If 26#14 = 80, 5#3 = 16 and 6#5 = 22 under a certain rule, then what is the value of 14#5 under the same rule?

Introduction / Context:
This problem defines a custom binary operator # acting on two numbers. Instead of ordinary addition or multiplication, # uses a simple formula that we must infer from the given examples and then apply to 14#5.

Given Data / Assumptions:

• 26#14 = 80.
• 5#3 = 16.
• 6#5 = 22.
• We must find 14#5.

Concept / Approach:
The outputs 80, 16, and 22 appear larger than either input individually, suggesting that the operator uses both numbers in an additive way and then scales the result. A very natural pattern is to add the two numbers and then multiply by a constant factor.

Step-by-Step Solution:
Step 1: Check 26#14 = 80. 26 + 14 = 40, and 40 × 2 = 80. Step 2: Check 5#3 = 16. 5 + 3 = 8, and 8 × 2 = 16. Step 3: Check 6#5 = 22. 6 + 5 = 11, and 11 × 2 = 22. Step 4: The consistent rule is: a # b = (a + b) × 2. Step 5: Apply this rule to 14#5. 14 + 5 = 19. 19 × 2 = 38.

Verification / Alternative check:
All three given examples perfectly match the rule (sum of the two numbers, then multiply by 2). No other simple linear operation fits them all this neatly, so the rule is clearly intended.

Why Other Options Are Wrong:
Values like 17, 4, and 9 do not equal 2 times the sum of 14 and 5. Any formula that yields those numbers would not work consistently for 26#14, 5#3, and 6#5, so such alternatives contradict the given pattern.

Common Pitfalls:
A common mistake is to multiply the numbers first and then add, or to apply the factor 2 incorrectly. Always check whether a proposed rule works for every example, not just one of them.

Final Answer:
Using the same rule, the value of 14#5 is 38.