Analyzing Number Triads to Find the Odd One Out

The question asks us to identify which triad of numbers does not follow the same mathematical pattern as the other three. We need to examine each given triad and look for a consistent relationship between the three numbers in each set.

Discovering the Pattern in the Triads

Let's represent a general triad as \(a : b : c\). We will test different mathematical operations to find a rule that applies to most of the given triads. Let's look closely at the numbers:

• Triad 1: 12 : 47 : 60
• Triad 2: 4 : 16 : 20
• Triad 3: 2 : 7 : 10
• Triad 4: 5 : 19 : 25

Observing Triads 1, 3, and 4, we can hypothesize a pattern relating \(a\) to \(b\) and \(a\) to \(c\).

• In Triad 3 (2 : 7 : 10), notice that \(2 \times 4 - 1 = 8 - 1 = 7\) and \(2 \times 5 = 10\).
• In Triad 4 (5 : 19 : 25), notice that \(5 \times 4 - 1 = 20 - 1 = 19\) and \(5 \times 5 = 25\).
• In Triad 1 (12 : 47 : 60), notice that \(12 \times 4 - 1 = 48 - 1 = 47\) and \(12 \times 5 = 60\).

It appears that the pattern for a triad \(a : b : c\) is:

• \(b = a \times 4 - 1\)
• \(c = a \times 5\)

Let's verify this pattern with each triad provided in the options.

Applying the Pattern to Each Triad

We will now check if each triad conforms to the discovered pattern: \(b = a \times 4 - 1\) and \(c = a \times 5\).

Analysis of Triad 1: 12 : 47 : 60

• Here, \(a = 12\), \(b = 47\), \(c = 60\).
• Check the first part of the pattern: \(a \times 4 - 1 = 12 \times 4 - 1 = 48 - 1 = 47\). This matches \(b\).
• Check the second part of the pattern: \(a \times 5 = 12 \times 5 = 60\). This matches \(c\).

• Conclusion: Triad 1 (12 : 47 : 60) follows the pattern.

Analysis of Triad 2: 4 : 16 : 20

• Here, \(a = 4\), \(b = 16\), \(c = 20\).
• Check the first part of the pattern: \(a \times 4 - 1 = 4 \times 4 - 1 = 16 - 1 = 15\). This does not match \(b = 16\).
• Check the second part of the pattern: \(a \times 5 = 4 \times 5 = 20\). This matches \(c\).

• Conclusion: Triad 2 (4 : 16 : 20) does not fully follow the pattern \(b = a \times 4 - 1\) and \(c = a \times 5\). Instead, it follows \(b = a \times 4\) and \(c = a \times 5\).

Analysis of Triad 3: 2 : 7 : 10

• Here, \(a = 2\), \(b = 7\), \(c = 10\).
• Check the first part of the pattern: \(a \times 4 - 1 = 2 \times 4 - 1 = 8 - 1 = 7\). This matches \(b\).
• Check the second part of the pattern: \(a \times 5 = 2 \times 5 = 10\). This matches \(c\).

• Conclusion: Triad 3 (2 : 7 : 10) follows the pattern.

Analysis of Triad 4: 5 : 19 : 25

• Here, \(a = 5\), \(b = 19\), \(c = 25\).
• Check the first part of the pattern: \(a \times 4 - 1 = 5 \times 4 - 1 = 20 - 1 = 19\). This matches \(b\).
• Check the second part of the pattern: \(a \times 5 = 5 \times 5 = 25\). This matches \(c\).

• Conclusion: Triad 4 (5 : 19 : 25) follows the pattern.

Identifying the Triad That Does Not Belong

Based on our analysis, Triads 1, 3, and 4 follow the mathematical pattern where the second number is four times the first number minus one (\(b = a \times 4 - 1\)), and the third number is five times the first number (\(c = a \times 5\)).

Triad 2 (4 : 16 : 20) follows the pattern \(c = a \times 5\), but the relationship for the second number is \(b = a \times 4\), not \(b = a \times 4 - 1\).

Therefore, Triad 2 does NOT belong to the group formed by the other three triads.

The triad that does NOT belong to the group is 4 : 16 : 20.

Revision Table: Triad Pattern Analysis

Reviewing the core concepts of number patterns and logical reasoning.

• Triad: A set of three numbers.
• Mathematical Pattern: A consistent rule or relationship between numbers in a set.
• Logical Reasoning: Using deduction to identify rules and exceptions.
• Outlier: An element that does not conform to the pattern of the group.

Additional Information: Finding Number Patterns

Finding patterns in number series or groups is a common type of logical reasoning question. Here are some strategies:

• Look for arithmetic progression (constant difference).
• Look for geometric progression (constant ratio).
• Check for relationships involving squares or cubes.
• Look for relationships between adjacent numbers (e.g., sum, difference, product).
• Look for relationships between the first number and others in a set (as in this problem).
• Consider combinations of operations (e.g., multiply and add/subtract).
• Test simple operations first before looking for complex ones.

Practicing with different types of number puzzles helps improve pattern recognition skills.