Understanding the Number Pair Problem

This question asks us to find the odd number pair among the given options. We are told that in most of the pairs, the second number is obtained by applying the same mathematical operation(s) on the first number. Our task is to identify this common operation and then find the pair that does not follow it.

Analyzing the Given Number Pairs

Let's look at the four number pairs provided:

• (21, 238)
• (17, 194)
• (13, 150)
• (23, 270)

We need to discover a rule or a mathematical pattern that connects the first number (let's call it $x$) to the second number (let's call it $y$) for most of these pairs. Once we find this pattern, we can test each pair to see which one breaks the rule.

Identifying the Mathematical Operation Pattern

Let's try to find a relationship between $x$ and $y$ for the first few pairs. Often, such patterns involve multiplication and addition or subtraction.

• For (21, 238): The second number (238) is roughly 11 times the first number (21). Let's check: $21 \times 11 = 231$. The difference is $238 - 231 = 7$. So, maybe the pattern is $y = 11x + 7$.

Let's test this potential pattern, $y = 11x + 7$, on the other pairs:

• For (17, 194): If $x = 17$, according to the pattern $y = 11 \times 17 + 7 = 187 + 7 = 194$. This matches the given pair (17, 194).
• For (13, 150): If $x = 13$, according to the pattern $y = 11 \times 13 + 7 = 143 + 7 = 150$. This also matches the given pair (13, 150).

The pattern $y = 11x + 7$ holds true for the first three number pairs. Now, let's test the last pair.

Checking the Last Number Pair

Let's test the pair (23, 270) against the pattern $y = 11x + 7$:

• For (23, 270): If $x = 23$, according to the pattern $y = 11 \times 23 + 7 = 253 + 7 = 260$.

The pattern suggests that for the first number 23, the second number should be 260. However, the given second number in the pair is 270. This means the pair (23, 270) does not follow the established mathematical operation.

Identifying the Odd Number Pair

Based on our analysis, three of the pairs follow the rule $y = 11x + 7$, but the pair (23, 270) does not.

We can summarize the checks in a table:

As the table clearly shows, the pair (23, 270) is the odd one out.

Conclusion

The mathematical operation followed by three of the number pairs is to multiply the first number by 11 and add 7 to get the second number ($y = 11x + 7$). The pair (23, 270) does not follow this rule, as $11 \times 23 + 7 = 260$, not 270.

Therefore, the odd number pair is (23, 270).

Revision Table for Number Pairs

Let's quickly review the key steps for solving such problems:

Additional Information on Pattern Recognition

Pattern recognition in number series and pairs is a common type of question in reasoning and aptitude tests. These questions assess your ability to identify logical or mathematical rules quickly. The patterns can be varied, including:

• Arithmetic progression (constant difference)
• Geometric progression (constant ratio)
• Combinations of arithmetic and geometric operations (like $ax + b$)
• Operations involving squares, cubes, or other powers
• Patterns based on digit properties (sum of digits, product of digits)
• Alternating patterns

Solving more problems helps you become familiar with common types of patterns and strategies for identifying them efficiently. Always start with simple relationships and gradually move to more complex ones if needed.