In a certain numerical code, 4 * 2 = 3 and 8 * 4 = 3. Using the same pattern, what is the value of 21 * 7?

Introduction / Context:
This problem presents an unusual operator, written as "*", which does not follow normal multiplication. Instead, it encodes a special relationship between two numbers. By using the given examples, you must decode the rule and then apply it to a new pair of numbers. Such puzzles test pattern recognition and logical reasoning rather than rote arithmetic.

Given Data / Assumptions:
- We are given 4 * 2 = 3.
- We are also given 8 * 4 = 3.
- We must find the value of 21 * 7 under the same rule.
- The operator "*" here represents some consistent arithmetic pattern involving the two numbers.

Concept / Approach:
The idea is to express the result as a function of the two given numbers. Since 4 * 2 and 8 * 4 both equal 3, we look for formulas such as (a + b) / b, (a + b) / a, or (a / b) plus or minus some adjustment. We then verify which expression gives the correct result for both known examples. Once we have a candidate rule that works for both, we apply it to the pair 21 and 7.

Step-by-Step Solution:
Step 1: Start with 4 * 2 = 3. Compute (4 + 2) / 2 = 6 / 2 = 3. This fits the given result. Step 2: Test the same expression on the second example: (8 + 4) / 4 = 12 / 4 = 3, which again matches the given value. Step 3: Therefore, we conclude that the rule is a * b = (a + b) / b. Step 4: Apply this rule to 21 * 7: (21 + 7) / 7 = 28 / 7 = 4. Step 5: So the correct value of 21 * 7 under this code is 4.

Verification / Alternative check:
To be sure, we can test other simple formulas. For example, (a + b) / a does not work, because (4 + 2) / 4 = 6 / 4 is not 3. Similarly, (a - b) or (a * b) divided by some number does not provide consistent results for both examples. The formula (a + b) / b is the only simple expression that correctly yields 3 in both given cases and therefore is the most reasonable coded rule.

Why Other Options Are Wrong:
- Option 3 would be the answer only if the pattern did not change with larger numbers, but (21 + 7) / 7 clearly equals 4, not 3.
- Option 8 might come from using direct division 21 / 7 = 3 and then guessing incorrectly, or from mixing up the pattern with normal multiplication.
- Option 16 cannot be constructed naturally from (21 + 7) using the pattern that fits the original examples, so it does not follow the consistent rule.

Common Pitfalls:
Many students assume that "*" always means multiplication and try to force the examples into that framework. Others guess a pattern after checking only one example rather than all of them. The safe strategy is to form a hypothesis from the first example and then rigorously test it on every other example given in the question.

Final Answer:
Using the decoded rule a * b = (a + b) / b, we obtain 21 * 7 = 4. Thus, the correct answer is 4.