Understanding Number Relationships and Analogies

This question asks us to identify a set of three numbers that share the same underlying relationship or pattern as the numbers in the given set: (11, 165, 209). These types of problems test your ability to find logical connections between numbers.

Analyzing the Given Number Set (11, 165, 209)

Let's examine the relationship between the numbers in the set (11, 165, 209). We need to find a rule that connects the first number (11) to the second number (165) and the third number (209).

Let the first number be $x$. Here, $x=11$.

How is 165 related to 11? Let's try division:

\( \frac{165}{11} = 15 \)

So, the second number is the first number multiplied by 15: \( 165 = 11 \times 15 \).

How is 209 related to 11? Let's try division again:

\( \frac{209}{11} = 19 \)

So, the third number is the first number multiplied by 19: \( 209 = 11 \times 19 \).

The relationship in the given set (11, 165, 209) appears to be \( (x, x \times 15, x \times 19) \).

Testing Options for Similar Number Relations

Now, let's apply this discovered relationship \( (x, x \times 15, x \times 19) \) to the given options to see which set follows the same pattern.

Option 1: (12, 180, 228)

• Here, the first number $x = 12$.
• According to the rule, the second number should be \( x \times 15 = 12 \times 15 \).
• \( 12 \times 15 = 180 \). This matches the second number in the set (180).
• According to the rule, the third number should be \( x \times 19 = 12 \times 19 \).
• \( 12 \times 19 = 228 \). This matches the third number in the set (228).

This set (12, 180, 228) follows the same relationship \( (x, x \times 15, x \times 19) \).

Option 2: (17, 245, 323)

• Here, the first number $x = 17$.
• Second number test: \( x \times 15 = 17 \times 15 = 255 \). The second number in the set is 245. This does not match.

This set does not follow the rule.

Option 3: (15, 225, 275)

• Here, the first number $x = 15$.
• Second number test: \( x \times 15 = 15 \times 15 = 225 \). This matches the second number in the set (225).
• Third number test: \( x \times 19 = 15 \times 19 = 285 \). The third number in the set is 275. This does not match.

This set does not follow the rule.

Option 4: (14, 210, 276)

• Here, the first number $x = 14$.
• Second number test: \( x \times 15 = 14 \times 15 = 210 \). This matches the second number in the set (210).
• Third number test: \( x \times 19 = 14 \times 19 = 266 \). The third number in the set is 276. This does not match.

This set does not follow the rule.

Step-by-Step Solution for the Correct Option

The set (12, 180, 228) from Option 1 is the only one that matches the relationship found in the given set (11, 165, 209).

Let the first number of the set be $x$.

The rule derived from the given set (11, 165, 209) is:

Second number = \( x \times 15 \)

Third number = \( x \times 19 \)

For the set (12, 180, 228), the first number \( x = 12 \).

Step 1: Check the second number:

\( 12 \times 15 = 180 \)

The second number in the set is 180, which matches the calculation.

Step 2: Check the third number:

\( 12 \times 19 = 228 \)

The third number in the set is 228, which matches the calculation.

Since both calculations match the numbers in the set (12, 180, 228), this set follows the same pattern as the given set (11, 165, 209).

Conclusion on Number Set Relationships

By analyzing the mathematical relationships between the numbers in the initial set (11, 165, 209), we found a rule based on multiplication by constants (15 and 19). Applying this rule to the options, only the set (12, 180, 228) exhibited the identical number relationship.

Number Analogy Revision Table

Additional Information on Number Pattern Problems

Number pattern and analogy problems are common in reasoning sections of competitive exams. They can involve various types of relationships, not just simple multiplication as seen here. Some other common patterns include:

• Arithmetic Progression: Adding a constant difference between consecutive terms.
• Geometric Progression: Multiplying by a constant ratio between consecutive terms.
• Squares or Cubes: Numbers related to the squares or cubes of the first number or other sequence numbers.
• Differences or Sums: Relationships based on the sum or difference of the numbers.
• Combinations of Operations: Patterns involving a mix of addition, subtraction, multiplication, or division.
• Digit Operations: Relationships based on the sum or product of the digits within the numbers.

Solving these problems requires careful observation, testing different possible relationships, and applying logical reasoning to find the consistent rule.