Analyzing the Relationship in Number Sets

The question asks us to identify the relationship between the numbers in the given sets (9, 55, 64) and (7, 43, 50) and then find which of the options follows the same rule. We must perform operations on the whole numbers themselves, not on their individual digits.

Examining the Example Sets

Let's look at the first example set: (9, 55, 64).

• First number is 9.
• Second number is 55.
• Third number is 64.

We need to find a relationship between these numbers.

Let's consider potential relationships:

• Addition: Is the third number the sum of the first two? \(9 + 55 = 64\). Yes, this works for the first set.
• Multiplication/Addition Combination: Is there a pattern involving multiplying the first number?
  • Relationship between first and second: \(9 \times 6 + 1 = 54 + 1 = 55\). This works.
  • Relationship between first and third: \(9 \times 7 + 1 = 63 + 1 = 64\). This works.

Now let's check these potential relationships with the second example set: (7, 43, 50).

• First number is 7.
• Second number is 43.
• Third number is 50.

Let's test the relationships:

• Addition: Is the third number the sum of the first two? \(7 + 43 = 50\). Yes, this also works for the second set.
• Multiplication/Addition Combination:
  • Relationship between first and second: \(7 \times 6 + 1 = 42 + 1 = 43\). This works.
  • Relationship between first and third: \(7 \times 7 + 1 = 49 + 1 = 50\). This also works.

Both the addition relationship (First + Second = Third) and the multiplication/addition relationship (Second = First * 6 + 1 and Third = First * 7 + 1) seem to work for the given example sets. We need to test the options to see which relationship holds true for exactly one of them.

Checking the Options

Let's check each option using the potential relationships.

Option 1: (7, 42, 50)

• Addition: \(7 + 42 = 49\). \(49 \neq 50\). The addition relationship does not hold.
• Multiplication/Addition:
  • Second number check: \(7 \times 6 + 1 = 42 + 1 = 43\). The second number is 42. \(42 \neq 43\). This relationship does not hold.

Option 1 does not follow either pattern.

Option 2: (8, 32, 41)

• Addition: \(8 + 32 = 40\). \(40 \neq 41\). The addition relationship does not hold.
• Multiplication/Addition:
  • Second number check: \(8 \times 6 + 1 = 48 + 1 = 49\). The second number is 32. \(32 \neq 49\). This relationship does not hold.

Option 2 does not follow either pattern.

Option 3: (4, 17, 21)

• Addition: \(4 + 17 = 21\). Yes, this follows the addition relationship.
• Multiplication/Addition:
  • Second number check: \(4 \times 6 + 1 = 24 + 1 = 25\). The second number is 17. \(17 \neq 25\). This relationship does not hold.

Option 3 follows the addition relationship, but not the multiplication/addition relationship.

Option 4: (5, 31, 36)

• Addition: \(5 + 31 = 36\). Yes, this follows the addition relationship.
• Multiplication/Addition:
  • First number is 5.
  • Second number check: \(5 \times 6 + 1 = 30 + 1 = 31\). The second number is 31. This holds.
  • Third number check: \(5 \times 7 + 1 = 35 + 1 = 36\). The third number is 36. This holds.

Option 4 follows both the addition relationship and the multiplication/addition relationship.

Since both Option 3 and Option 4 satisfy the First + Second = Third relationship, but only Option 4 satisfies the Second = First * 6 + 1 and Third = First * 7 + 1 relationship derived consistently from both example sets, the multiplication/addition pattern is the intended one that uniquely identifies the correct option among the choices.

Let's verify the multiplication/addition relationship with the original examples:

• Set (9, 55, 64):
  • \(55 = 9 \times 6 + 1\) (True)
  • \(64 = 9 \times 7 + 1\) (True)
• Set (7, 43, 50):
  • \(43 = 7 \times 6 + 1\) (True)
  • \(50 = 7 \times 7 + 1\) (True)

This confirms the consistent pattern across the example sets: If the numbers are \(a, b, c\), then \(b = 6a + 1\) and \(c = 7a + 1\).

Applying this rule to Option 4 (5, 31, 36), where \(a=5\):

• Check for the second number: \(6 \times 5 + 1 = 30 + 1 = 31\). The second number is 31. This matches.
• Check for the third number: \(7 \times 5 + 1 = 35 + 1 = 36\). The third number is 36. This matches.

Option 4 follows the established pattern.

Conclusion

The relationship in the given sets is that if the first number is \(a\), the second number is \(6a+1\), and the third number is \(7a+1\). Option (5, 31, 36) is the only set among the options that follows this specific pattern.

For (5, 31, 36):

\(a=5\)

Second number \(b = 6 \times 5 + 1 = 30 + 1 = 31\)

Third number \(c = 7 \times 5 + 1 = 35 + 1 = 36\)

This matches the numbers in option 4.

Therefore, the set in which the numbers are related in the same way as the given sets is (5, 31, 36).

As the table shows, only Option 4 satisfies both relationships consistently derived from the example sets.

Revision Table: Number Set Relationships

Additional Information: Solving Logical Reasoning Number Questions

Logical reasoning number set questions require careful observation and systematic testing of potential patterns. Here are some tips:

• Look for simple arithmetic operations first: addition, subtraction, multiplication, division between pairs of numbers or all numbers in the set.
• Consider relationships between the first and second, first and third, and second and third numbers.
• Test for common mathematical sequences or properties (e.g., squares, cubes, prime numbers).
• Look for patterns involving multiplication/division combined with addition/subtraction.
• The pattern must be consistent across all provided example sets.
• Apply the derived pattern to each option. The correct option is the one that uniquely fits the pattern. If multiple options fit one simple pattern (like addition here), look for a more complex or specific pattern that also fits the examples but differentiates the options.
• Pay close attention to any constraints mentioned in the question, such as the rule about not breaking down digits.

Practicing with various types of number set analogies helps in quickly recognizing common patterns and developing a systematic approach to testing them.