Understanding Number Analogy and Reasoning

This question asks us to find a missing term in a number analogy based on the relationship between the given pairs. The analogy is presented as:

\(169 : 14 :: ? : 16 :: 289 : 18\)

This format means "169 is to 14 as ? is to 16 as 289 is to 18". We need to identify the pattern connecting the numbers in the first pair (169 and 14) and the third pair (289 and 18), and then apply that same pattern to the second pair to find the missing number.

Analyzing the Given Number Pairs

Let's examine the relationship between the numbers in the known pairs:

• Pair 1: 169 and 14
• We can observe that 169 is a perfect square. Specifically, \(169 = 13 \times 13 = 13^2\).
• The second number in this pair is 14.
• What is the relationship between 13 (the base of the square) and 14? It seems to be \(13 + 1 = 14\).
• So, the pattern for this pair could be: If the first term is \(N^2\), the second term is \(N+1\).

• Pair 3: 289 and 18
• Let's check if the same pattern applies here. Is 289 a perfect square? Yes, \(289 = 17 \times 17 = 17^2\).
• The second number in this pair is 18.
• According to our hypothesized pattern, the second term should be the base of the square plus 1. The base is 17. Is \(17 + 1 = 18\)? Yes, it is.
• The pattern \(N^2 : N+1\) holds true for this pair as well.

Applying the Pattern to Find the Missing Term

Now that we've identified the consistent pattern \(N^2 : N+1\), we can apply it to the middle pair:

\( ? : 16 \)

In this pair, the second term is 16. According to the pattern, the second term is \(N+1\).

So, we have the equation: \(N + 1 = 16\).

To find the value of \(N\), we subtract 1 from both sides:

\(N = 16 - 1\)

\(N = 15\)

The first term in this pair is \(N^2\). Since \(N = 15\), the missing term is \(15^2\).

Let's calculate \(15^2\):

\(15^2 = 15 \times 15 = 225\)

Therefore, the missing term is 225.

Verifying the Answer with Options

We found the missing term to be 225. Let's look at the given options:

1. 205
2. 230
3. 225
4. 125

Our calculated value, 225, matches option 3.

Final Solution

The relationship in the analogy is that the first term is the square of a number, and the second term is that number plus one. Applying this pattern to the pair \(? : 16\), where 16 is \(N+1\), we find \(N=15\). The missing term is \(N^2\), which is \(15^2 = 225\).

Revision Table: Key Concepts in Number Analogy

Additional Information: Solving Reasoning Problems

Number analogy questions are common in reasoning and quantitative aptitude tests. To excel at these, consider the following strategies:

• Look for simple arithmetic operations: Addition, subtraction, multiplication, division.
• Check for squares or cubes: Numbers might be squares, cubes, or related to squares/cubes (e.g., \(N^2 \pm k\), \(N^3 \pm k\)).
• Consider prime numbers: The sequence might involve prime numbers.
• Look for patterns in digits: Sometimes the relationship is based on the sum, product, or manipulation of the digits within the numbers.
• Check for sequences: The numbers (or their bases/roots) might form an arithmetic or geometric progression. In this problem, the bases (13, 15, 17) form an arithmetic progression with a common difference of 2.
• Practice regularly: Familiarity with common patterns (like squares, cubes, basic series) helps in quickly identifying the relationship.

This problem involved recognizing perfect squares and a simple additive relationship with the base of the square, highlighting the importance of knowing common squares and basic arithmetic operations.