In a certain numerical code, the operation "$" between two numbers is defined so that 11$25 = 18 and 12$20 = 16. Using the same pattern, what is the value of 4$50?

Introduction / Context:
This numerical reasoning problem defines a custom operation "$" that combines two numbers in a special way. The examples 11$25 = 18 and 12$20 = 16 hint that the operation may involve an average or similar combination of the two numbers. Your objective is to uncover the underlying rule and then apply it to 4$50.

Given Data / Assumptions:
- 11$25 = 18.
- 12$20 = 16.
- We must compute 4$50 using the same rule.
- The operation is consistent across all examples and involves both numbers in a simple arithmetic fashion.

Concept / Approach:
Because the outputs 18 and 16 lie between the two numbers in each pair (11 and 25, 12 and 20), it is natural to suspect that the operation involves averaging. Therefore, we test the arithmetic mean (a + b) / 2 for both examples. If this matches, then we apply the same formula to 4 and 50 to find the required value.

Step-by-Step Solution:
Step 1: For 11$25 = 18, compute the arithmetic mean (11 + 25) / 2. Step 2: 11 + 25 = 36, and 36 / 2 = 18, which matches the given result. Step 3: For 12$20 = 16, compute (12 + 20) / 2. Step 4: 12 + 20 = 32, and 32 / 2 = 16, again matching the given result. Step 5: This confirms the rule a$b = (a + b) / 2. Step 6: Apply this rule to 4$50: (4 + 50) / 2 = 54 / 2 = 27.

Verification / Alternative check:
Alternative possibilities like a + b or b - a do not produce the results 18 and 16 for the given examples. The arithmetic mean works perfectly for both and is an intuitive and simple symmetric function of a and b. Checking the final calculation for 4 and 50 confirms that (4 + 50) / 2 is 27, so the rule applies cleanly to the new pair as well.

Why Other Options Are Wrong:
- Option 42 or 37 do not fall exactly halfway between 4 and 50, contradicting the average based rule.
- Option 17 does not match any simple symmetric pattern of 4 and 50 and cannot be justified using the confirmed operation definition.

Common Pitfalls:
An error some students make is to try to construct a very complicated formula when a simple average explains the data perfectly. Others check the rule on only one example and then assume it holds without testing the second example. Always validate your rule on all given pairs to avoid incorrect generalizations.

Final Answer:
Using the rule a$b = (a + b) / 2, the value of 4$50 is 27.