Jane bought 36 pies and Kelly bought 36 muffins. The total cost of all pies and muffins was $576. Jane exchanged 6 pies with Kelly for 6 muffins. After the exchange, the total cost of the items Jane had to the total cost of the items Kelly had was 2:1.

1. Determine the total number of units representing the total cost after the exchange.

The total cost of all items remains unchanged after the exchange, which is $\char36{}576$.
The ratio of the total cost of Jane's items to Kelly's items is $2:1$.
Total units = $2 + 1 = 3$ units.

2. Calculate the cost of the items Kelly had after the exchange.

Kelly has 1 unit out of the 3 units.
Cost of Kelly's items = $\frac{1}{3} \times 576 = \char36{}192$.

1. Find the total cost of 1 pie and 1 muffin from the initial total.

Total cost of 36 pies and 36 muffins = $\char36{}576$
Cost of 1 pie + 1 muffin = $576 \div 36 = \char36{}16$

2. Find the difference in cost between 1 pie and 1 muffin using the items after the exchange.

After the exchange:
Jane has 30 pies and 6 muffins, which cost $\char36{}384$ (since $576 \times \frac{2}{3} = 384$).
Kelly has 6 pies and 30 muffins, which cost $\char36{}192$ (from part a).
Difference in cost = $\char36{}384 - \char36{}192 = \char36{}192$
Difference in items = $(30 \text{ pies} + 6 \text{ muffins}) - (6 \text{ pies} + 30 \text{ muffins}) = 24 \text{ pies} - 24 \text{ muffins}$
So, 24 pies - 24 muffins = $\char36{}192$
Cost of 1 pie - 1 muffin = $192 \div 24 = \char36{}8$

3. Solve for the cost of 1 pie using the sum and difference.

We have:
1 pie + 1 muffin = $\char36{}16$
1 pie - 1 muffin = $\char36{}8$
Adding the two equations:
2 pies = $16 + 8 = \char36{}24$
1 pie = $24 \div 2 = \char36{}12$