If 23@5 = 56, 6@7 = 26 and 5@2 = 14 under a certain rule, then what is the value of 7@5 under the same rule?

Introduction / Context:
Another custom operator @ is defined on two numbers, but not as ordinary addition or multiplication. We are given three examples and must use them to deduce the rule and then apply it to 7@5.

Given Data / Assumptions:

• 23@5 = 56.
• 6@7 = 26.
• 5@2 = 14.
• We must compute 7@5.

Concept / Approach:
We look for a simple arithmetic relationship between the two numbers and the result. Notice that 6@7 = 26 and 5@2 = 14 are both exactly twice the sum of the two numbers. We can check if the same pattern applies to 23@5 as well.

Step-by-Step Solution:
Step 1: Check 6@7 = 26. 6 + 7 = 13. Twice this sum: 2 × 13 = 26, which matches. Step 2: Check 5@2 = 14. 5 + 2 = 7. Twice this sum: 2 × 7 = 14, which matches. Step 3: Check 23@5 = 56. 23 + 5 = 28. Twice this sum: 2 × 28 = 56, which matches. Step 4: Therefore, the rule is a @ b = 2 × (a + b). Step 5: Apply this rule to 7@5. 7 + 5 = 12. 2 × 12 = 24.

Verification / Alternative check:
All three given examples are consistent with this simple formula. No other straightforward operation, such as product or difference, fits everything so neatly. Hence we are confident about the rule.

Why Other Options Are Wrong:
Values like 26, 19, and 52 require different formulas that would contradict at least one of the given examples. For instance, 26 is correct for 6@7 but not for 7@5 when the rule is fixed.

Common Pitfalls:
Some learners are tempted to mix different rules for different examples or treat the first case (23@5) separately. In all such coding puzzles, a single consistent rule must work for every case.

Final Answer:
Using the same rule, the value of 7@5 is 24.