Let's assume that x servers were full-time and y servers were part-time. Then we can write two equations based on the given information:
x + y = 18 (since there were 18 servers in total)
21x + 15.5y = 312 (since the labor cost was $312 per hour)
To solve for x and y, we can use elimination or substitution. Here's one way to use elimination:
Multiply the first equation by 15.5 to get 15.5x + 15.5y = 279.
Subtract the first equation from the second equation to get 5.5x = 33.
Divide both sides by 5.5 to get x = 6.
Now we know that there were 6 full-time servers. We can substitute this value into either of the two equations to solve for y:
6 + y = 18
y = 12
Therefore, there were 6 full-time servers and 12 part-time servers at the banquet.