Finding the Number Relationship in Sets

The problem asks us to identify the mathematical relationship between the numbers in the given sets and then find an option set that shares the same relationship. A crucial rule is that operations must be performed on the whole numbers as they are, without breaking them down into individual digits.

Analyzing the Given Number Sets

We are given two sets:

• Set 1: (7, 41, 211)
• Set 2: (12, 66, 336)

Let's analyze the relationship between the numbers in these sets. We need to find a pattern that connects the first number to the second, and the second number to the third, that holds true for both sets.

Examining Set 1: (7, 41, 211)

Let's look at the relationship between 7 and 41. We can try multiplication and addition/subtraction:

• $7 \times 5 = 35$. To get to 41, we need $35 + 6 = 41$. So, the relationship could be First number $\times$ 5 + 6.

Now let's check the relationship between 41 and 211 using a similar pattern:

• $41 \times 5 = 205$. To get to 211, we need $205 + 6 = 211$. So, the relationship could be Second number $\times$ 5 + 6.

This suggests a potential relationship: If the set is (A, B, C), then $B = A \times 5 + 6$ and $C = B \times 5 + 6$.

Verifying with Set 2: (12, 66, 336)

Let's test this proposed relationship with the second set (12, 66, 336):

• Apply the first part of the relationship: First number $\times$ 5 + 6. $12 \times 5 + 6 = 60 + 6 = 66$. This matches the second number in the set.
• Apply the second part of the relationship: Second number $\times$ 5 + 6. $66 \times 5 + 6 = 330 + 6 = 336$. This matches the third number in the set.

Since the relationship $B = A \times 5 + 6$ and $C = B \times 5 + 6$ holds true for both given sets, this is the relationship we need to find in the options.

Checking the Options for the Same Relationship

Now, we will examine each option to see which set of numbers follows the identified relationship.

Let the option set be (A, B, C). We need to check if $B = A \times 5 + 6$ and $C = B \times 5 + 6$.

Option 1: (10, 45, 205)

• Check the first relationship: $10 \times 5 + 6 = 50 + 6 = 56$. The second number in the option is 45. $56 \neq 45$. This option does not follow the relationship.

Option 2: (9, 51, 255)

• Check the first relationship: $9 \times 5 + 6 = 45 + 6 = 51$. The second number in the option is 51. This matches.
• Check the second relationship: $51 \times 5 + 6 = 255 + 6 = 261$. The third number in the option is 255. $261 \neq 255$. This option does not follow the relationship.

Option 3: (14, 76, 385)

• Check the first relationship: $14 \times 5 + 6 = 70 + 6 = 76$. The second number in the option is 76. This matches.
• Check the second relationship: $76 \times 5 + 6 = 380 + 6 = 386$. The third number in the option is 385. $386 \neq 385$. This option does not follow the relationship.

Option 4: (11, 61, 311)

• Check the first relationship: $11 \times 5 + 6 = 55 + 6 = 61$. The second number in the option is 61. This matches.
• Check the second relationship: $61 \times 5 + 6 = 305 + 6 = 311$. The third number in the option is 311. This matches.

Option 4 follows the same relationship as the given sets.

Conclusion

The relationship shared by the numbers in the given sets (7, 41, 211) and (12, 66, 336) is that the second number is obtained by multiplying the first number by 5 and adding 6, and the third number is obtained by multiplying the second number by 5 and adding 6. Only option (11, 61, 311) follows this pattern.

Revision Table: Number Set Relationships

Understanding number set relationships is a key part of logical reasoning and quantitative aptitude tests. These problems often involve identifying patterns based on arithmetic operations (addition, subtraction, multiplication, division), powers, roots, or combinations of these. The key is to test potential relationships systematically across all given examples before applying them to the options.

Additional Information: Solving Number Analogy Problems

Number analogy questions require you to find the rule that connects a pair or set of numbers and apply that rule to another number or set. Here are some common types of relationships to look for:

• Arithmetic Operations: Addition, subtraction, multiplication, division (e.g., y = ax + b).
• Powers and Roots: Squares, cubes, square roots, cube roots (e.g., y = x² + c).
• Prime Numbers: The numbers might be prime or related to prime numbers.
• Digit Operations: Although explicitly disallowed in this specific problem, sometimes the sum, product, or other operations on digits are involved in other questions.
• Sequential Patterns: The pattern might depend on the position of the number in the set or a sequence.

Always start by examining the relationship between the first two numbers, then see if a similar relationship holds for the subsequent numbers in the set(s). Verify the pattern with all given examples before checking the options.