To simplify the expression \( \frac{5}{8}x + \frac{1}{2} \left(\frac{1}{4}x + 10\right) \), let's start by distributing \( \frac{1}{2} \) across the expression inside the parentheses.
\[ \frac{1}{2} \left( \frac{1}{4}x + 10 \right) = \frac{1}{2} \cdot \frac{1}{4}x + \frac{1}{2} \cdot 10 \] \[ = \frac{1}{8}x + 5 \]
Now, we can substitute this back into the original expression:
\[ \frac{5}{8}x + \left(\frac{1}{8}x + 5\right) \]
Next, we can combine the terms containing \( x \):
\[ \frac{5}{8}x + \frac{1}{8}x = \left(\frac{5}{8} + \frac{1}{8}\right)x = \frac{6}{8}x \]
Now, simplify \( \frac{6}{8} \):
\[ \frac{6}{8} = \frac{3}{4} \]
Thus, we have:
\[ \frac{3}{4}x + 5 \]
So, the final simplified expression is:
\[ \frac{3}{4}x + 5 \]