Understanding Number Relations in Sets

This question asks us to find the underlying mathematical relationship between the numbers in the given sets and then identify the option that follows the same relationship. We are given two sets: (9, 7, 189) and (4, 13, 156). The rule is to perform operations on the whole numbers themselves, not their individual digits.

Analyzing the Given Number Sets

Let's look closely at the numbers in the first set (9, 7, 189). Let's call the numbers A, B, and C respectively. So, A=9, B=7, and C=189.

We need to find how 189 is related to 9 and 7. Let's try some basic arithmetic operations:

• A + B = 9 + 7 = 16 (Not 189)
• A × B = 9 × 7 = 63 (Not 189)

The third number, 189, is significantly larger than the product of the first two numbers, 63. This suggests that the product might be multiplied by another factor or some other operation is involved.

Let's see if 189 is a multiple of 63:

So, it seems that \( C = (A \times B) \times 3 \). Let's verify this relationship with the first set:

This relationship holds true for the first set.

Verifying the Relationship with the Second Set

Now, let's check if the same relationship \( C = (A \times B) \times 3 \) applies to the second set (4, 13, 156). Here, A=4, B=13, and C=156.

Applying the potential relationship:

This also holds true for the second set. The relationship seems to be \( C = A \times B \times 3 \).

Applying the Relation to the Options

Now, we will check each option to see which set follows the relationship \( C = A \times B \times 3 \).

Option 1: (5, 9, 160)

• A = 5, B = 9, C = 160
• Calculate A × B × 3: \( 5 \times 9 \times 3 = 45 \times 3 = 135 \)
• Is C equal to 135? 160 ≠ 135.
• Option 1 does not follow the relationship.

Option 2: (6, 9, 162)

• A = 6, B = 9, C = 162
• Calculate A × B × 3: \( 6 \times 9 \times 3 = 54 \times 3 = 162 \)
• Is C equal to 162? 162 = 162.
• Option 2 follows the relationship.

Option 3: (8, 5, 125)

• A = 8, B = 5, C = 125
• Calculate A × B × 3: \( 8 \times 5 \times 3 = 40 \times 3 = 120 \)
• Is C equal to 120? 125 ≠ 120.
• Option 3 does not follow the relationship.

Option 4: (7, 8, 188)

• A = 7, B = 8, C = 188
• Calculate A × B × 3: \( 7 \times 8 \times 3 = 56 \times 3 = 168 \)
• Is C equal to 168? 188 ≠ 168.
• Option 4 does not follow the relationship.

Conclusion

Only Option 2 follows the discovered relationship \( C = A \times B \times 3 \).

The set (6, 9, 162) is related in the same way as the given sets.

Revision Table: Key Learnings on Number Sets

Additional Information: Strategies for Finding Number Patterns

When faced with number relation or set-based reasoning questions, consider these common strategies:

• Basic Arithmetic: Check for sums, differences, products, or quotients between the numbers.
• Squares and Cubes: See if any number is a square or cube, or if the numbers are related to squares or cubes of other numbers in the set.
• Multiples/Factors: Check if one number is a multiple or factor of another, or of a combination of others.
• Constants: Sometimes, a fixed number is added, subtracted, multiplied, or divided.
• Combinations: The pattern might involve a combination of operations (e.g., \(A \times B + k\), \(A^2 - B\), \((A+B) \times k\)).
• Positional Logic: The relationship might specifically connect the first, second, and third numbers in a set.

Systematically testing these possibilities on the given examples helps in identifying the correct pattern efficiently.