To solve the expression \( \frac{1}{2} \times \frac{4}{3} + \frac{5}{6} \div \frac{7}{8} \), we'll follow the order of operations (parentheses, exponents, multiplication and division from left to right, addition and subtraction from left to right).
First, we compute each part step by step:
- Calculate \( \frac{1}{2} \times \frac{4}{3} \):
\[ \frac{1}{2} \times \frac{4}{3} = \frac{1 \times 4}{2 \times 3} = \frac{4}{6} = \frac{2}{3} \]
- Calculate \( \frac{5}{6} \div \frac{7}{8} \) (which is the same as multiplying by the reciprocal):
\[ \frac{5}{6} \div \frac{7}{8} = \frac{5}{6} \times \frac{8}{7} = \frac{5 \times 8}{6 \times 7} = \frac{40}{42} = \frac{20}{21} \]
- Now, add the results of the two calculations:
\[ \frac{2}{3} + \frac{20}{21} \]
To add these fractions, we need a common denominator. The least common multiple of 3 and 21 is 21.
Convert \( \frac{2}{3} \) to a fraction with a 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 final answer is:
\[ \frac{34}{21} \text{ or } 1 \frac{13}{21} \]
So, the result of the expression \( \frac{1}{2} \times \frac{4}{3} + \frac{5}{6} \div \frac{7}{8} \) is:
\[ \frac{34}{21} \]