The question asks us to find the number that relates to 24 in the same way that 445 relates to 21. This type of problem is known as a number analogy, where we need to identify the pattern or relationship between the first pair of numbers and apply it to the third number to find the missing fourth number.

Analyzing the Number Relationship

Let's examine the relationship between the first pair of numbers: 21 and 445.

We need to think about common mathematical operations or patterns that might connect 21 to 445.

• Is it simple addition or subtraction? $445 - 21 = 424$. This doesn't seem to reveal an obvious pattern.
• Is it multiplication? $21 \times N = 445$. $445 / 21$ is not a whole number, so it's not a simple multiplication by an integer.
• What about powers or squares of the number? Let's consider the square of 21.

Calculating the square of 21:

$\qquad 21^2 = 21 \times 21 = 441$

Now, let's compare this result to 445. The difference is $445 - 441 = 4$.

This suggests a possible relationship: The second number is obtained by squaring the first number and then adding 4.

Relationship discovered: $\text{Second Number} = (\text{First Number})^2 + 4$

Applying the Pattern to Find the Missing Number

Now, we apply this same relationship to the second pair of numbers. The third number is 24. We need to find the fourth number using the pattern:

$\qquad \text{Fourth Number} = (\text{Third Number})^2 + 4$

Substitute the third number (24) into the formula:

$\qquad \text{Fourth Number} = 24^2 + 4$

Calculate the square of 24:

$\qquad 24^2 = 24 \times 24 = 576$

Now, add 4 to this result:

$\qquad \text{Fourth Number} = 576 + 4 = 580$

So, the missing number is 580.

Verifying the Result with Options

Let's check if 580 is one of the given options:

• Option 1: 495
• Option 2: 505
• Option 3: 523
• Option 4: 580

The calculated value, 580, matches Option 4.

Summary of the Analogy Pattern

The analogy follows the pattern where the second number is the square of the first number plus 4. We applied this rule to the third number to find the fourth number.

$\qquad 21 \to 21^2 + 4 = 441 + 4 = 445$

$\qquad 24 \to 24^2 + 4 = 576 + 4 = 580$

Therefore, the missing number in the analogy 21 ∶ 445 ∶∶ 24 ∶ ? is 580.

Revision Table: Number Analogy Review

Additional Information: Types of Number Analogies

Number analogies in reasoning questions can involve various mathematical operations and patterns. Some common types include:

• Addition or Subtraction: A constant value is added or subtracted.
• Multiplication or Division: Numbers are related by multiplication or division.
• Squaring or Cubing: Numbers are related by squares or cubes of the number.
• Combination of Operations: A mix of operations like squaring/cubing followed by addition/subtraction, or multiplication followed by addition/subtraction.
• Digit Operations: Patterns based on the digits of the number (e.g., sum of digits, product of digits).
• Prime, Composite, Even, Odd Numbers: Patterns based on the properties of the numbers.

Solving number analogies requires careful observation, knowledge of basic arithmetic operations, squares, cubes, and the ability to test different possible patterns quickly.