The problem here is to add the mixed numbers $7 \frac{4}{9}$ and $4 \frac{9}{18}$.

Let's break down the solution step by step:

1. Convert the mixed numbers to improper fractions:

   • For $7 \frac{4}{9}$: $7 \frac{4}{9} = \frac{7 \times 9 + 4}{9} = \frac{63 + 4}{9} = \frac{67}{9}$
   • For $4 \frac{9}{18}$: $4 \frac{9}{18} = \frac{4 \times 18 + 9}{18} = \frac{72 + 9}{18} = \frac{81}{18}$ Since $\frac{9}{18}$ simplifies to $\frac{1}{2}$, this expression could also be written as $4.5$.

2. Convert fractions to a common denominator (if necessary):

   • Here, the denominators are 9 and 18. The least common multiple is 18.
   • Convert $\frac{67}{9}$ to have a denominator of 18: $\frac{67}{9} = \frac{67 \times 2}{9 \times 2} = \frac{134}{18}$

3. Add the fractions: $\frac{134}{18} + \frac{81}{18} = \frac{134 + 81}{18} = \frac{215}{18}$

4. Convert back to a mixed number (if required):

   • Divide 215 by 18: $215 \div 18 = 11 \text{ remainder } 17$
   • So, $\frac{215}{18} = 11 \frac{17}{18}$.

Final Answer:

$7 \frac{4}{9} + 4 \frac{9}{18} = 11 \frac{17}{18}$

Would you like more details on any part of this solution?

Here are five questions to explore related concepts:

1. How can you convert mixed numbers to improper fractions?
2. What is the least common multiple, and why is it used in adding fractions?
3. How do you simplify fractions to their lowest terms?
4. How can you convert an improper fraction back to a mixed number?
5. What other methods exist for adding mixed numbers?

Tip: Always simplify fractions as much as possible before adding or subtracting them for easier calculations.