Finding the Number Set Relationship

The question asks us to identify the option set that shares the same mathematical relationship as the two given sets of numbers. We are given two sets: (8, 9, 431) and (11, 8, 1267). The rule states that operations must be performed on the whole numbers themselves, not on their individual digits.

Analyzing the Given Number Sets

Let's look for a pattern or relationship between the three numbers in each set. Let the numbers in a set be represented as \(A\), \(B\), and \(C\), where the set is \((A, B, C)\). We need to find a formula or rule that connects \(A\), \(B\), and \(C\) and holds true for both given sets.

Given Sets:

• Set 1: (8, 9, 431) → \(A=8\), \(B=9\), \(C=431\)
• Set 2: (11, 8, 1267) → \(A=11\), \(B=8\), \(C=1267\)

Discovering the Pattern

Observing the numbers, the third number \(C\) is significantly larger than \(A\) and \(B\). This suggests operations involving multiplication or powers are likely involved. Let's try combining powers of \(A\) and \(B\).

Consider Set 1 (8, 9, 431):

• \(8^2 = 64\), \(9^2 = 81\). \(64 + 81 = 145\) (Not 431)
• \(8^3 = 512\), \(9^3 = 729\).

Let's explore combinations using cubes and squares. What if we use \(A^3\) and \(B^2\)?

• Try \(A^3 - B^2\): \(8^3 - 9^2 = 512 - 81 = 431\). This matches the third number \(C\) in Set 1!

Let's test this potential relationship \(C = A^3 - B^2\) on Set 2 (11, 8, 1267):

• \(A = 11\), \(B = 8\)
• Calculate \(A^3 - B^2\): \(11^3 - 8^2 = 1331 - 64 = 1267\). This also matches the third number \(C\) in Set 2!

The relationship \(C = A^3 - B^2\) holds true for both given number sets. This is the pattern we need to find in the options.

Testing the Options

Now, we apply the relationship \(C = A^3 - B^2\) to each option set to see which one satisfies it.

• Option 1: (9, 12, 595)
  • \(A = 9\), \(B = 12\)
  • \(A^3 - B^2 = 9^3 - 12^2 = 729 - 144 = 585\).
  • This is not equal to 595. This option does not share the same relationship.
• Option 2: (13, 14, 2011)
  • \(A = 13\), \(B = 14\)
  • \(A^3 - B^2 = 13^3 - 14^2 = 2197 - 196 = 2001\).
  • This is not equal to 2011. This option does not share the same relationship.
• Option 3: (7, 5, 318)
  • \(A = 7\), \(B = 5\)
  • \(A^3 - B^2 = 7^3 - 5^2 = 343 - 25 = 318\).
  • This is equal to 318. This option shares the same relationship!
• Option 4: (12, 10, 1528)
  • \(A = 12\), \(B = 10\)
  • \(A^3 - B^2 = 12^3 - 10^2 = 1728 - 100 = 1628\).
  • This is not equal to 1528. This option does not share the same relationship.

Conclusion

The option set (7, 5, 318) is the only one that follows the same relationship \(C = A^3 - B^2\) as the given number sets (8, 9, 431) and (11, 8, 1267).

Revision Table - Number Set Relationships

Additional Information - Number Analogy and Patterns

Number analogy questions, like this one involving number sets, are common in logical reasoning and quantitative aptitude tests. The key is to find the underlying mathematical relationship or pattern that connects the numbers in the given set(s).

Common types of relationships include:

• Arithmetic operations (addition, subtraction, multiplication, division)
• Powers and roots (squares, cubes, square roots, cube roots)
• Combinations of operations (e.g., \(A \times B + C\), \(A^2 + B^2\), \(A \times k \pm B \times m\))
• Sequences or series within the set (less common for three numbers)

When trying to find the pattern in a number set \((A, B, C)\), especially when \(C\) is much larger or smaller than \(A\) and \(B\), consider:

• Operations involving powers of \(A\) and \(B\).
• Multiplying \(A\) and \(B\) and then adding/subtracting/multiplying by a constant or another function of \(A\) or \(B\).
• Checking if there's a difference or sum of powers (\(A^n \pm B^m\)).

It's helpful to know the squares and cubes of small numbers by heart, as these often appear in such problems.