Perform the following arithmetic operations. Reduce your final answer to lowest terms. Express a negative fraction by including "-" in the numerator. \( -\left(\frac{2}{3} \times 2 \frac{1}{2}\right)-\left(1 \frac{1}{8} \div-6\right)+\left(-1 \frac{5}{6}\right)\left(-\frac{9}{11}\right) \)
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To compute the expression step by step: 1. **Calculate \( -\left(\frac{2}{3} \times 2 \frac{1}{2}\right) \)**: First, convert \( 2 \frac{1}{2} \) to an improper fraction: \( 2 \frac{1}{2} = 2 \times 2 + 1 = 5/2 \). Now multiply: \( \frac{2}{3} \times \frac{5}{2} = \frac{2 \times 5}{3 \times 2} = \frac{10}{6} = \frac{5}{3} \). Applying the negative: \( -\frac{5}{3} \). 2. **Calculate \( -\left(1 \frac{1}{8} \div -6\right) \)**: Convert \( 1 \frac{1}{8} \): \( 1 \frac{1}{8} = 1 \times 8 + 1 = \frac{9}{8} \). Performing the division: \( \frac{9}{8} \div -6 = \frac{9}{8} \times -\frac{1}{6} = -\frac{9}{48} = -\frac{3}{16} \). Applying the negative: \( -\left(-\frac{3}{16}\right) = \frac{3}{16} \). 3. **Calculate \(\left(-1 \frac{5}{6}\right)\left(-\frac{9}{11}\right)\)**: Convert \( -1 \frac{5}{6} \) to an improper fraction: \( -1 \frac{5}{6} = -\frac{11}{6} \). Now multiply: \(-\frac{11}{6} \cdot -\frac{9}{11} = \frac{11 \cdot 9}{6 \cdot 11} = \frac{9}{6} = \frac{3}{2}\). Putting it all together: \[ -\frac{5}{3} + \frac{3}{16} + \frac{3}{2} \] To combine the fractions, we need a common denominator, which is 48: - Convert \( -\frac{5}{3} \): \( -\frac{5}{3} = -\frac{5 \times 16}{3 \times 16} = -\frac{80}{48} \). - Convert \( \frac{3}{16} \): \( \frac{3}{16} = \frac{3 \times 3}{16 \times 3} = \frac{9}{48} \). - Convert \( \frac{3}{2} \): \( \frac{3}{2} = \frac{3 \times 24}{2 \times 24} = \frac{72}{48} \). Now, add them together: \[ -\frac{80}{48} + \frac{9}{48} + \frac{72}{48} = \frac{-80 + 9 + 72}{48} = \frac{1}{48} \] Final answer: \(\frac{1}{48}\)
