\frac{1}{2}.\ \frac{4}{3\ }\ +\ \frac{5}{6}\ \div\ \frac{7}{8\ }\ =
A. 1 27/48
B.1 13/21
C. 1 33/42
D. 1 19/48
To evaluate the expression, we need to simplify each fraction and perform the operations in the given order.
First, we simplify the fraction $\frac{4}{3}$:
$\frac{4}{3} = \frac{4 \div 1}{3 \div 1} = \frac{4}{3}$.
Next, we simplify the fraction $\frac{7}{8}$:
$\frac{7}{8} = \frac{7 \div 1}{8 \div 1} = \frac{7}{8}$.
Now, we perform the division $\frac{5}{6} \div \frac{7}{8}$:
$\frac{5}{6} \div \frac{7}{8} = \frac{5}{6} \times \frac{8}{7} = \frac{5 \times 8}{6 \times 7} = \frac{40}{42} = \frac{20}{21}$.
Finally, we add the fractions $\frac{1}{2}$ and $\frac{20}{21}$:
$\frac{1}{2} + \frac{20}{21} = \frac{1 \times 21}{2 \times 21} + \frac{20 \times 2}{21 \times 2} = \frac{21}{42} + \frac{40}{42} = \frac{61}{42} = 1 \frac{19}{42}$.
Therefore, the answer is $\boxed{\text{(D) } 1 \: \frac{19}{42}}$.