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

검증된 단계별 안내

Identify the complex fraction as a division of two fractions: the numerator is \(\frac{5}{8} + \frac{2}{3}\) and the denominator is \(\frac{7}{3} - \frac{1}{4}\).

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.

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.

After simplifying both the numerator and denominator to single fractions, rewrite the complex fraction as a division of two fractions: \(\frac{\text{numerator fraction}}{\text{denominator fraction}}\).

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.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.