Among the pairs (16, 32), (20, 40), (64, 81) and (81, 162), identify the odd pair based on the relationship between the two numbers.

Introduction / Context:
This question presents four ordered pairs of numbers and asks us to find the odd one out. A natural idea is to check whether there is a simple multiplicative relationship between the first and second numbers in each pair. In many aptitude tests, the most common relationship is that the second number is double the first number. If three pairs follow this rule and one does not, the pair that breaks the rule is the odd one out.

Given Data / Assumptions:
The given pairs are (16, 32), (20, 40), (64, 81) and (81, 162). We test whether the second number equals 2 times the first number. All numbers are positive integers.

Concept / Approach:
The core concept is the doubling relationship. If in a pair (a, b) we have b = 2 * a, then the second number is exactly twice the first. We compute this relationship for each pair. If three pairs satisfy b = 2 * a and one pair does not, then the one pair that fails the condition is the odd one out. This is a direct and easily tested pattern and is typically what exam setters intend when they give such simple number pairs.

Step-by-Step Solution:
Step 1: For (16, 32), 2 * 16 = 32, so the second number is double the first. Step 2: For (20, 40), 2 * 20 = 40, so this pair also satisfies the doubling rule. Step 3: For (81, 162), 2 * 81 = 162, so this pair again follows the same rule. Step 4: For (64, 81), 2 * 64 = 128, which is not equal to 81. Thus, this pair does not obey the doubling relationship. Step 5: Therefore, (64, 81) is the only pair that does not have the second number equal to twice the first number and is the odd one out.

Verification / Alternative check:
An alternative way to think is in terms of ratios. For the first, second and fourth pairs, the ratio b / a equals 2, while for (64, 81), the ratio 81 / 64 is greater than 1 but not equal to 2. This confirms that the multiplicative pattern is broken only in the third pair. No other simple relationship such as difference or sum works so neatly for three pairs and fails for only one, so doubling is clearly the intended logic.

Why Other Options Are Wrong:
(16, 32) is not the odd one out because 32 is exactly double 16 and fits the main pattern. (20, 40) is not the odd one out because 40 is exactly double 20, again consistent with the pattern. (81, 162) is not the odd one out because 162 is exactly double 81 and follows the same rule. Only (64, 81) fails the condition second equals twice the first, which makes it different.

Common Pitfalls:
Some candidates may be distracted by the fact that 64 and 81 are themselves perfect squares (8^2 and 9^2) and look for a square based pattern, but the exam setter here is focusing on the relation between the two members of each pair. Others may try to match differences between numbers instead of ratios, which does not reveal as clean a pattern in this set. When you see number pairs where three are simple multiples and one is not, always test for the multiplicative relationship first.

Final Answer:
The pair in which the second number is not double the first and is therefore the odd one out is (64, 81).