If 9 # 3 = 120, 3 # 4 = 70 and 2 # 9 = 110 in a certain numerical code, then using the same pattern what is the value of 5 # 2?

Introduction / Context:
This is a number coding question where a special symbol, here the hash sign, represents a custom operation between two numbers. The examples 9 # 3 = 120, 3 # 4 = 70, and 2 # 9 = 110 illustrate the rule. Our task is to observe the pattern and then apply it to evaluate 5 # 2. These problems strengthen the ability to search for simple arithmetic relationships such as sums, products, and scalar multiples.

Given Data / Assumptions:

• 9 # 3 equals 120.
• 3 # 4 equals 70.
• 2 # 9 equals 110.
• The same rule must be used to compute 5 # 2.
• The operation is deterministic and uses basic arithmetic like addition and multiplication.

Concept / Approach:
The natural way is to test simple combinations of the two numbers: their sum, difference, and product, and see how they relate to the result. Observing that 9 plus 3 equals 12, and 120 is 10 times 12, suggests that the result might be ten times the sum. Checking the other examples for the same relationship validates this idea. Once we confirm that a # b is equal to 10 times the sum of a and b, we can safely apply this rule to the new pair 5 and 2.

Step-by-Step Solution:
Step 1: For 9 # 3, compute the sum 9 + 3 = 12.Step 2: Multiply this sum by 10 to get 12 * 10 = 120, which matches the given result.Step 3: For 3 # 4, compute the sum 3 + 4 = 7.Step 4: Multiply by 10 to get 7 * 10 = 70, again matching the given result.Step 5: For 2 # 9, compute the sum 2 + 9 = 11, and 11 * 10 = 110, which is also consistent.Step 6: Now apply the rule to 5 # 2. The sum is 5 + 2 = 7, and multiplying by 10 gives 7 * 10 = 70.

Verification / Alternative check:
We can check whether any alternative rule like a^2 plus b^2 or a times b plus a constant fits all three examples. Most such patterns will fail to match all the given values. The rule of 10 times the sum is the simplest and uniquely works in every case, making it the most reasonable coding pattern. Since 5 plus 2 equals 7 and 7 times 10 equals 70, the answer is clearly consistent with the established code.

Why Other Options Are Wrong:
The value 40 might appear if someone incorrectly uses the product 5 * 2 and then applies an arbitrary constant factor. The value 50 could come from mistakenly multiplying only one of the numbers by 10. The value 80 does not correspond to any simple combination of 5 and 2 that matches the earlier examples. The distractor 90 is also not related to 10 times the sum or any pattern consistent with all three initial equations. Only 70 satisfies the pattern derived from every given code pair.

Common Pitfalls:
A common mistake is to try overly complicated formulas before examining the simplest ideas like sum, difference, or product. Another error is to infer a rule from only one example without checking the remaining ones, which can lead to a rule that fits one case but fails others. Always validate a candidate pattern against all provided examples before using it to find the unknown value.

Final Answer:
Using the rule that a # b equals 10 times the sum of a and b, we have 5 # 2 = 10 * (5 + 2) = 70.