Simplify each complex fraction. See Examples 5 and 6. 5⁄8 + 2⁄3———— 7⁄3 - 1⁄4

Verified step by step guidance

1. Step 1: Simplify the numerator by finding a common denominator for the fractions \( \frac{5}{8} \) and \( \frac{2}{3} \).
2. Step 2: Simplify the denominator by finding a common denominator for the fractions \( \frac{7}{3} \) and \( \frac{1}{4} \).
3. Step 3: Add the fractions in the numerator after converting them to have a common denominator.
4. Step 4: Subtract the fractions in the denominator after converting them to have a common denominator.
5. Step 5: Divide the simplified numerator by the simplified denominator to get the final simplified complex fraction.

Key Concepts

Complex Fractions

A complex fraction is a fraction where the numerator, the denominator, or both contain fractions themselves. To simplify complex fractions, one typically finds a common denominator for the fractions involved, allowing for easier addition or subtraction before simplifying the overall fraction.

Common Denominator

A common denominator is a shared multiple of the denominators of two or more fractions. When adding or subtracting fractions, converting them to a common denominator is essential, as it allows for direct combination of the numerators, leading to a simplified result.

Order of Operations

The order of operations is a set of rules that dictates the sequence in which mathematical operations should be performed to ensure consistent results. In the context of simplifying fractions, it is crucial to follow these rules—parentheses first, then exponents, followed by multiplication and division, and finally addition and subtraction—to accurately simplify complex expressions.