To simplify the expression \( \frac{3}{10}x + \frac{1}{6}y + \frac{3}{10}x + \frac{3}{6}y \), we can start by combining like terms.
First, let's combine the \( x \) terms:
\[ \frac{3}{10}x + \frac{3}{10}x = \left(\frac{3}{10} + \frac{3}{10}\right)x = \frac{6}{10}x \]
Next, we simplify \( \frac{6}{10} \):
\[ \frac{6}{10} = \frac{3}{5} \quad \text{(by dividing both the numerator and denominator by 2)} \]
Now let's combine the \( y \) terms:
\[ \frac{1}{6}y + \frac{3}{6}y = \left(\frac{1}{6} + \frac{3}{6}\right)y = \frac{4}{6}y \]
Next, we simplify \( \frac{4}{6} \):
\[ \frac{4}{6} = \frac{2}{3} \quad \text{(by dividing both the numerator and denominator by 2)} \]
Now we can write the simplified expression:
\[ \frac{3}{5}x + \frac{2}{3}y \]
Thus, the final simplified expression is:
\[ \frac{3}{5}x + \frac{2}{3}y \]