In a certain numerical code, 13 ∆ 3 = 20, 12 ∆ 7 = 10 and 12 ∆ 5 = 14. Using the same operation, what is the value of 16 ∆ 2?

Introduction / Context:
This is a custom operator question where the symbol delta defines a particular calculation between two numbers. We are provided with three examples and must find the underlying rule so that we can compute the value of 16 ∆ 2. Questions like this are designed to make students think about simple algebraic patterns and test flexibility in exploring possibilities such as sums, products, and differences.

Given Data / Assumptions:

• 13 ∆ 3 equals 20.
• 12 ∆ 7 equals 10.
• 12 ∆ 5 equals 14.
• The same rule must be applied to 16 ∆ 2.
• The rule is assumed to be the same for all pairs and based on basic arithmetic combinations.

Concept / Approach:
To uncover the rule, we treat the expression a ∆ b as some function of a and b. A manageable approach is to examine simple linear combinations of a and b, such as 2 times a minus 2 times b, a plus b, or a minus b, and see if any of them match the given outputs. By solving a small system of equations for a form like p times a plus q times b plus r, we can discover a consistent pattern. In this case the pattern turns out to be especially neat: the result is 2 times the difference between the first and second number.

Step-by-Step Solution:
Step 1: Suppose the rule is of the form k * (a − b) where k is a constant.Step 2: Use the first example: 13 ∆ 3 = 20. The difference 13 − 3 is 10, so k * 10 must equal 20, giving k = 2.Step 3: Check the second example: 12 ∆ 7. The difference is 12 − 7 = 5, and 2 * 5 equals 10, which matches the given value.Step 4: Check the third example: 12 ∆ 5. The difference is 12 − 5 = 7, and 2 * 7 equals 14, again matching the given value.Step 5: Conclude that a ∆ b = 2 * (a − b) is the rule.Step 6: Now compute 16 ∆ 2. The difference is 16 − 2 = 14.Step 7: Multiply by 2 to get 2 * 14 = 28.

Verification / Alternative check:
We could also check other candidate rules, such as 2 times a plus b or a minus 2 times b, but they will fail at least one of the given examples. The simplicity and full consistency of 2 times (a minus b) across all three test cases make it the most reliable rule. Since it correctly predicts 28 for 16 ∆ 2, we can be confident this is the correct value.

Why Other Options Are Wrong:
The option 14 is just the simple difference 16 − 2 and omits the multiplication by 2. The value 4 might be guessed by subtracting 16 from 2 or using an incorrect coefficient. The option 8 could arise if someone mistakenly multiplies the difference by one half instead of two. The distractor 30 does not correspond to any plausible simple pattern that matches the earlier examples. Only 28 respects the established rule a ∆ b = 2 * (a − b).

Common Pitfalls:
Students might focus only on the sums or products of the numbers and overlook the role of the difference. Another common error is to infer the rule from only one example and not verify it against the rest, leading to a pattern that fails later. A systematic approach that checks both sums and differences, and confirms the chosen rule against all given data, is key to success in such problems.

Final Answer:
Using the rule a ∆ b = 2 * (a − b), we get 16 ∆ 2 = 2 * (16 − 2) = 28.