Jimmy runs a successful pizza shop.  In the middle of a busy day, he realizes that he is running low on ingredients.  Each pizza must have 1 lb of dough, $\frac14$ lb of cheese, $\frac16$ lb of sauce, and $\frac13$ lb of toppings, which include pepperonis, mushrooms, olives, and sausages.  Given that Jimmy currently has 200 lbs of dough, 20 lbs of cheese, 20 lbs of sauce, 15 lbs of pepperonis, 5 lbs of mushrooms, 5 lbs of olives, and 10 lbs of sausages, what is the maximum number of pizzas that JImmy can make?
1. **Determine the limiting ingredient for each component:**
   - Dough: Each pizza requires 1 lb of dough. Jimmy has 200 lbs of dough.
     \[
     \text{Number of pizzas from dough} = \frac{200 \text{ lbs}}{1 \text{ lb/pizza}} = 200 \text{ pizzas}
     \]
   - Cheese: Each pizza requires \(\frac{1}{4}\) lb of cheese. Jimmy has 20 lbs of cheese.
     \[
     \text{Number of pizzas from cheese} = \frac{20 \text{ lbs}}{\frac{1}{4} \text{ lb/pizza}} = 20 \times 4 = 80 \text{ pizzas}
     \]
   - Sauce: Each pizza requires \(\frac{1}{6}\) lb of sauce. Jimmy has 20 lbs of sauce.
     \[
     \text{Number of pizzas from sauce} = \frac{20 \text{ lbs}}{\frac{1}{6} \text{ lb/pizza}} = 20 \times 6 = 120 \text{ pizzas}
     \]
   - Toppings: Each pizza requires \(\frac{1}{3}\) lb of toppings. Jimmy has a total of 15 lbs of pepperonis, 5 lbs of mushrooms, 5 lbs of olives, and 10 lbs of sausages.
     \[
     \text{Total toppings} = 15 \text{ lbs} + 5 \text{ lbs} + 5 \text{ lbs} + 10 \text{ lbs} = 35 \text{ lbs}
     \]
     \[
     \text{Number of pizzas from toppings} = \frac{35 \text{ lbs}}{\frac{1}{3} \text{ lb/pizza}} = 35 \times 3 = 105 \text{ pizzas}
     \]

2. **Identify the limiting factor:**
   - Dough: 200 pizzas
   - Cheese: 80 pizzas
   - Sauce: 120 pizzas
   - Toppings: 105 pizzas

   The limiting factor is the cheese, which allows for a maximum of 80 pizzas.

Conclusion:
The maximum number of pizzas Jimmy can make is \(\boxed{80}\).