Analyzing the Number Pattern Series

The question asks us to identify the pattern relating the two numbers in each pair provided and then use that pattern to find the missing number in the last pair.

We are given the following pairs:

1. 72 : 8
2. 156 : 12
3. 272 : 16
4. 600 : ?

Let's examine the relationship between the numbers in the first three pairs to find a consistent pattern.

Finding the Pattern in the Number Pairs

Let's denote the first number in a pair as $A$ and the second number as $B$. We need to find a rule that connects $A$ and $B$ for pairs (A), (B), and (C).

• Pair (A): 72 : 8
• Pair (B): 156 : 12
• Pair (C): 272 : 16

Let's try simple arithmetic operations. Division is often a good starting point for ratio-like questions.

• For 72 : 8, \( \frac{72}{8} = 9 \)
• For 156 : 12, \( \frac{156}{12} = 13 \)
• For 272 : 16, \( \frac{272}{16} = 17 \)

The results of the division are 9, 13, and 17. Let's look at the sequence of these results: 9, 13, 17. The difference between consecutive terms is \( 13 - 9 = 4 \) and \( 17 - 13 = 4 \). This sequence is an arithmetic progression with a common difference of 4.

Now let's compare these results (9, 13, 17) with the second number in each pair (8, 12, 16).

• For 72 : 8, the result 9 is \( 8 + 1 \).
• For 156 : 12, the result 13 is \( 12 + 1 \).
• For 272 : 16, the result 17 is \( 16 + 1 \).

This suggests a strong pattern: \( \frac{\text{First Number}}{\text{Second Number}} = \text{Second Number} + 1 \).

Let's rewrite this pattern in terms of the first number:

First Number = (Second Number) \( \times \) (Second Number + 1)

Or, if the second number is \( n \), the first number is \( n(n+1) = n^2 + n \).

Verifying the Discovered Pattern

Let's check if this pattern holds for the given pairs:

• For 72 : 8, Second Number \( n=8 \). \( n^2 + n = 8^2 + 8 = 64 + 8 = 72 \). This matches the first number.
• For 156 : 12, Second Number \( n=12 \). \( n^2 + n = 12^2 + 12 = 144 + 12 = 156 \). This matches the first number.
• For 272 : 16, Second Number \( n=16 \). \( n^2 + n = 16^2 + 16 = 256 + 16 = 272 \). This matches the first number.

The pattern First Number = (Second Number)\(^2\) + Second Number is consistent across all the given pairs.

Applying the Pattern to Find the Missing Number

Now we apply this pattern to the last pair: 600 : ? Let the missing number be \( x \). According to the pattern, the first number (600) should be equal to \( x^2 + x \).

So, we have the equation: \( x^2 + x = 600 \).

To solve for \( x \), we can rearrange the equation into a standard quadratic form:

\( x^2 + x - 600 = 0 \)

We need to find a positive integer value for \( x \) that satisfies this equation. We can solve this by factoring the quadratic equation or by testing the given options.

Let's try factoring. We look for two numbers that multiply to -600 and add up to 1. The numbers are 25 and -24.

So, the equation can be factored as \( (x + 25)(x - 24) = 0 \).

This gives two possible solutions for \( x \):

• \( x + 25 = 0 \implies x = -25 \)
• \( x - 24 = 0 \implies x = 24 \)

Since the numbers in the given pairs are positive, the missing number is likely the positive solution, which is \( x = 24 \).

Checking the Options

Let's check if \( x=24 \) is among the options and if it fits the pattern with 600.

If \( x = 24 \), then Second Number \( n=24 \). According to the pattern, the first number should be \( n^2 + n \).

\( 24^2 + 24 = 576 + 24 = 600 \).

This perfectly matches the first number in the pair 600 : ?. So, the missing number is 24.

Let's quickly check the other options using the pattern \( n^2 + n \):

• Option 1: 26. If \( n=26 \), \( 26^2 + 26 = 676 + 26 = 702 \ne 600 \).
• Option 2: 32. If \( n=32 \), \( 32^2 + 32 = 1024 + 32 = 1056 \ne 600 \).
• Option 3: 24. If \( n=24 \), \( 24^2 + 24 = 576 + 24 = 600 \). This matches.
• Option 4: 22. If \( n=22 \), \( 22^2 + 22 = 484 + 22 = 506 \ne 600 \).

Only 24 satisfies the pattern.

Conclusion

The pattern relating the numbers in each pair is that the first number is the square of the second number plus the second number itself (\( \text{First Number} = \text{Second Number}^2 + \text{Second Number} \)). Applying this pattern to the pair 600 : ?, we find that the missing number is 24.

Revision Table: Key Concepts in Number Patterns

Additional Information: Solving Quadratic Equations and Number Series Logic

Solving equations like \( x^2 + x - 600 = 0 \) is a common task in mathematics and logic puzzles. The quadratic formula is a general method to solve such equations: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). For \( x^2 + x - 600 = 0 \), we have \( a=1, b=1, c=-600 \). Using the formula would give the same solutions, 24 and -25.

Number series and pattern recognition questions are frequent in aptitude tests. They require observing the relationships between numbers, which can involve arithmetic operations (addition, subtraction, multiplication, division), powers, sequences (arithmetic, geometric), or combinations of these. Often, looking at the difference or ratio between terms, or the relationship between corresponding terms in related sequences, can reveal the underlying logic.

In this specific logic puzzle, the relationship First Number = Second Number \( \times \) (Second Number + 1) or First Number = Second Number\(^2\) + Second Number is the key. This type of pattern, involving the square of a number, is quite common in number series problems.

Understanding how to identify these patterns and translate them into mathematical expressions is crucial for solving such quantitative reasoning questions. Testing the options using the suspected pattern is also a very effective strategy when dealing with multiple-choice questions.