Understanding Number Set Relationships

The question asks us to identify the relationship between the numbers in the given sets and find an option set that shares the same relationship. We are given two sets: \( (4, 12, 2) \) and \( (7, 42, 14) \). The key rule is that operations must be performed on the whole numbers as they are, not by breaking them down into individual digits.

Analyzing the Given Sets to Find the Pattern

Let's examine the first given set: \( (4, 12, 2) \).

• The numbers are 4, 12, and 2.
• We need to find a relationship between these three numbers. Let's try combining the first and third numbers to get the second number.
• Possible operations involving 4 and 2 to get 12:
• Addition: \( 4 + 2 = 6 \). If we multiply the sum by 2, we get \( 6 \times 2 = 12 \). This looks like a potential pattern. Let's test the relationship: (First Number + Third Number) × 2 = Second Number.

Let's test the potential relationship \( ( \text{First Number} + \text{Third Number} ) \times 2 = \text{Second Number} \) with the second given set: \( (7, 42, 14) \).

• The numbers are 7, 42, and 14.
• First Number = 7, Third Number = 14.
• Apply the rule: \( (7 + 14) \times 2 = 21 \times 2 = 42 \).
• The result, 42, matches the second number in the set.

The relationship identified, (First Number + Third Number) × 2 = Second Number, holds true for both given sets.

Testing the Options Against the Identified Relationship

Now we will apply this relationship rule to each of the given options to find the set that follows the same pattern.

Option 1: (19, 62, 11)

• First Number = 19, Second Number = 62, Third Number = 11
• Apply the rule: \( (19 + 11) \times 2 = 30 \times 2 = 60 \).
• The calculated value (60) is not equal to the second number in the option (62).
• This option does not follow the relationship.

Option 2: (25, 29, 5)

• First Number = 25, Second Number = 29, Third Number = 5
• Apply the rule: \( (25 + 5) \times 2 = 30 \times 2 = 60 \).
• The calculated value (60) is not equal to the second number in the option (29).
• This option does not follow the relationship.

Option 3: (9, 26, 4)

• First Number = 9, Second Number = 26, Third Number = 4
• Apply the rule: \( (9 + 4) \times 2 = 13 \times 2 = 26 \).
• The calculated value (26) is equal to the second number in the option (26).
• This option follows the relationship.

Option 4: (15, 44, 9)

• First Number = 15, Second Number = 44, Third Number = 9
• Apply the rule: \( (15 + 9) \times 2 = 24 \times 2 = 48 \).
• The calculated value (48) is not equal to the second number in the option (44).
• This option does not follow the relationship.

Conclusion

Only Option 3, \( (9, 26, 4) \), satisfies the relationship \( ( \text{First Number} + \text{Third Number} ) \times 2 = \text{Second Number} \) observed in the given sets \( (4, 12, 2) \) and \( (7, 42, 14) \).

Revision Table: Key Relationship Summary

Additional Information: Number Pattern Recognition

Number pattern recognition questions are common in logical reasoning and quantitative aptitude tests. These questions require you to identify the underlying rule or relationship governing a series of numbers, a set of numbers, or a sequence. The relationship can involve basic arithmetic operations (addition, subtraction, multiplication, division), squares, cubes, roots, or combinations of these. Sometimes the relationship might link the numbers' positions or specific properties.

Strategies for solving number pattern questions:

• Look for simple arithmetic relationships between adjacent numbers or numbers in corresponding positions within sets.
• Consider the difference or ratio between consecutive numbers.
• Check for squares or cubes of numbers.
• Try combinations of operations, like the sum or difference of two numbers influencing a third number.
• Always test your hypothesized rule on all the given examples before applying it to the options.
• Pay attention to any specific constraints mentioned in the question, such as the rule about not breaking down numbers into digits.

Practicing with various types of number patterns helps improve your ability to quickly identify the correct relationship.