In a number analogy, 19 (36) 13 and 37 (81) 28 follow the same pattern. Using the same rule, what is the value of A in 43 (A) 38?

Introduction / Context:
This problem presents pairs of numbers with a value in brackets between them. The bracketed value is not a normal arithmetic result, but comes from a pattern that links the two outer numbers. We must find the rule that converts each pair into the middle number and then apply that rule to the pair 43 and 38 in order to determine A.

Given Data / Assumptions:

• 19 (36) 13 is a valid example.
• 37 (81) 28 is another valid example.
• The same rule is applied to 43 (A) 38.
• We assume all operations involve simple integer arithmetic.

Concept / Approach:
A natural approach is to examine each example and try basic relationships such as sums, differences, products or squares. For 19 and 13, the difference is 19 - 13 = 6, and 6^2 = 36, which matches the middle number. For 37 and 28, the difference is 37 - 28 = 9, and 9^2 = 81, again matching the middle number. This suggests a consistent rule: the bracketed value is the square of the difference between the two outer numbers.

Step-by-Step Solution:
Step 1: Confirm the rule with the first pair. 19 - 13 = 6 and 6^2 = 36, which matches 19 (36) 13. Step 2: Confirm with the second pair. 37 - 28 = 9 and 9^2 = 81, which matches 37 (81) 28. Step 3: Apply the same rule to the pair 43 and 38. Step 4: Compute the difference: 43 - 38 = 5. Step 5: Square the difference: 5^2 = 25. Step 6: Therefore A must be 25, so the completed relation is 43 (25) 38.

Verification / Alternative check:
We can check that no other simple pattern, such as sums, products, or combination of both, matches both given examples as neatly. The "square of the difference" rule fits perfectly in both cases, which strongly indicates that it is the intended pattern. Applying it to 43 and 38 fits well and produces a clean square number, 25.

Why Other Options Are Wrong:
Values 16, 36, 49 and 64 are all perfect squares, but they do not correspond to the square of the difference between 43 and 38. The difference is 5, and its square is 25, not 4, 6, 7 or 8 squared. Therefore only 25 is consistent with the discovered rule.

Common Pitfalls:
Some learners look at sums like 19 + 13 or 37 + 28, or attempt to combine sum and difference in complicated ways, which makes the pattern unnecessarily complex. When tackling such questions, always start with the simplest observations: difference, square, cube, or small multiples. Often the correct rule is based on these straightforward relationships.

Final Answer:
Using the rule "square of the difference between the two outer numbers", the value of A in 43 (A) 38 is 25.