If 18 (9) 3 and 36 (30) 5 follow the same numerical rule, then what value of A will satisfy 19 (A) 18?

Introduction / Context:
This is a number puzzle where the expression a (b) c represents a certain relationship between the outer numbers and the inner number. You are given two examples where the middle value is already supplied, and you must determine the rule that connects the three numbers. Once that rule is known, you apply it to a new pair of outer numbers to find the missing middle value A.

Given Data / Assumptions:

- 18 (9) 3 satisfies the hidden rule.
- 36 (30) 5 satisfies the same rule.
- 19 (A) 18 must satisfy that rule as well.
- The outer numbers play a symmetric role in the pattern, and the middle number is derived from them through some arithmetic relation.

Concept / Approach:
We look for a simple function of the outer numbers that yields the middle one. Trying with the first example:

- Outer numbers: 18 and 3.
- Middle number: 9.

One neat observation is that 18 × 3 = 54, and 54 ÷ 6 = 9. For the second example, 36 × 5 = 180, and 180 ÷ 6 = 30. Thus in both cases, the middle number equals the product of the outer numbers divided by 6. We assume the same constant factor applies in the third case.

Step-by-Step Solution:
Step 1: Verify the pattern for 18 (9) 3. Compute the product of the outer numbers: 18 × 3 = 54. Divide by 6: 54 ÷ 6 = 9, which matches the middle number. Step 2: Verify the pattern for 36 (30) 5. Compute 36 × 5 = 180. Divide by 6: 180 ÷ 6 = 30, which again matches the middle number. Step 3: Apply the pattern to 19 (A) 18. Outer numbers: 19 and 18. Product: 19 × 18 = 342. Divide by 6: 342 ÷ 6 = 57. Therefore, A must be 57.

Verification / Alternative check:
The factor 6 is consistent across both given examples, so there is no need to change it for the third case. Also, 57 is present among the options, which confirms that the puzzle is well aligned with this rule. Any attempt to change the divisor or to use a more complicated formula would either fail for one of the examples or not match any answer choice.

Why Other Options Are Wrong:

- 33, 75, and 96 cannot be obtained by multiplying the outer numbers 19 and 18 and dividing by any constant that also works for both examples.
- For instance, 19 × 18 ÷ 6 is exactly 57 and not 33, 75, or 96.

Hence those values are inconsistent with the established pattern.

Common Pitfalls:
Some students may try to combine the outer numbers using addition or subtraction only, or may attempt to involve digits separately in a complicated way. While such methods might incidentally fit one example, they will fail when checked against both. Always validate any discovered pattern against all provided examples before applying it to the unknown case.

Final Answer:
Using the rule middle = (first × third) ÷ 6, the value of A in 19 (A) 18 is 57.