Solving the Number Analogy Problem

The question asks us to find the relationship between the first pair of numbers (12 and 140) and then apply that same relationship to the third number (15) to find the missing fourth number. This type of problem is known as a number analogy.

Analyzing the Relationship: 12 and 140

Let's look for a pattern connecting 12 and 140. Often, these relationships involve operations like squaring, cubing, multiplication, addition, subtraction, or a combination of these. Let's try squaring the first number:

• Calculate the square of 12: $12^2 = 144$.
• Compare this to 140. The difference is $144 - 140 = 4$.
• So, it appears the relationship might be "square the number and subtract 4".

Let's verify this pattern:

For the first pair (12 ∶ 140):

Relationship: $n \rightarrow n^2 - 4$

Applying this to 12: $12^2 - 4 = 144 - 4 = 140$. This matches the given second number.

Applying the Pattern to Find the Missing Number

Now, we apply the same relationship ($n^2 - 4$) to the third number, which is 15.

For the second pair (15 ∶ ?):

Let the missing number be $X$. The relationship is $15 \rightarrow 15^2 - 4$.

Calculate the square of 15: $15^2 = 225$.

Apply the subtraction: $15^2 - 4 = 225 - 4 = 221$.

So, the missing number is 221.

Checking the Options

Let's compare our calculated result with the given options:

• Option 1: 231
• Option 2: 115
• Option 3: 135
• Option 4: 221

Our calculated value, 221, matches Option 4.

Final Answer Derivation

The relationship found between the first pair of numbers (12 and 140) is $n \rightarrow n^2 - 4$. Applying this same relationship to the third number (15) gives $15^2 - 4 = 225 - 4 = 221$. Therefore, the missing number in the analogy is 221.

Revision Table: Key Concepts in Number Analogy

Additional Information: Types of Number Patterns

Number analogy problems often use various mathematical patterns. Some common types include:

• Arithmetic Operations: Simple addition, subtraction, multiplication, or division.
• Squaring or Cubing: Using the square ($n^2$) or cube ($n^3$) of the number, often with an added or subtracted constant.
• Prime Numbers: The numbers might be prime numbers or related to prime numbers.
• Sequential Patterns: The relationship might involve the next number in a sequence (like the next prime, the next square, etc.).
• Digit Operations: Sometimes the pattern involves the sum, product, or manipulation of the digits of the number.

Solving number analogies requires careful observation, testing different mathematical operations, and logical deduction to find the consistent rule.