Understanding Number Analogy Reasoning

Number analogy questions test your ability to find a relationship or pattern between a pair of numbers and apply that same relationship to another number to find a missing value. In this question, we are given two complete pairs and one incomplete pair, and we need to determine the rule connecting the numbers in the complete pairs.

Analyzing the Given Number Pairs

We are given the following relationships:

• 8 : 49
• 16 : ?
• 12 : 121

Let's look closely at the complete pairs to find a consistent pattern.

Pair 1: 8 : 49

We need to find a relationship between 8 and 49. Let's consider common mathematical operations:

• Is it addition/subtraction? \( 49 - 8 = 41 \). Unlikely to be a simple constant difference.
• Is it multiplication/division? \( 49 / 8 \) is not a whole number.
• Is it related to squares or cubes? \( 8^2 = 64 \), \( 8^3 = 512 \). Neither is 49. \( 7^2 = 49 \). How can we get 7 from 8? By subtracting 1.

Let's test the idea that the second number is the square of the first number minus 1: \( (8-1)^2 = 7^2 = 49 \). This fits for the first pair.

Pair 2: 12 : 121

Now let's test the same pattern on the second complete pair (which is the third pair in the sequence). The numbers are 12 and 121. Applying the pattern: \( (12-1)^2 = 11^2 \).

We know that \( 11 \times 11 = 121 \). So, \( 11^2 = 121 \). This also fits the pattern perfectly.

Identifying the Underlying Relationship

Based on the analysis of the two complete pairs, the consistent pattern is that the second number in each pair is the square of the value obtained by subtracting 1 from the first number. Mathematically, if the pair is \( x : y \), the relationship is \( y = (x-1)^2 \).

Applying the Pattern to Find the Missing Number

We need to find the missing number in the pair 16 : ?. Let the missing number be \( N \). The first number is 16.

Using the identified pattern, we subtract 1 from the first number and then square the result:

\( N = (16-1)^2 \)

\( N = (15)^2 \)

Now, we calculate the square of 15:

\( 15^2 = 15 \times 15 \)

\( 15 \times 15 = 225 \)

So, the missing number is 225.

Comparing with Options

Let's check the given options:

• 215
• 220
• 250
• 225

The calculated missing number is 225, which matches one of the options.

Conclusion

The relationship in the number analogy is that the second number is the square of the first number minus 1. Applying this pattern to 16 gives \( (16-1)^2 = 15^2 = 225 \). Therefore, the missing number is 225.

Revision Table: Number Analogy Pattern

Additional Information on Number Analogy Types

Number analogy questions can follow various patterns. Some common types of relationships include:

• Arithmetic Operations: Addition, subtraction, multiplication, division, or a combination of these.
• Squares and Cubes: The second number is the square or cube of the first number, or of the first number plus/minus a constant.
• Prime Numbers: The numbers might follow a sequence related to prime numbers.
• Digits: The relationship might involve the sum, difference, product, or other operations on the digits of the number.
• Positional Value: Sometimes related to alphabetical position if letters are involved in other analogy types.

Solving number analogies requires careful observation and testing different possible mathematical relationships between the given numbers.