(f) \( 5 \frac{1}{6} \cdot\left(-\frac{7}{9}\right) \)

To calculate the product \( 5 \frac{1}{6} \times \left(-\frac{7}{9}\right) \), follow these steps: 1. **Convert the Mixed Number to an Improper Fraction:** The mixed number \( 5 \frac{1}{6} \) can be converted to an improper fraction as follows: \[ 5 \frac{1}{6} = 5 + \frac{1}{6} = \frac{30}{6} + \frac{1}{6} = \frac{31}{6} \] 2. **Multiply the Fractions:** Now, multiply the improper fraction by \(-\frac{7}{9}\): \[ \frac{31}{6} \times \left(-\frac{7}{9}\right) = -\frac{31 \times 7}{6 \times 9} = -\frac{217}{54} \] 3. **Simplify the Result (Optional):** The fraction \(-\frac{217}{54}\) can be simplified or converted back to a mixed number: - **Simplified Fraction:** \(-\frac{217}{54}\) already has no common factors other than 1, so it cannot be simplified further. - **Mixed Number:** Divide 217 by 54: \[ 217 \div 54 = 4 \text{ with a remainder of } 1 \] So, the mixed number form is: \[ -4 \frac{1}{54} \] **Final Answer:** \[ 5 \frac{1}{6} \times \left(-\frac{7}{9}\right) = -\frac{217}{54} = -4 \frac{1}{54} \]