In a number pattern, 12 (20) 16 and 21 (35) 28 are given. Using the same relationship between the outer numbers and the middle number, what is the value of A in 48 (80) A?

Introduction / Context:
This question presents a three-number pattern in the form X (Y) Z, where Y is derived from X and Z by a hidden rule. The first two examples are given, and we must determine the corresponding value for A in the third instance. It tests recognition of simple arithmetic relationships involving differences and progressions.

Given Data / Assumptions:

• 12 (20) 16 is a valid pattern.
• 21 (35) 28 follows the same pattern.
• We are given 48 (80) A and must determine A.
• The same rule connects the pair of outer numbers to the number in brackets.

Concept / Approach:
Look at how the middle number might relate to the first and third numbers. The difference between the outer numbers is constant in each pattern: 16 − 12 = 4 and 28 − 21 = 7. The middle numbers 20 and 35 can be formed by moving forward another step in an arithmetic progression based on this difference. Specifically, we can investigate whether the middle term equals the first term plus two times the difference, or equivalently 2Z − X.

Step-by-Step Solution:
For 12 (20) 16, compute the difference Z − X = 16 − 12 = 4. Check if middle term Y equals X + 2 * difference: 12 + 2 * 4 = 12 + 8 = 20, which matches Y. So a possible rule is Y = X + 2 * (Z − X), or Y = 2Z − X. Verify this with 21 (35) 28. Difference: 28 − 21 = 7. Compute Y = 21 + 2 * 7 = 21 + 14 = 35, which matches the given middle number. The rule is therefore confirmed: Y = 2Z − X. Now apply this to 48 (80) A. We know X = 48 and Y = 80, so use 80 = 2A − 48. Then 2A = 80 + 48 = 128. So A = 128 ÷ 2 = 64.

Verification / Alternative check:
We can also think of the pattern as three consecutive terms in an arithmetic sequence: for the first example, terms are 12, 16, 20; for the second, 21, 28, 35. In each case, the common difference is Z − X. Continuing the pattern from 48 with the same idea leads to A = 64, consistent with the rule we derived.

Why Other Options Are Wrong:
If A were 50, 56 or 72, the relation 80 = 2A − 48 would not hold. These values would break the consistent pattern established by the first two pairs, so they are not acceptable answers.

Common Pitfalls:
A common error is to assume that the middle number is a simple average of the outer numbers. In this question, the average of 12 and 16 is 14, not 20. Another mistake is to try overly complex formulas without checking simple difference-based patterns first.

Final Answer:
The value of A that maintains the same pattern is 64.