**Example 1: Adding Fractions with Common Denominator**
Add: 1/3 + 2/3

- Since the denominators are the same, simply add the numerators: 1 + 2 = 3.
- The denominator remains 3.
- The sum is 3/3, which is equivalent to 1.

**Example 2: Multiplying Fractions**
Multiply: 3/4 * 5/6

- Multiply the numerators: 3 * 5 = 15.
- Multiply the denominators: 4 * 6 = 24.
- The product is 15/24.
- Simplify the fraction: Divide both the numerator and denominator by their greatest common divisor, which is 3.
- The simplified fraction is 5/8.

**Example 3: Dividing Fractions**
Divide: 2/5 ÷ 3/4

- Multiply the first fraction by the reciprocal of the second: 2/5 * 4/3.
- Multiply the numerators: 2 * 4 = 8.
- Multiply the denominators: 5 * 3 = 15.
- The result is 8/15.

**Example 4: Comparing Fractions**
Compare: 7/12 and 5/8

- Find a common denominator for comparison. In this case, the least common multiple of 12 and 8 is 24.
- Convert both fractions to have the common denominator: 7/12 becomes 14/24, and 5/8 becomes 15/24.
- Now it's easy to see that 15/24 is larger than 14/24.

**Example 5: Solving a Fraction Equation**
Solve: 3/x = 1/4

- Cross-multiply: 3 * 4 = 12, so x = 12.

**Example 6: Estimating Fractions**
Add: 7/8 + 3/5

- Estimate that 7/8 is close to 1 and 3/5 is close to 1/2.
- The sum should be approximately 1 + 1/2 = 1.5.

These examples demonstrate how to apply various techniques to solve fractions quickly and efficiently. With practice, you'll become more comfortable with these methods and be able to solve fractions rapidly in various situations.