Understanding Number Pair Patterns

In this question, we are presented with four pairs of numbers. The core idea is that the second number in each pair is derived from the first number through a consistent mathematical operation or a sequence of operations. However, this consistency holds true for all pairs except one. Our task is to pinpoint this unique pair, the 'odd one out', which deviates from the common rule.

Analyzing the Number Pairs and Identifying the Pattern

Let's examine each number pair individually to uncover the relationship between the first and the second numbers.

• Pair 1: 7 ∶ 46
• We need to find a mathematical link between 7 and 46. Let's consider common operations like multiplication or powers.
• \(7 \times 6 = 42\)
• \(7 \times 7 = 49\)
• Neither multiplication directly results in 46. However, \(7^2 = 49\). Notice that \(49 - 3 = 46\). This suggests a potential pattern: the second number is the square of the first number minus 3, i.e., \(x \ratio; x^2 - 3\).

Let's test this potential pattern \(x^2 - 3\) with the other given pairs to see if it holds true.

• Pair 2: 13 ∶ 166
• Applying the pattern \(x^2 - 3\) to the first number, 13: \(13^2 - 3\).
• \(13^2 = 169\).
• \(169 - 3 = 166\). This result exactly matches the second number in the pair. So, this pair follows the \(x^2 - 3\) pattern.

• Pair 3: 9 ∶ 78
• Applying the pattern \(x^2 - 3\) to the first number, 9: \(9^2 - 3\).
• \(9^2 = 81\).
• \(81 - 3 = 78\). This result also perfectly matches the second number in the pair. This pair also follows the \(x^2 - 3\) pattern.

At this point, it's clear that the first three number pairs follow the same mathematical operation: the second number is obtained by squaring the first number and then subtracting 3.

• Pair 4: 48 ∶ 144
• Let's check if this pair follows the established pattern \(x^2 - 3\). Applying it to the first number, 48: \(48^2 - 3\).
• \(48^2 = 2304\).
• \(2304 - 3 = 2301\).
• The second number in the pair is given as 144, which is significantly different from 2301. Therefore, this number pair, 48 ∶ 144, does NOT follow the \(x^2 - 3\) pattern common to the other three pairs.

Identifying the Odd Number Pair Out

Since the pairs (7 ∶ 46), (13 ∶ 166), and (9 ∶ 78) all follow the rule \(x \ratio; x^2 - 3\), the pair (48 ∶ 144) which does not follow this rule is the odd number pair.

It's interesting to note the relationship in the odd pair: \(144 \div 48 = 3\). The second number is simply three times the first number (\(x \ratio; x \times 3\)). This confirms it follows a different rule.

Summary of Number Pair Pattern Analysis

The pair that deviates from the pattern observed in the others is 48 ∶ 144.

Revision Table: Understanding Odd Number Pairs

Additional Information: Solving Number Pattern Questions

Questions involving number pairs or series are common in logical reasoning and aptitude tests. To solve them effectively, consider the following approaches:

• Look for simple arithmetic operations: addition, subtraction, multiplication, or division.
• Consider squares, cubes, or other powers of the numbers.
• Look for combined operations (e.g., square and add/subtract, multiply and add/subtract).
• Check relationships between digits (sum of digits, product of digits).
• Test sequences based on prime numbers, Fibonacci series, etc., if the numbers are in a series format.
• Systematically test potential patterns starting with simpler ones. Once a pattern fits multiple examples, verify if it fits all except one.

Practice with different types of number pattern questions helps in quickly identifying the underlying logic.