Analyzing Number Pairs for Patterns

The question asks us to identify the number-pair that is different from the other three based on a hidden pattern or rule connecting the two numbers in each pair. We are given four number pairs:

• 27 : 85
• 33 : 109
• 44 : 136
• 30 : 94

We need to look for a consistent mathematical relationship between the first number and the second number in each pair. Let's test common relationships like multiplication and addition/subtraction.

Discovering the Number Pair Pattern

Let's examine each pair to find a potential rule. Often, these patterns involve multiplying the first number by a constant and then adding or subtracting another constant. Let's try multiplying the first number by a small integer and see if we can reach the second number by adding a constant.

Let's test the first number pair, 27 : 85. We can see that 85 is roughly three times 27 ($3 \times 27 = 81$). The difference is $85 - 81 = 4$. So, the relationship could be $3 \times \text{First Number} + 4 = \text{Second Number}$.

Let's test this potential pattern ($y = 3x + 4$) on the other pairs:

• Pair 1: 27 : 85

  Applying the rule: $3 \times 27 + 4 = 81 + 4 = 85$.

  This matches the given second number.

• Pair 2: 33 : 109

  Applying the rule: $3 \times 33 + 4 = 99 + 4 = 103$.

  The given second number is 109. $103 \neq 109$. This pair does not follow the pattern $y = 3x + 4$.

• Pair 3: 44 : 136

  Applying the rule: $3 \times 44 + 4 = 132 + 4 = 136$.

  This matches the given second number.

• Pair 4: 30 : 94

  Applying the rule: $3 \times 30 + 4 = 90 + 4 = 94$.

  This matches the given second number.

Based on this analysis, three of the number pairs (27:85, 44:136, and 30:94) follow the rule where the second number is obtained by multiplying the first number by 3 and adding 4. The number pair 33 : 109 does not follow this rule; instead, $109 = 3 \times 33 + 10$.

Identifying the Different Pair

Since the pair 33 : 109 is the only one that does not fit the identified pattern ($y = 3x + 4$), it is the number-pair that is different from the rest.

The analysis confirms that 33 : 109 is the pair that is different.

Revision Table: Number Pair Patterns

Additional Information: Types of Number Patterns

Problems involving finding the odd one out based on number pairs often rely on recognizing common mathematical patterns. Here are a few types you might encounter:

• Linear Relationships: $y = ax + b$ (like in this problem)
• Quadratic Relationships: $y = ax^2 + bx + c$ or involving squares/cubes
• Multiplicative Relationships: $y = a \times x$
• Divisibility Rules: Both numbers share a common factor, or one is a multiple of the other.
• Sum/Difference/Product of Digits: Relationship based on the digits of the numbers.
• Prime/Composite Numbers: One number is prime, the other composite, or both belong to a specific set.
• Specific Number Properties: Perfect squares, perfect cubes, triangular numbers, etc.

Solving these problems requires systematic testing of potential rules and careful calculation.