Understanding the Number Set Similarity Question

This question asks us to find a set of numbers from the given options that follows a similar pattern or rule as the provided set: (8, 12, 18). To solve this, we first need to analyze the relationship between the numbers in the given set and then check which option exhibits a comparable relationship.

Analyzing the Given Set (8, 12, 18)

Let's look for common patterns:

• Differences: We can find the difference between consecutive numbers.
  • Difference between the second and first number: \(12 - 8 = 4\)
  • Difference between the third and second number: \(18 - 12 = 6\)

  The sequence of differences is (4, 6).

• Ratios: We can find the ratio between consecutive numbers.
  • Ratio of the second to the first number: \(12 / 8 = 3/2 = 1.5\)
  • Ratio of the third to the second number: \(18 / 12 = 3/2 = 1.5\)

  The sequence of ratios is (1.5, 1.5). This indicates a geometric progression with a common ratio of 1.5.

• Other Patterns: Let's consider adding amounts: Start with 8, add 4 to get 12, then add 6 to get 18. (Add 4, Add 6).

Based on this initial analysis, the set (8, 12, 18) could be following a geometric progression with a ratio of 1.5, or a pattern of adding 4 then adding 6.

Examining the Options for a Similar Pattern

Now, let's analyze each option to see if it follows the same pattern as the given set (8, 12, 18).

Option (a) (6, 12, 18)

• Differences: \(12 - 6 = 6\), \(18 - 12 = 6\). Differences are (6, 6).
• Ratios: \(12 / 6 = 2\), \(18 / 12 = 1.5\). Ratios are (2, 1.5).
• Addition: Start with 6, add 6 to get 12, then add 6 to get 18. (Add 6, Add 6).

This set follows an arithmetic progression. It does not match the difference pattern (4, 6) or the ratio pattern (1.5, 1.5) exactly, although the second ratio is 1.5 and the second difference is 6.

Option (b) (7, 11, 15)

• Differences: \(11 - 7 = 4\), \(15 - 11 = 4\). Differences are (4, 4).
• Ratios: \(11 / 7 \approx 1.57\), \(15 / 11 \approx 1.36\). Ratios are (approx 1.57, approx 1.36).
• Addition: Start with 7, add 4 to get 11, then add 4 to get 15. (Add 4, Add 4).

This set follows an arithmetic progression. It does not match the difference pattern (4, 6) or the ratio pattern (1.5, 1.5).

Option (c) (6, 11, 17)

• Differences: \(11 - 6 = 5\), \(17 - 11 = 6\). Differences are (5, 6).
• Ratios: \(11 / 6 \approx 1.83\), \(17 / 11 \approx 1.54\). Ratios are (approx 1.83, approx 1.54).
• Addition: Start with 6, add 5 to get 11, then add 6 to get 17. (Add 5, Add 6).

Comparing this to the given set (8, 12, 18) with differences (4, 6) or addition (Add 4, Add 6):

• The sequence of differences (5, 6) shares the second difference (6) with the given set (4, 6).
• The addition pattern (Add 5, Add 6) shares the second addition amount (Add 6) with the given set (Add 4, Add 6).

This similarity in the second step of the pattern (adding 6 to get the third term) suggests a potential match.

Option (d) (9, 13, 20)

• Differences: \(13 - 9 = 4\), \(20 - 13 = 7\). Differences are (4, 7).
• Ratios: \(13 / 9 \approx 1.44\), \(20 / 13 \approx 1.54\). Ratios are (approx 1.44, approx 1.54).
• Addition: Start with 9, add 4 to get 13, then add 7 to get 20. (Add 4, Add 7).

This set does not clearly match the pattern of the given set (4, 6) differences or (1.5, 1.5) ratios, or (Add 4, Add 6) additions.

Identifying the Most Similar Set

Let's compare the difference/addition patterns side-by-side:

While the geometric ratio pattern (1.5, 1.5) for the given set is clear, none of the options follow this strictly. Looking at the differences or addition pattern, the given set uses the sequence (Add 4, Add 6).

Option (c) uses the sequence (Add 5, Add 6). Both the given set and option (c) have 'Add 6' as the second step to get the third number from the second number. This common characteristic in the second operation makes option (c) the most similar set based on this type of pattern analysis.

Conclusion

The given set (8, 12, 18) shows a pattern of adding 4, then adding 6. Option (c) (6, 11, 17) shows a pattern of adding 5, then adding 6. The similarity lies in the second step of the pattern, which is adding 6 in both cases. Therefore, option (c) is the set most similar to the given set.

Revision Table: Analyzing Number Set Patterns

Comparing the patterns, option (c) uniquely shares the '+6' addition (or difference of 6) for the step from the second to the third number with the given set (8, 12, 18), apart from option (a). However, option (c)'s sequence of differences (5, 6) has one element (6) exactly matching the second element of the given set's difference sequence (4, 6). Option (a)'s difference sequence (6, 6) has both elements different from the first element of (4, 6).

Additional Information on Number Patterns and Sequences

Number pattern questions are common in reasoning tests. They require identifying the rule that connects the numbers in a series or set. Common types of patterns include:

• Arithmetic Progression (AP): Each term after the first is obtained by adding a constant difference (common difference) to the preceding term. Example: 2, 4, 6, 8... (common difference is 2).
• Geometric Progression (GP): Each term after the first is obtained by multiplying the preceding term by a constant ratio (common ratio). Example: 3, 6, 12, 24... (common ratio is 2).
• Difference Series: The differences between consecutive terms form a separate pattern (like an AP or GP). Example: 1, 2, 4, 7, 11... Differences are 1, 2, 3, 4... (an AP). The given set (8, 12, 18) has differences (4, 6), where the differences themselves form a sequence (4, 6).
• Quadratic Sequences: Sequences where the second differences (differences between the differences) are constant. The given set (8, 12, 18) with differences (4, 6) has a second difference of \(6-4=2\), indicating it is a quadratic sequence following the rule \(n^2 + n + 6\).
• Mixed Patterns: Combinations of arithmetic and geometric operations, or other rules involving squares, cubes, etc.

Solving similar set questions involves testing these common patterns against the given set and the options to find the most consistent rule.