To solve the expression \( \frac{1}{2} \times \frac{4}{3} + \frac{5}{6} \div \frac{7}{8} \), we will follow the order of operations: first multiplication and division, then addition.
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Multiplication: \[ \frac{1}{2} \times \frac{4}{3} = \frac{1 \times 4}{2 \times 3} = \frac{4}{6} = \frac{2}{3} \]
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Division: When dividing by a fraction, we multiply by its reciprocal. Thus: \[ \frac{5}{6} \div \frac{7}{8} = \frac{5}{6} \times \frac{8}{7} = \frac{5 \times 8}{6 \times 7} = \frac{40}{42} = \frac{20}{21} \]
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Addition: Now, we add \( \frac{2}{3} \) and \( \frac{20}{21} \). To do this, we need a common denominator. The least common multiple of 3 and 21 is 21.
Convert \( \frac{2}{3} \) to have a denominator of 21: \[ \frac{2}{3} = \frac{2 \times 7}{3 \times 7} = \frac{14}{21} \]
Now, add the fractions: \[ \frac{14}{21} + \frac{20}{21} = \frac{14 + 20}{21} = \frac{34}{21} \]
So, the final result of the calculation \( \frac{1}{2} \times \frac{4}{3} + \frac{5}{6} \div \frac{7}{8} \) is: \[ \frac{34}{21} \]
This can also be expressed as a mixed number: \[ 1 \frac{13}{21} \]