Understanding Number Analogies

Number analogy questions test your ability to find the relationship between a pair of numbers and apply that same relationship to another number to find a missing value. In this question, we are given the analogy 23 : 441 : : 28 : ? We need to determine the pattern connecting 23 and 441 and then use that pattern to find the number related to 28.

Analyzing the Relationship between 23 and 441

Let's look at the first pair of numbers: 23 and 441.

We need to figure out how 441 is derived from 23. Often, relationships involve basic arithmetic operations like addition, subtraction, multiplication, division, squaring, cubing, or combinations of these.

• Could it be simple multiplication? $23 \times \text{something} = 441$? $441 / 23 \approx 19.17$, not a simple integer.
• Could it involve squaring? Let's consider numbers close to 23.

Let's think about squares of numbers near 23.

• $20^2 = 400$
• $21^2 = 441$
• $22^2 = 484$

We see that $21^2$ is exactly 441. Now, how is 21 related to 23?

The difference is $23 - 21 = 2$.

So, the relationship between 23 and 441 appears to be: subtract 2 from the first number and then square the result.

Let's verify this pattern:

Relationship: $(\text{First Number} - 2)^2 = \text{Second Number}$

Applying this to the first pair (23 : 441):

$(23 - 2)^2 = 21^2 = 441$.

This confirms the pattern for the first pair.

Applying the Pattern to Find the Missing Number

Now we apply the same pattern to the third number, 28, to find the fourth number (the missing number).

The third number is 28.

Following the pattern: Subtract 2 from the third number and then square the result.

$(\text{Third Number} - 2)^2 = \text{Missing Number}$

$(28 - 2)^2 = \text{Missing Number}$

$26^2 = \text{Missing Number}$

Calculating the Square of 26

To find the missing number, we need to calculate $26^2$.

$26^2 = 26 \times 26$

We can calculate this as:

So, $26^2 = 676$.

The missing number is 676.

Comparing with Options

Let's check the given options:

• 692
• 676
• 494
• 528

Our calculated number, 676, matches option 2.

Conclusion

The relationship between the numbers in the analogy 23 : 441 : : 28 : ? is that the second number is the square of (the first number minus 2). Applying this pattern to 28, we get $(28 - 2)^2 = 26^2 = 676$.

Revision Table: Number Analogy Pattern

Additional Information on Number Reasoning

Number analogy and number series questions are common in reasoning sections of competitive exams. They require identifying the rule or pattern that connects the numbers. Common patterns include:

• Arithmetic Progression (constant difference)
• Geometric Progression (constant ratio)
• Squares or cubes of numbers (or numbers plus/minus a constant)
• Operations on digits of the number
• Alternating patterns
• Combined operations (e.g., multiply by 2 then add 1)

Practicing various types of patterns helps in quickly identifying the relationship during an exam.