This question asks us to identify a set of numbers among the given options that shares the same relationship as the provided sets: (2, 8, 36) and (4, 9, 25). We are instructed to perform operations only on the whole numbers themselves, not on their individual digits.

Analyzing the Given Number Set Patterns

Let's first examine the relationship between the numbers in the given sets. We have two example sets:

• Set 1: (2, 8, 36)
• Set 2: (4, 9, 25)

Our goal is to find a rule or pattern that connects the first number, the second number, and the third number in both of these sets.

Discovering the Relationship in Set (2, 8, 36)

Consider the numbers 2, 8, and 36. Let's see how they might be related. The third number, 36, is the square of 6 (\( 6^2 \)). Can we obtain the number 6 using the first two numbers, 2 and 8?

• If we add them: \( 2 + 8 = 10 \)
• If we subtract them: \( 8 - 2 = 6 \)

The difference between the second number (8) and the first number (2) is 6. If we square this difference, we get \( 6^2 = 36 \), which is the third number in the set.

So, a potential rule is: (Second Number - First Number)$^2$ = Third Number.

Verifying the Pattern with Set (4, 9, 25)

Let's check if this rule holds true for the second given set (4, 9, 25).

First number is 4, second number is 9, and third number is 25.

Applying our rule: \( (9 - 4)^2 = 5^2 = 25 \)

This matches the third number (25). Therefore, the pattern we identified, \( (b - a)^2 = c \) where 'a' is the first number, 'b' is the second, and 'c' is the third, is the correct relationship governing these number sets.

Checking Options to Find the Matching Number Set

Now we will apply the rule \( (b - a)^2 = c \) to each of the given options to find the set that follows the same pattern.

Option 1: (14, 18, 9)

• First Number (a) = 14
• Second Number (b) = 18
• Third Number (c) = 9

Applying the rule: \( (18 - 14)^2 = 4^2 = 16 \)

Since \( 16 \neq 9 \), this set does not follow the pattern.

Option 2: (13, 20, 49)

• First Number (a) = 13
• Second Number (b) = 20
• Third Number (c) = 49

Applying the rule: \( (20 - 13)^2 = 7^2 = 49 \)

Since \( 49 = 49 \), this set follows the pattern.

Option 3: (21, 15, 35)

• First Number (a) = 21
• Second Number (b) = 15
• Third Number (c) = 35

Applying the rule: \( (15 - 21)^2 = (-6)^2 = 36 \)

Since \( 36 \neq 35 \), this set does not follow the pattern.

Option 4: (12, 14, 26)

• First Number (a) = 12
• Second Number (b) = 14
• Third Number (c) = 26

Applying the rule: \( (14 - 12)^2 = 2^2 = 4 \)

Since \( 4 \neq 26 \), this set does not follow the pattern.

Conclusion

Our analysis shows that only Option 2, the set (13, 20, 49), matches the relationship found in the original sets (2, 8, 36) and (4, 9, 25), which is \( (\text{Second Number} - \text{First Number})^2 = \text{Third Number} \).

Number Pattern Analogy Revision Table

Additional Information on Number Analogies and Reasoning

Number analogy questions are a key part of logical reasoning and quantitative ability tests. They require you to quickly identify mathematical or logical relationships between numbers in a given set and apply that same rule to find a corresponding set.

Common strategies for solving number analogy problems include:

• Looking for simple arithmetic progressions or differences.
• Checking for relationships involving multiplication, division, squares, cubes, or roots.
• Considering the position of numbers in the set (e.g., relationship between 1st and 2nd, 2nd and 3rd, or 1st and 3rd).
• Sometimes, relationships involve the sum, difference, or product of the numbers in specific positions.
• Always check the identified pattern against all the given examples before applying it to the options.

Practice with various types of number patterns will improve your ability to spot the underlying rule quickly.