Understanding Number Relations in Sets-

The question asks us to identify the set of numbers that shares the same relationship as the numbers in the two given sets: (12, 40, 14) and (15, 33, 9). We are told that operations must be performed on the whole numbers themselves, not on their individual digits.

Analyzing the Given Number Sets

Let's examine the two provided sets to discover the underlying rule or relationship between the numbers within each set.

• Set 1: (12, 40, 14)
• Set 2: (15, 33, 9)

Let's denote the numbers in a set as (A, B, C), where A is the first number, B is the second (middle) number, and C is the third number.

• For Set 1: A = 12, B = 40, C = 14
• For Set 2: A = 15, B = 33, C = 9

We need to find a rule that connects A, B, and C that is consistent for both sets.

Finding the Relationship or Pattern

Let's look for possible relationships between the numbers. Often, the middle number is related to the first and third numbers through addition, subtraction, multiplication, or division, or a combination of these operations.

Consider the relationship between the first (A) and third (C) numbers and how they might result in the middle number (B).

• In Set 1: A = 12, C = 14. The middle number B = 40.
• In Set 2: A = 15, C = 9. The middle number B = 33.

Let's try simple combinations of A and C:

• Sum of A and C:
  • Set 1: \(12 + 14 = 26\). How does 26 relate to 40? \(40 - 26 = 14\).
  • Set 2: \(15 + 9 = 24\). How does 24 relate to 33? \(33 - 24 = 9\).

Verifying the Rule on Given Sets

Let's check if the rule \(B = A + 2C\) holds true for both initial sets.

• For Set 1: (12, 40, 14)
  • A = 12, C = 14
  • Applying the rule: \(B = 12 + 2 \times 14 = 12 + 28 = 40\).
  • This matches the middle number in Set 1. The rule works for Set 1.
• For Set 2: (15, 33, 9)
  • A = 15, C = 9
  • Applying the rule: \(B = 15 + 2 \times 9 = 15 + 18 = 33\).
  • This matches the middle number in Set 2. The rule works for Set 2.

The rule \(B = A + 2C\) is consistent for both given sets.

Applying the Rule to the Options

Now, we apply the rule \(B = A + 2C\) to each of the given options to find the set that follows the same pattern.

• Option 1: (11, 105, 42)
  • A = 11, C = 42
  • Calculated B = \(11 + 2 \times 42 = 11 + 84 = 95\).
  • The middle number in the option is 105. Since \(95 \neq 105\), this option does not follow the rule.
• Option 2: (21, 35, 41)
  • A = 21, C = 41
  • Calculated B = \(21 + 2 \times 41 = 21 + 82 = 103\).
  • The middle number in the option is 35. Since \(103 \neq 35\), this option does not follow the rule.
• Option 3: (21, 95, 35)
  • A = 21, C = 35
  • Calculated B = \(21 + 2 \times 35 = 21 + 70 = 91\).
  • The middle number in the option is 95. Since \(91 \neq 95\), this option does not follow the rule.
• Option 4: (21, 91, 35)
  • A = 21, C = 35
  • Calculated B = \(21 + 2 \times 35 = 21 + 70 = 91\).
  • The middle number in the option is 91. Since \(91 = 91\), this option follows the rule.

Conclusion

Based on the analysis, Option 4 (21, 91, 35) is the only set that follows the identified rule \(B = A + 2C\).

Revision Table: Number Analogy Rule \(B = A + 2C\)

This table summarizes the application of the derived rule to the given sets and options.

Additional Information: Number Analogy Questions

Number analogy questions are common in reasoning tests. They require you to find a specific relationship or pattern that connects a given set of numbers and then apply that same relationship to find a missing number or a set of numbers that follows the same pattern.

Common types of relationships include:

• Arithmetic operations (addition, subtraction, multiplication, division)
• Operations on sums or differences of numbers
• Operations involving squares, cubes, or roots
• Sequences (arithmetic progression, geometric progression, etc.)
• Digit-based operations (although restricted in this specific question)
• Combinations of multiple operations

The key is to carefully analyze the given examples, test potential rules systematically, and apply the validated rule to the options while adhering to any specific constraints mentioned in the question, such as using whole numbers only.