Understanding Number Set Relations

This question asks us to find a set of numbers from the options that shares the same relationship pattern as the two given sets: (111, 117, 123) and (169, 200, 231). We need to analyze the relationship between the numbers within each given set first, then apply that logic to the options. The rule specifies that operations must be performed on the whole numbers as given, not on their individual digits.

Analyzing Given Number Sets

Let's examine the first given set: (111, 117, 123).

• Find the difference between the second number and the first number: \(117 - 111\).
• \(117 - 111 = 6\).
• Find the difference between the third number and the second number: \(123 - 117\).
• \(123 - 117 = 6\).

In the first set, the difference between consecutive numbers is constant, which is 6. This means the numbers form an arithmetic progression.

Now, let's look at the second given set: (169, 200, 231).

• Find the difference between the second number and the first number: \(200 - 169\).
• \(200 - 169 = 31\).
• Find the difference between the third number and the second number: \(231 - 200\).
• \(231 - 200 = 31\).

In the second set, the difference between consecutive numbers is also constant, which is 31. This set also forms an arithmetic progression.

The common relationship or pattern observed in both given sets is that the numbers within each set are in an arithmetic progression, meaning there is a constant difference between consecutive terms.

Evaluating Option Number Sets

We will now check each option to see which set follows the pattern of an arithmetic progression.

Option 1: (169, 188, 199)

• Difference 1: \(188 - 169 = 19\).
• Difference 2: \(199 - 188 = 11\).

The differences (19 and 11) are not equal. This is not an arithmetic progression.

Option 2: (126, 138, 148)

• Difference 1: \(138 - 126 = 12\).
• Difference 2: \(148 - 138 = 10\).

The differences (12 and 10) are not equal. This is not an arithmetic progression.

Option 3: (125, 135, 145)

• Difference 1: \(135 - 125 = 10\).
• Difference 2: \(145 - 135 = 10\).

The differences (10 and 10) are equal. This is an arithmetic progression with a common difference of 10.

Option 4: (129, 133, 139)

• Difference 1: \(133 - 129 = 4\).
• Difference 2: \(139 - 133 = 6\).

The differences (4 and 6) are not equal. This is not an arithmetic progression.

Identifying Matching Number Set Pattern

Based on the analysis, the given sets (111, 117, 123) and (169, 200, 231) both show a pattern of arithmetic progression (constant difference between consecutive numbers).

Comparing this pattern to the options, only Option 3: (125, 135, 145) also shows a constant difference between consecutive numbers (a common difference of 10).

Conclusion

The set (125, 135, 145) is related in the same way as the given sets because all three sets are arithmetic progressions, even though the common difference is different in each set.

Revision Table: Key Concepts in Number Series Reasoning

When solving number set or number series problems, remember these key concepts:

• Arithmetic Progression (AP): A sequence where the difference between consecutive terms is constant. This constant difference is called the common difference.
• Geometric Progression (GP): A sequence where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio.
• Other Patterns: Look for patterns involving squares, cubes, prime numbers, Fibonacci sequence, or a combination of operations (addition, subtraction, multiplication, division) applied sequentially or alternatingly.
• Whole Number Operations: As specified in the question, focus on operations applied to the numbers themselves, not their individual digits, unless the problem explicitly states otherwise.

Additional Information: Arithmetic Progressions in Reasoning

Arithmetic progressions are a fundamental pattern often tested in logical reasoning and quantitative aptitude questions. Recognizing this pattern quickly can help solve many problems.

A sequence \(a_1, a_2, a_3, \dots\) is an arithmetic progression if \(a_{n+1} - a_n = d\) for any \(n \ge 1\), where \(d\) is the common difference. In a set of three numbers \((a, b, c)\), they form an arithmetic progression if \(b - a = c - b\).

In this specific question, the pattern was that the three numbers in the set formed an arithmetic progression. The common difference could vary between different valid sets, as seen in the two given examples (difference 6 and 31) and the correct option (difference 10).