Understanding the Number Analogy Problem

This question is a number analogy, which means we need to find the relationship or pattern that connects the first pair of numbers (\(20\) and \(480\)). Once we understand this pattern, we apply it to the third number (\(25\)) to find the missing fourth number in the analogy \(20 : 480 :: 25 : ?\).

Analyzing the Relationship between 20 and 480

Let's try to figure out how 20 is related to 480. We can consider several mathematical operations:

• Could it be simple multiplication? \(20 \times ? = 480\). If we divide 480 by 20, we get \(480 \div 20 = 24\). So, \(20 \times 24 = 480\). This is a direct multiplication. Let's examine the multiplier, 24. How is 24 related to the original number 20? \(24 = 20 + 4\).
• This suggests a potential pattern: the second number is obtained by multiplying the first number (\(n\)) by (\(n + 4\)). The formula would be \(n \times (n+4)\). Let's test this with the first pair: For \(n=20\), \(20 \times (20 + 4) = 20 \times 24 = 480\). This works perfectly for the first pair.
• Another way to express \(n \times (n+4)\) is by expanding it: \(n^2 + 4n\). Let's also test this form with \(n=20\): \(20^2 + 4 \times 20 = 400 + 80 = 480\). This also fits the relationship between 20 and 480.

Both forms of the pattern, \(n \times (n+4)\) and \(n^2 + 4n\), describe the same relationship. We will use this pattern to find the missing number.

Applying the Pattern to Find the Fourth Number

Now, we apply the pattern \(n \times (n+4)\) or \(n^2 + 4n\) to the third number, which is 25. Here, \(n = 25\).

Using the formula \(n \times (n+4)\):

For \(n=25\), the fourth number will be \(25 \times (25 + 4)\).

First, calculate the value inside the parentheses: \(25 + 4 = 29\).

Next, multiply 25 by 29: \(25 \times 29\).

Calculation:

\(25 \times 29 = 25 \times (30 - 1) = (25 \times 30) - (25 \times 1) = 750 - 25 = 725\)

Alternatively, using the formula \(n^2 + 4n\):

For \(n=25\), the fourth number will be \(25^2 + 4 \times 25\).

Calculate the square of 25: \(25^2 = 25 \times 25 = 625\).

Calculate 4 times 25: \(4 \times 25 = 100\).

Add the results: \(625 + 100 = 725\).

Both methods confirm that the missing number is 725.

Verification and Final Answer

Let's summarize the pattern and its application:

The pattern \(n^2 + 4n\) (or \(n(n+4)\)) holds for the first pair and gives 725 when applied to the third number 25. Therefore, the missing number in the analogy is 725.

Revision Table: Solving Number Analogies

Additional Information: Types of Number Patterns

Number analogies can be based on various mathematical patterns. Recognizing common patterns helps in solving these problems faster. Some common types include:

• Arithmetic Progression: Adding/subtracting a constant.
• Geometric Progression: Multiplying/dividing by a constant.
• Square/Cube Patterns: Relations involving \(n^2\), \(n^3\), \(n^2 \pm C\), \(n^3 \pm C\), \(n^2 \pm n\), \(n^3 \pm n\), etc.
• Product/Sum of Digits: Patterns based on the digits of the number.
• Factor/Multiple Relationships: One number being a factor or multiple of the other.
• Combination of Operations: Patterns involving more than one operation.

Practicing different types of analogy problems is key to improving your pattern recognition skills.