To solve the calculation \( \frac{1}{2} \times \frac{4}{3} + \frac{5}{6} \div \frac{7}{8} \), we will break it down into two parts.
-
Calculate \( \frac{1}{2} \times \frac{4}{3} \): \[ \frac{1}{2} \times \frac{4}{3} = \frac{1 \times 4}{2 \times 3} = \frac{4}{6} = \frac{2}{3} \quad (\text{by simplifying } \frac{4}{6} \text{ to } \frac{2}{3}) \]
-
Calculate \( \frac{5}{6} \div \frac{7}{8} \): Dividing by a fraction is the same as multiplying by its reciprocal. So, \[ \frac{5}{6} \div \frac{7}{8} = \frac{5}{6} \times \frac{8}{7} = \frac{5 \times 8}{6 \times 7} = \frac{40}{42} = \frac{20}{21} \quad (\text{by simplifying } \frac{40}{42} \text{ to } \frac{20}{21}) \]
-
Now we add \( \frac{2}{3} + \frac{20}{21} \): First, we need a common denominator for \( \frac{2}{3} \) and \( \frac{20}{21} \). The least common multiple of 3 and 21 is 21.
Convert \( \frac{2}{3} \) to have the denominator of 21: \[ \frac{2}{3} = \frac{2 \times 7}{3 \times 7} = \frac{14}{21} \]
Now add \( \frac{14}{21} + \frac{20}{21} \): \[ \frac{14 + 20}{21} = \frac{34}{21} \]
Thus, the result of the calculation is: \[ \frac{34}{21} \]
This can also be expressed as a mixed number: \[ 1 \frac{13}{21} \]
So the final answer is: \[ \frac{34}{21} \text{ or } 1 \frac{13}{21} \]