Finding the Odd Number Pair with Mathematical Operations

The question asks us to identify the number pair where the relationship between the first and second number is different from the other pairs. A consistent mathematical operation is applied to the first number in each pair to get the second number, except for one pair.

We need to find the rule that connects the first number to the second number in the majority of the pairs. Let's examine the given options:

• Pair 1: 32 : 99
• Pair 2: 42 : 123
• Pair 3: 24 : 75
• Pair 4: 64 : 195

Let's try to find a pattern by applying simple mathematical operations (addition, subtraction, multiplication, division, or a combination) to the first number to obtain the second number. We are told operations should be performed on the whole number itself, not its individual digits.

Testing Potential Mathematical Rules

Let's consider common relationships, often involving multiplication and addition/subtraction:

• If we multiply the first number by a constant and add/subtract another constant, does it give the second number?

Let's test a common multiplier, like 3 or 4, often seen in such pattern questions.

Applying the Rule: Multiply by 3 and Add a Constant

Let's try multiplying the first number by 3 and see what needs to be added or subtracted to get the second number.

• Pair 1 (32 : 99): If we multiply 32 by 3, we get $32 \times 3 = 96$. To get 99 from 96, we need to add 3 ($96 + 3 = 99$). So, the rule could be $n \times 3 + 3$.
• Pair 2 (42 : 123): Let's apply the potential rule $n \times 3 + 3$. If we multiply 42 by 3, we get $42 \times 3 = 126$. Applying the rule, $126 + 3 = 129$. This is not 123. This pair might be the odd one out.
• Pair 3 (24 : 75): Let's apply the potential rule $n \times 3 + 3$. If we multiply 24 by 3, we get $24 \times 3 = 72$. Applying the rule, $72 + 3 = 75$. This matches 75.
• Pair 4 (64 : 195): Let's apply the potential rule $n \times 3 + 3$. If we multiply 64 by 3, we get $64 \times 3 = 192$. Applying the rule, $192 + 3 = 195$. This matches 195.

The mathematical operation that connects the first number to the second number in three of the four pairs is: Multiply the first number by 3 and then add 3. This can be expressed as: Second Number $= (\text{First Number} \times 3) + 3$.

Analyzing Each Number Pair Against the Rule

Let's summarize the results of applying the rule $(\text{First Number} \times 3) + 3$ to each pair:

As shown in the table, the pair 42 : 123 does not follow the rule $(\text{First Number} \times 3) + 3$. The calculation for 42 gives 129, but the given second number is 123. All other pairs follow this rule.

Therefore, the odd number pair is 42 : 123.

Conclusion: Identifying the Odd Number Pair

Based on the consistent mathematical operation ($n \times 3 + 3$) observed in three of the pairs, the pair that deviates from this pattern is 42 : 123. This makes it the odd number pair.

Revision Table: Number Pattern Analysis

Additional Information: Number Pattern Recognition Skills

Solving problems involving number pairs and sequences requires developing strong number pattern recognition skills. Here are some tips:

• Start by looking for simple arithmetic progressions (constant difference).
• Check for geometric progressions (constant ratio).
• Consider squares, cubes, or other powers of the numbers.
• Try combining operations, such as multiplying by a constant and then adding/subtracting a constant.
• Look at the difference between the numbers in each pair. Is there a pattern in these differences?
• Look at the ratio between the numbers in each pair. Is there a pattern in these ratios?
• Sometimes, the rule might involve the position of the number in a sequence, although that's not the case here.
• Always test a potential rule on all given examples to confirm its consistency before concluding it is the correct rule for the majority.