In a special operation %, it is given that 10 % 25 = 8 and 12 % 25 = 10. Using the same operation, what is the value of 14 % 53?

Introduction / Context:
This problem defines a custom operation represented by the symbol %. The exact formula is not given, but we have two examples showing how it works. Our task is to deduce the rule and then use it to compute 14 % 53. The structure and digits of the numbers are the key clues.

Given Data / Assumptions:

• 10 % 25 = 8.
• 12 % 25 = 10.
• We must find 14 % 53.
• The same underlying rule is used in all three cases.

Concept / Approach:
Since the right hand results are smaller numbers and the examples share the same second operand 25 in two cases, it is natural to suspect the operation uses digit sums rather than direct multiplication or division. A simple and elegant pattern is that a % b equals the sum of the digit sums of a and b. We test this against the given expressions and, if it works, apply it to the third pair.

Step-by-Step Solution:
Consider 10 % 25 = 8. Digit sum of 10 is 1 + 0 = 1. Digit sum of 25 is 2 + 5 = 7. Their sum is 1 + 7 = 8, which matches the given result. Now consider 12 % 25 = 10. Digit sum of 12 is 1 + 2 = 3. Digit sum of 25 remains 7. 3 + 7 = 10, again matching the given value. Therefore, the rule a % b = (digit sum of a) + (digit sum of b) is consistent. Now apply this rule to 14 % 53. Digit sum of 14 is 1 + 4 = 5. Digit sum of 53 is 5 + 3 = 8. Add them: 5 + 8 = 13. Hence, 14 % 53 = 13.

Verification / Alternative check:
The same simple rule fits all known examples perfectly and uses only the digits of the numbers, which is typical in these reasoning questions. There is no need for more complex or different formulas, since a single pattern explains both given equations and the required one.

Why Other Options Are Wrong:
If we tried 15, 11 or 9, we would not be able to build a consistent rule that simultaneously satisfies 10 % 25 = 8 and 12 % 25 = 10. For example, digit sums cannot add up to 15, 11 or 9 for the pair 14 and 53; the correct total is 13.

Common Pitfalls:
Learners often look for direct arithmetic like products or quotients instead of exploring digit-based relationships. Another mistake is to force a complicated formula to fit only one example without checking against all the given ones for consistency.

Final Answer:
Using the same digit sum rule, the value of 14 % 53 is 13.