Let the number of pounds of Type A coffee used be denoted as \( x \). According to the problem, the number of pounds of Type B coffee used is three times the pounds of Type A, which can be expressed as \( 3x \).
Next, we can calculate the total cost for both types of coffee:
- The cost for Type A coffee is \( 5.95x \) (since it costs $5.95 per pound).
- The cost for Type B coffee is \( 4.65(3x) = 13.95x \) (since it costs $4.65 per pound and there are \( 3x \) pounds).
The total cost for the mixture is given by the sum of the costs of Type A and Type B coffee, which equals $796. This leads to the following equation:
\[ 5.95x + 13.95x = 796 \]
Combining like terms, we have:
\[ 19.90x = 796 \]
To find \( x \), we divide both sides of the equation by 19.90:
\[ x = \frac{796}{19.90} \approx 40 \]
Thus, the number of pounds of Type A coffee used is \( \boxed{40} \).