To find out how much ham Sam will use for his sandwiches, we need to multiply the amount of ham he uses per sandwich by the number of sandwiches he makes.
Sam uses \( \frac{11}{4} \) ounces of ham for each sandwich, and he makes \( 4 \frac{1}{2} \) sandwiches. First, we convert \( 4 \frac{1}{2} \) into an improper fraction:
\[ 4 \frac{1}{2} = 4 + \frac{1}{2} = \frac{8}{2} + \frac{1}{2} = \frac{9}{2} \]
Now we multiply the amount of ham per sandwich by the total number of sandwiches:
\[ \text{Total ham} = \left( \frac{11}{4} \right) \times \left( \frac{9}{2} \right) \]
To perform this multiplication, we multiply the numerators together and the denominators together:
\[ \text{Total ham} = \frac{11 \times 9}{4 \times 2} = \frac{99}{8} \]
Now, to convert \( \frac{99}{8} \) into a mixed number, we divide 99 by 8:
\[ 99 \div 8 = 12 \quad \text{remainder} \quad 3 \]
This gives us:
\[ \frac{99}{8} = 12 \frac{3}{8} \]
Thus, Sam will use \( 12 \frac{3}{8} \) ounces of ham for 4 and a half sandwiches.
The final answer is:
\[ \text{Sam will use } 12 \frac{3}{8} \text{ ounces of ham.} \]