Understanding Number Set Relationships

This question asks us to identify the underlying relationship between the numbers in the given sets and then find an option set that shares the same relationship. We are told to perform operations on whole numbers and not on their constituent digits.

Analyzing the Given Number Sets

We are provided with two sets:

1. (36, 90, 18)
2. (15, 35, 8)

Let's examine the numbers in the first set (36, 90, 18). We need to find a rule or pattern that connects these three numbers. Let's consider the positions of the numbers: First (36), Middle (90), and Third (18).

Let's try some simple arithmetic operations involving the First and Third numbers to see if we can arrive at the Middle number.

• Sum of First and Third: \(36 + 18 = 54\). \(54\) is not \(90\).
• Difference between First and Third: \(|36 - 18| = 18\). Is \(18\) related to \(90\)? Yes, \(18 \times 5 = 90\).

Let's hypothesize a relationship based on this observation: The Middle number is equal to the absolute difference between the First and Third numbers, multiplied by 5.

Let's write this as a formula:

\[ \text{Middle} = |\text{First} - \text{Third}| \times 5 \]

Verifying the Relationship with the Second Set

Now, let's test this proposed relationship with the second given set (15, 35, 8).

• First number = 15
• Third number = 8
• Absolute difference: \(|15 - 8| = 7\)
• Apply the rule: \(7 \times 5 = 35\)

The calculated value is \(35\), which is exactly the Middle number in the set (15, 35, 8). This confirms that the relationship "Middle = |First - Third| \(\times\) 5" holds true for both the given sets.

Testing the Options

Now, we will apply this verified rule to each of the options provided to find the set that follows the same relationship.

Option 1: (14, 45, 5)

• First number = 14
• Third number = 5
• Absolute difference: \(|14 - 5| = 9\)
• Apply the rule: \(9 \times 5 = 45\)

The calculated value is \(45\), which matches the Middle number in the set (14, 45, 5). This option follows the relationship.

Option 2: (14, 42, 7)

• First number = 14
• Third number = 7
• Absolute difference: \(|14 - 7| = 7\)
• Apply the rule: \(7 \times 5 = 35\)

The calculated value is \(35\). The Middle number in the set is \(42\). Since \(35 \neq 42\), this option does not follow the relationship.

Option 3: (13, 78, 6)

• First number = 13
• Third number = 6
• Absolute difference: \(|13 - 6| = 7\)
• Apply the rule: \(7 \times 5 = 35\)

The calculated value is \(35\). The Middle number in the set is \(78\). Since \(35 \neq 78\), this option does not follow the relationship.

Option 4: (12, 60, 8)

• First number = 12
• Third number = 8
• Absolute difference: \(|12 - 8| = 4\)
• Apply the rule: \(4 \times 5 = 20\)

The calculated value is \(20\). The Middle number in the set is \(60\). Since \(20 \neq 60\), this option does not follow the relationship.

Conclusion

Based on our analysis, only Option 1 follows the same number set relationship as the given sets.

Revision Table: Applying the Number Set Rule

Additional Information: Logical Reasoning with Number Sets

Problems involving finding relationships in number sets are common in logical reasoning and aptitude tests. These problems test your ability to observe patterns and apply basic arithmetic operations (addition, subtraction, multiplication, division) and sometimes concepts like squares, cubes, or prime numbers.

Key strategies for solving such number set problems include:

• Look for relationships between the first and third numbers to get the second.
• Consider relationships between any two numbers to get the third.
• Check for common factors or multiples.
• Explore operations like sum, difference, product, or quotient, or combinations of these.
• Always test your hypothesized relationship on all the given examples before checking the options.
• Remember the constraint given in the question, such as operating on whole numbers versus individual digits.

Practicing different types of number set problems helps in quickly identifying potential patterns during an exam.