In a certain operator puzzle, the symbol "$" is defined so that 10 $ 25 = 8 and 12 $ 25 = 10. Using the same rule, what is the value of 14 $ 53?

Introduction / Context:
This question defines a special operation "$" using examples and asks you to extend the pattern to a new pair of numbers. The mapping involves the numbers 10, 25, 12, 25 and the results 8 and 10. Since the operation seems small relative to the inputs, it suggests that the rule might involve the digits of the numbers rather than the numbers themselves directly.

Given Data / Assumptions:

• 10 $ 25 = 8.
• 12 $ 25 = 10.
• The same rule must be applied to 14 $ 53.
• We assume a single consistent operation for all examples.

Concept / Approach:
Check the digits of the numbers rather than their full values. For 10 and 25, the digits are 1, 0, 2 and 5. The result 8 can be obtained by adding these digits: 1 + 0 + 2 + 5 = 8. For 12 and 25, the digits are 1, 2, 2 and 5. Their sum is 1 + 2 + 2 + 5 = 10, which matches the given result. So a natural rule is that a $ b equals the sum of all decimal digits in a and b together.

Step-by-Step Solution:
Step 1: Verify the rule with 10 $ 25: digits 1, 0, 2 and 5 give 1 + 0 + 2 + 5 = 8.Step 2: Verify the rule with 12 $ 25: digits 1, 2, 2 and 5 give 1 + 2 + 2 + 5 = 10.Step 3: The rule is therefore "sum of all digits in both numbers".Step 4: For 14 $ 53, the digits are 1, 4, 5 and 3.Step 5: Add them: 1 + 4 + 5 + 3 = 13.

Verification / Alternative check:
Try alternative rules like absolute differences, ratios or simple algebraic combinations. None of them produce both 8 from (10, 25) and 10 from (12, 25) in a consistent way, while the digit sum rule works cleanly for both. Because 14 and 53 are similar two digit numbers, it is natural to apply the same digit sum rule to them as well.

Why Other Options Are Wrong:

• 15, 11, 9 and 12 do not equal the sum of the digits 1, 4, 5 and 3.
• They could only arise from incorrect partial addition or from using a different inconsistent formula.

Common Pitfalls:
Some examinees immediately look for a formula involving the full numbers, such as (a + b) / something or a * b. Those approaches quickly fail when checked against both given examples. When dealing with two digit numbers, always consider digit level patterns like sums and differences, since they are very popular in reasoning exams.

Final Answer:
Using the digit sum rule, 14 $ 53 equals 1 + 4 + 5 + 3 = 13.