Understanding Number Relationships in Sets

This question asks us to identify the relationship between the numbers in the given sets and then find an option set that follows the same relationship. The rule is important: operations must be performed on the whole numbers themselves, not by breaking them down into individual digits.

Analyzing the Given Number Sets

We are given two sets:

• Set 1: (9, 3, 30)
• Set 2: (10, 3, 33)

Let's look at the first set (9, 3, 30). We need to find a way to combine the first two numbers (9 and 3) to get the third number (30). Let's try basic arithmetic operations.

• Addition: $9 + 3 = 12$. This is not 30.
• Subtraction: $9 - 3 = 6$ or $3 - 9 = -6$. Neither is 30.
• Multiplication: $9 \times 3 = 27$. This is close to 30. What if we add or subtract something from 27 to get 30? $27 + 3 = 30$.

Notice that the number added (3) is the second number in the set (9, 3, 30). Let's propose a relationship: the first number multiplied by the second number, plus the second number, equals the third number.

Let the three numbers in a set be a, b, and c. The proposed relationship is: $a \times b + b = c$.

Checking the Relationship with the Second Given Set

Now, let's verify this relationship with the second set (10, 3, 33).

• Here, a = 10, b = 3, and c = 33.
• Using the relationship $a \times b + b$: $10 \times 3 + 3 = 30 + 3 = 33$.

The result, 33, matches the third number in the set. So, the relationship $a \times b + b = c$ holds true for both given sets.

Evaluating the Options

We will now apply the discovered relationship $a \times b + b = c$ to each of the options to see which one follows the same pattern.

• Option 1: (12, 3, 39)
  • Here, a = 12, b = 3, c = 39.
  • Let's calculate $a \times b + b$: $12 \times 3 + 3 = 36 + 3 = 39$.
  • This matches the third number, 39. Option 1 follows the relationship.
• Option 2: (9, 7, 98)
  • Here, a = 9, b = 7, c = 98.
  • Let's calculate $a \times b + b$: $9 \times 7 + 7 = 63 + 7 = 70$.
  • This does not match the third number, 98. Option 2 does not follow the relationship.
• Option 3: (6, 9, 40)
  • Here, a = 6, b = 9, c = 40.
  • Let's calculate $a \times b + b$: $6 \times 9 + 9 = 54 + 9 = 63$.
  • This does not match the third number, 40. Option 3 does not follow the relationship.
• Option 4: (8, 8, 74)
  • Here, a = 8, b = 8, c = 74.
  • Let's calculate $a \times b + b$: $8 \times 8 + 8 = 64 + 8 = 72$.
  • This does not match the third number, 74. Option 4 does not follow the relationship.

Conclusion

Based on our analysis, only Option 1 (12, 3, 39) exhibits the same numerical relationship ($a \times b + b = c$) as the given sets (9, 3, 30) and (10, 3, 33).

Revision Table: Checking Number Set Relationships

To revise this concept, remember the process:

• Analyze the given sets to find a common pattern or rule linking the numbers (usually the first two to the third).
• Test the identified rule with all the given sets to ensure it is consistent.
• Apply the confirmed rule to each of the options provided.
• The option that satisfies the rule is the correct answer.

Additional Information on Number Relationships

Number relationship or number analogy questions in reasoning test your ability to find patterns. The patterns can involve various mathematical operations. Some common relationships include:

• Basic arithmetic operations: addition, subtraction, multiplication, division.
• Combined operations: like the one found here ($a \times b + b$), or combinations of adding, subtracting, multiplying, dividing.
• Squares, cubes, or other powers of the numbers.
• Sum or difference of squares/cubes.
• Operations involving constants.

Always remember the rule specified in the question, like operating on whole numbers and not individual digits, as this guides your approach to finding the correct numerical relationship.