Solve the following arithmetic problems: 1. 1/2 + 2 3/4 - 1/8 2. 2 1/2 4 / 5/6 3. 4 1/3 + 2 1/2 4 1/2

$$2\frac{3}{4} = \frac{(2 \times 4) + 3}{4} = \frac{8 + 3}{4} = \frac{11}{4}$$

The expression becomes:

$$\frac{1}{2} + \frac{11}{4} - \frac{1}{8}$$

$$\frac{25}{8} = 3\frac{1}{8}$$

The final answer is $\boxed{\mathbf{3\frac{1}{8}}}$.

2. $2\frac{1}{2} \times 4 \div \frac{5}{6}$

$$2\frac{1}{2} = \frac{(2 \times 2) + 1}{2} = \frac{5}{2}$$

The expression becomes:

$$\frac{5}{2} \times 4 \div \frac{5}{6}$$

$$\frac{5}{2} \times 4 = \frac{5 \times 4}{2} = \frac{20}{2} = 10$$

The expression becomes:

$$10 \div \frac{5}{6}$$

$$10 \times \frac{6}{5} = \frac{10 \times 6}{5} = \frac{60}{5} = 12$$

The final answer is $\boxed{\mathbf{12}}$.

3. $4\frac{1}{3} + 2\frac{1}{2} \times 4\frac{1}{2}$

$$4\frac{1}{3} = \frac{(4 \times 3) + 1}{3} = \frac{13}{3}$$ $$2\frac{1}{2} = \frac{(2 \times 2) + 1}{2} = \frac{5}{2}$$ $$4\frac{1}{2} = \frac{(4 \times 2) + 1}{2} = \frac{9}{2}$$

The expression becomes:

$$\frac{13}{3} + \frac{5}{2} \times \frac{9}{2}$$

$$\frac{5}{2} \times \frac{9}{2} = \frac{5 \times 9}{2 \times 2} = \frac{45}{4}$$

The expression becomes:

$$\frac{13}{3} + \frac{45}{4}$$

$$\frac{187}{12} = 15\frac{7}{12}$$

The final answer is $\boxed{\mathbf{15\frac{7}{12}}}$.

4. $16\frac{1}{8} + 8\frac{1}{4} \div 3\frac{2}{3}$

$$16\frac{1}{8} = \frac{(16 \times 8) + 1}{8} = \frac{128 + 1}{8} = \frac{129}{8}$$ $$8\frac{1}{4} = \frac{(8 \times 4) + 1}{4} = \frac{32 + 1}{4} = \frac{33}{4}$$ $$3\frac{2}{3} = \frac{(3 \times 3) + 2}{3} = \frac{9 + 2}{3} = \frac{11}{3}$$

The expression becomes:

$$\frac{129}{8} + \frac{33}{4} \div \frac{11}{3}$$

$$\frac{33}{4} \div \frac{11}{3} = \frac{33}{4} \times \frac{3}{11} = \frac{33 \times 3}{4 \times 11} = \frac{3 \times 11 \times 3}{4 \times 11} = \frac{3 \times 3}{4} = \frac{9}{4}$$

The expression becomes:

$$\frac{129}{8} + \frac{9}{4}$$

$$\frac{147}{8} = 18\frac{3}{8}$$

The final answer is $\boxed{\mathbf{18\frac{3}{8}}}$.

5. $\frac{3}{4} - \frac{1}{2} \times \frac{1}{8}$

$$\frac{1}{2} \times \frac{1}{8} = \frac{1 \times 1}{2 \times 8} = \frac{1}{16}$$

The expression becomes:

$$\frac{3}{4} - \frac{1}{16}$$

$$\frac{12 - 1}{16} = \frac{11}{16}$$

The final answer is $\boxed{\mathbf{\frac{11}{16}}}$.

6. $5\frac{4}{7} - 16\frac{2}{3} \div 4\frac{1}{6}$

$$5\frac{4}{7} = \frac{(5 \times 7) + 4}{7} = \frac{35 + 4}{7} = \frac{39}{7}$$ $$16\frac{2}{3} = \frac{(16 \times 3) + 2}{3} = \frac{48 + 2}{3} = \frac{50}{3}$$ $$4\frac{1}{6} = \frac{(4 \times 6) + 1}{6} = \frac{24 + 1}{6} = \frac{25}{6}$$

The expression becomes:

$$\frac{39}{7} - \frac{50}{3} \div \frac{25}{6}$$

$$\frac{50}{3} \div \frac{25}{6} = \frac{50}{3} \times \frac{6}{25} = \frac{50 \times 6}{3 \times 25} = \frac{2 \times 25 \times 2 \times 3}{3 \times 25} = 2 \times 2 = 4$$

The expression becomes:

$$\frac{39}{7} - 4$$

$$\frac{11}{7} = 1\frac{4}{7}$$

The final answer is $\boxed{\mathbf{1\frac{4}{7}}}$.

7. Subtract $7\frac{3}{8}$ from the sum of $1\frac{11}{12}$ and $10\frac{1}{6}$.

Convert mixed numbers to improper fractions:

$$1\frac{11}{12} = \frac{(1 \times 12) + 11}{12} = \frac{23}{12}$$ $$10\frac{1}{6} = \frac{(10 \times 6) + 1}{6} = \frac{61}{6}$$

Add the fractions:

$$\frac{23}{12} + \frac{61}{6} = \frac{23}{12} + \frac{61 \times 2}{6 \times 2} = \frac{23}{12} + \frac{122}{12} = \frac{23 + 122}{12} = \frac{145}{12}$$

Convert $7\frac{3}{8}$ to an improper fraction:

$$7\frac{3}{8} = \frac{(7 \times 8) + 3}{8} = \frac{56 + 3}{8} = \frac{59}{8}$$

Now subtract:

$$\frac{145}{12} - \frac{59}{8}$$

Find a common denominator for 12 and 8, which is 24.

$$\frac{145 \times 2}{12 \times 2} - \frac{59 \times 3}{8 \times 3} = \frac{290}{24} - \frac{177}{24}$$ $$\frac{290 - 177}{24} = \frac{113}{24}$$

$$\frac{113}{24} = 4\frac{17}{24}$$

The final answer is $\boxed{\mathbf{4\frac{17}{24}}}$.

8. Divide $4\frac{2}{7}$ by $7\frac{1}{2}$ and multiply the result by $\frac{1}{2}$.

$$4\frac{2}{7} = \frac{(4 \times 7) + 2}{7} = \frac{28 + 2}{7} = \frac{30}{7}$$ $$7\frac{1}{2} = \frac{(7 \times 2) + 1}{2} = \frac{14 + 1}{2} = \frac{15}{2}$$

$$\frac{4}{14} = \frac{2}{7}$$

The final answer is $\boxed{\mathbf{\frac{2}{7}}}$.

9. Divide the difference between $10\frac{1}{12}$ and $2\frac{7}{8}$ by $\frac{3}{4}$.

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