1. **Problem:** Calculate $$4 \frac{1}{6} - \frac{1}{2} \times 9$$. 2. **Formula and rules:** Follow order of operations (PEMDAS): multiplication before subtraction. 3. **Step-by-step:** - Convert mixed number: $$4 \frac{1}{6} = \frac{25}{6}$$. - Multiply: $$\frac{1}{2} \times 9 = \frac{9}{2}$$. - Subtract: $$\frac{25}{6} - \frac{9}{2} = \frac{25}{6} - \frac{27}{6} = -\frac{2}{6} = -\frac{1}{3}$$. 4. **Answer:** $$-\frac{1}{3}$$. --- 1. **Problem:** Calculate $$1 \frac{1}{2} \div \left(-1 \frac{1}{3}\right) - 3 \frac{1}{4}$$. 2. **Formula and rules:** Division and subtraction; convert mixed numbers to improper fractions. 3. **Step-by-step:** - Convert: $$1 \frac{1}{2} = \frac{3}{2}$$, $$-1 \frac{1}{3} = -\frac{4}{3}$$, $$3 \frac{1}{4} = \frac{13}{4}$$. - Divide: $$\frac{3}{2} \div -\frac{4}{3} = \frac{3}{2} \times -\frac{3}{4} = -\frac{9}{8}$$. - Subtract: $$-\frac{9}{8} - \frac{13}{4} = -\frac{9}{8} - \frac{26}{8} = -\frac{35}{8} = -4 \frac{3}{8}$$. 4. **Answer:** $$-4 \frac{3}{8}$$. --- 1. **Problem:** Calculate $$1 \frac{1}{2} \times \left(-\frac{2}{3}\right) - \left(-\frac{4}{15}\right) \div \frac{2}{3}$$. 2. **Formula and rules:** Multiplication and division with negative fractions. 3. **Step-by-step:** - Convert: $$1 \frac{1}{2} = \frac{3}{2}$$. - Multiply: $$\frac{3}{2} \times -\frac{2}{3} = -1$$. - Divide: $$-\left(-\frac{4}{15}\right) \div \frac{2}{3} = \frac{4}{15} \times \frac{3}{2} = \frac{12}{30} = \frac{2}{5}$$. - Subtract: $$-1 - \frac{2}{5} = -\frac{5}{5} - \frac{2}{5} = -\frac{7}{5} = -1 \frac{2}{5}$$. 4. **Answer:** $$-1 \frac{2}{5}$$. --- 1. **Problem:** Calculate $$\frac{12 \times (-6)^2}{(-16) \times (-3)^3}$$. 2. **Formula and rules:** Calculate powers first, then multiplication and division. 3. **Step-by-step:** - Powers: $$(-6)^2 = 36$$, $$(-3)^3 = -27$$. - Numerator: $$12 \times 36 = 432$$. - Denominator: $$-16 \times -27 = 432$$. - Divide: $$\frac{432}{432} = 1$$. 4. **Answer:** $$1$$.