Perform the division inside the parentheses. To divide by a fraction, multiply by its reciprocal.

Here are the solutions to the fraction problems:

1) $(\frac{1}{2} \div \frac{1}{3}) \times \frac{1}{3}$

Step 1: Perform the division inside the parentheses. To divide by a fraction, multiply by its reciprocal.

$$\frac{1}{2} \div \frac{1}{3} = \frac{1}{2} \times \frac{3}{1} = \frac{3}{2}$$

Step 2: Multiply the result by $\frac{1}{3}$.

$$\frac{3}{2} \times \frac{1}{3} = \frac{3}{6}$$

Step 3: Simplify the fraction.

$$\frac{3}{6} = \frac{1}{2}$$

The final answer is $\boxed{\frac{1}{2}}$.

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

Step 1: Perform the multiplication first.

$$\frac{2}{3} \times \frac{1}{4} = \frac{2 \times 1}{3 \times 4} = \frac{2}{12}$$

Simplify the fraction:

$$\frac{2}{12} = \frac{1}{6}$$

Step 2: Perform the addition. Find a common denominator for $\frac{1}{2}$ and $\frac{1}{6}$, which is 6.

$$\frac{1}{2} + \frac{1}{6} = \frac{1 \times 3}{2 \times 3} + \frac{1}{6} = \frac{3}{6} + \frac{1}{6} = \frac{3+1}{6} = \frac{4}{6}$$

Step 3: Simplify the fraction.

$$\frac{4}{6} = \frac{2}{3}$$

The final answer is $\boxed{\frac{2}{3}}$.

3) $\frac{1}{4} - \frac{2}{3} + \frac{5}{6}$

Step 1: Find a common denominator for 4, 3, and 6, which is 12. Convert each fraction.

$$\frac{1}{4} = \frac{1 \times 3}{4 \times 3} = \frac{3}{12}$$ $$\frac{2}{3} = \frac{2 \times 4}{3 \times 4} = \frac{8}{12}$$ $$\frac{5}{6} = \frac{5 \times 2}{6 \times 2} = \frac{10}{12}$$

Step 2: Perform the operations from left to right.

$$\frac{3}{12} - \frac{8}{12} + \frac{10}{12} = \frac{3-8}{12} + \frac{10}{12} = \frac{-5}{12} + \frac{10}{12} = \frac{-5+10}{12} = \frac{5}{12}$$

The final answer is $\boxed{\frac{5}{12}}$.

4) $\frac{3}{4} \div \frac{1}{2} \times \frac{1}{6}$

Step 1: Perform the division first (from left to right). To divide by a fraction, multiply by its reciprocal.

$$\frac{3}{4} \div \frac{1}{2} = \frac{3}{4} \times \frac{2}{1} = \frac{6}{4}$$

Simplify the fraction:

$$\frac{6}{4} = \frac{3}{2}$$

Step 2: Multiply the result by $\frac{1}{6}$.

$$\frac{3}{2} \times \frac{1}{6} = \frac{3}{12}$$

Step 3: Simplify the fraction.

$$\frac{3}{12} = \frac{1}{4}$$

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

5) $\frac{2}{3} \text{ of } 2\frac{1}{4} \div \frac{1}{4}$

Step 1: Convert the mixed number to an improper fraction.

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

Step 2: Replace "of" with multiplication and perform the multiplication.

$$\frac{2}{3} \times \frac{9}{4} = \frac{18}{12}$$

Simplify the fraction:

$$\frac{18}{12} = \frac{3}{2}$$

Step 3: Perform the division. To divide by a fraction, multiply by its reciprocal.

$$\frac{3}{2} \div \frac{1}{4} = \frac{3}{2} \times \frac{4}{1} = \frac{12}{2}$$

Step 4: Simplify the result.

$$\frac{12}{2} = 6$$

The final answer is $\boxed{6}$.

6) $\frac{2}{3} - \frac{5}{6} + \frac{1}{2}$

Step 1: Find a common denominator for 3, 6, and 2, which is 6. Convert each fraction.

$$\frac{2}{3} = \frac{2 \times 2}{3 \times 2} = \frac{4}{6}$$ $$\frac{1}{2} = \frac{1 \times 3}{2 \times 3} = \frac{3}{6}$$

Step 2: Perform the operations from left to right.

$$\frac{4}{6} - \frac{5}{6} + \frac{3}{6} = \frac{4-5}{6} + \frac{3}{6} = \frac{-1}{6} + \frac{3}{6} = \frac{-1+3}{6} = \frac{2}{6}$$

Step 3: Simplify the fraction.

$$\frac{2}{6} = \frac{1}{3}$$

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

7) $\frac{1}{2} \div \frac{4}{3}$

Step 1: To divide by a fraction, multiply by its reciprocal.

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