To solve the given expression in a reasonable way, we need to follow the order of operations and simplify step by step.

The given expression is:
\[ \left(\frac{5}{9}\right) \cdot \frac{3}{11} + \left(\frac{13}{18}\right) \cdot \frac{3}{11} \]

First, let's simplify each term separately:

1. Simplify \(\left(\frac{5}{9}\right) \cdot \frac{3}{11}\):
\[ \frac{5}{9} \cdot \frac{3}{11} = \frac{5 \cdot 3}{9 \cdot 11} = \frac{15}{99} \]

2. Simplify \(\left(\frac{13}{18}\right) \cdot \frac{3}{11}\):
\[ \frac{13}{18} \cdot \frac{3}{11} = \frac{13 \cdot 3}{18 \cdot 11} = \frac{39}{198} \]

Now, we need to add the two fractions:
\[ \frac{15}{99} + \frac{39}{198} \]

To add these fractions, we need a common denominator. The least common multiple of 99 and 198 is 198. So, we convert \(\frac{15}{99}\) to a fraction with a denominator of 198:
\[ \frac{15}{99} = \frac{15 \cdot 2}{99 \cdot 2} = \frac{30}{198} \]

Now we can add the fractions:
\[ \frac{30}{198} + \frac{39}{198} = \frac{30 + 39}{198} = \frac{69}{198} \]

Finally, we simplify \(\frac{69}{198}\):
\[ \frac{69}{198} = \frac{69 \div 3}{198 \div 3} = \frac{23}{66} \]

So, the simplified result of the given expression is:
\[ \frac{23}{66} \]