In a certain number code, the operator "%" does not mean the usual percentage. Instead, 14 % 32 is given as 46 and 52 % 20 is given as 72. Using the same hidden rule, what is the value of 35 % 14?

Introduction:
This question involves a nonstandard numeric operator, represented by the "%" sign. Here it does not mean percentage. Instead, two example equations are provided, and we must discover the pattern or rule connecting the numbers. Once that rule is clear, we apply it to a new pair to find the required value of 35 % 14. It is a straightforward test of pattern recognition in arithmetic puzzles.

Given Data / Assumptions:
We are given two sample relations:
1) 14 % 32 = 46
2) 52 % 20 = 72
We must determine 35 % 14. We assume:
1) The operator % represents the same operation in all three cases.
2) The operation is likely simple and based on addition, subtraction, or another basic arithmetic combination of the two numbers.

Concept / Approach:
When unknown operators appear in reasoning questions, a good first step is to check basic operations: sum, difference, product, or ratios of the given numbers. Since the results 46 and 72 are modest and close to the sum of each pair, examining the sum is a natural starting point. If the same rule explains both examples, we accept it as the underlying pattern and then use it to compute the third case.

Step-by-Step Solution:
Step 1: For 14 % 32, compute the sum 14 + 32 = 46. This exactly matches the given result 46.Step 2: For 52 % 20, compute the sum 52 + 20 = 72. This also matches the given result 72.Step 3: Since both examples fit the simple rule "a % b = a + b", we adopt this rule.Step 4: Apply the rule to the required expression: 35 % 14 = 35 + 14.Step 5: Add the numbers: 35 + 14 = 49.Step 6: Therefore, the value of 35 % 14 is 49.

Verification / Alternative check:
We can confirm by checking whether any alternative simple pattern, such as difference or product, fits both examples. The difference 32 - 14 equals 18, which does not match 46, and 52 - 20 equals 32, which does not match 72. The products 14 * 32 and 52 * 20 are far larger than the results. So the sum rule is the most natural and consistent explanation.

Why Other Options Are Wrong:
The values 39, 29, and 19 are much smaller and would require subtracting something from the sum without any hint from the examples. The value 59 is larger than the sum and would need unjustified addition. None of these candidate values can be obtained from both given examples with a single simple arithmetic rule.

Common Pitfalls:
Learners sometimes look for overly complex patterns involving squares, cubes, or digit manipulations when a very simple rule suffices. Another error is to test a rule on only one example and assume it is correct without verifying it on the second given relation. Always ensure that the same rule explains all provided examples before using it on the target expression.

Final Answer:
Using the pattern, the value of 35 % 14 is 49.