There are seven different dishes (4 main dishes and 3 side dishes), each with a different integer cost; we cannot use just two variables to represent the two types of dishes. Each of the four main dishes is more expensive than any of the side dishes. The seven dishes add up to $89. The question asks for the price of the most expensive side dish.

(1) SUFFICIENT: Since this problem involves 7 different integers, some of which must be larger than others, it's a good idea to test extreme possibilities. We're told that the most expensive main dish is $16 so we'll also try maximizing the cost of the other three: $15, $14, and $13. In this case, the total cost of the four main dishes is $58, leaving $31 to split among the three side dishes. The prices of the three side dishes must be different integers and all must cost less than $13. Again, try maximum values first: if two side dishes cost $12 and $11, then the third must cost $8. What other possibilities exist that will still add to $31? 11 + 10 + 9 adds only to $30, so we have to keep $12 in the set. It turns out that there's one other possible combination: the three side dishes could cost $12, $10, and $9. In either case, the most expensive side dish costs $12.

Is it possible to get a value other than $12? Let's try to change the values for the main dishes. We can't make the main dishes more expensive, but we can take away $1 from the least expensive main dish; perhaps the most expensive side dish will become more expensive than $12. The next largest possible set of main dish prices are $16, $15, $14, and $12, for a total of $57. This leaves $32 for the three side dishes - but that value is impossible to achieve. The side dishes must cost less than the least-expensive main dish ($12), so the largest possible cost for the three side dishes is $11 + $10 + $9 = $30.

It's impossible, therefore, for the main dishes to cost anything other than $16, $15, $14, and $13. As a result, the most expensive side dish must cost $12.

(2) NOT SUFFICIENT: Since this problem involves 7 different integers, some of which must be larger than others, it's a good idea to test extreme possibilities. We're told that the least expensive side dish costs $9, so this time we'll try minimizing the cost of the other two: $10, and $11. The side dishes cost a total of $30, leaving $59 for the main dishes. That total can be achieved in many ways; for instance, the main dishes could cost $12, $13, $14, and $20. In this scenario, the most expensive side dish is $11. Are there any other possible values?

Let's try increasing the price of the most expensive side dish by $1: say the side dishes now cost $9, $10, and $12. That's a total of $31 for the side dishes, leaving $58 for the main dishes. That total can be achieved, exactly, if the main dishes cost $13, $14, $15, and $16. Therefore, the most expensive side dish can also cost $12. We have at least two possible values, $11 and $12, so this information is not sufficient to answer the question.

The correct answer is A.