Simplify each complex fraction. See Examples 5 and 6. (5/8 + 2/3) ÷ (7/1 − 1/4)

Verified step by step guidance

1. Identify the complex fraction as a division of two fractions: the numerator is \(\frac{5}{8} + \frac{2}{3}\) and the denominator is \(\frac{7}{3} - \frac{1}{4}\).
2. Find a common denominator for the fractions in the numerator to combine them: for \(\frac{5}{8}\) and \(\frac{2}{3}\), the common denominator is 24. Rewrite each fraction with denominator 24 and add them.
3. Similarly, find a common denominator for the fractions in the denominator: for \(\frac{7}{3}\) and \(\frac{1}{4}\), the common denominator is 12. Rewrite each fraction with denominator 12 and subtract them.
4. After simplifying both the numerator and denominator to single fractions, rewrite the complex fraction as a division of two fractions: \(\frac{\text{numerator fraction}}{\text{denominator fraction}}\).
5. Recall that dividing by a fraction is the same as multiplying by its reciprocal. Multiply the numerator fraction by the reciprocal of the denominator fraction and simplify the resulting fraction if possible.

Key Concepts

Complex Fractions

A complex fraction is a fraction where the numerator, denominator, or both contain fractions themselves. Simplifying involves rewriting the complex fraction as a single simple fraction by performing operations on the numerator and denominator separately before dividing.

Addition and Subtraction of Fractions

To add or subtract fractions, they must have a common denominator. Find the least common denominator (LCD), convert each fraction accordingly, then add or subtract the numerators while keeping the denominator the same.

Division of Fractions

Dividing by a fraction is equivalent to multiplying by its reciprocal. After simplifying the numerator and denominator of a complex fraction, divide by multiplying the numerator by the reciprocal of the denominator to get the final simplified fraction.