaluate \( 4^{1/2} \), which is simply 2 because the square root of 4 is 2. Thus, we can rewrite the expression: \[ \frac{67/4 \times 2}{85/2} \] Now, we can rewrite \( 2 \) as \( \frac{2}{1} \): \[ \frac{67}{4} \times \frac{2}{1} = \frac{134}{4} \] Now we have: \[ \frac{\frac{134}{4}}{\frac{85}{2}} \] To divide fractions, we multiply by the reciprocal: \[ \frac{134}{4} \times \frac{2}{85} \] Next, we can simplify this. First, multiply the numerators and the denominators: \[ \frac{134 \times 2}{4 \times 85} = \frac{268}{340} \] Now, we simplify \( \frac{268}{340} \). Both 268 and 340 can be divided by 4: \[ \frac{268 \div 4}{340 \div 4} = \frac{67}{85} \] So, the simplified form of the original expression is \[ \frac{67}{85} \]