In this number analogy question, select the related number from the given alternatives: 42 : 20 :: 64 : ?

Introduction / Context:
This number analogy question checks your ability to detect a consistent arithmetic operation linking one number to another. The pair 42 and 20 is provided, and you must apply the same rule to 64 to find the correct related number. Identifying patterns such as halving, dividing, and small adjustments is an essential skill for quantitative reasoning sections.

Given Data / Assumptions:
- The first pair is 42 and 20, where 20 is generated from 42 by a certain arithmetic rule.
- The second starting value is 64 and must be processed by the same rule.
- The answer choices are 31, 32, 33, and 34, which are all close enough to require careful checking.
- We assume standard integer arithmetic without tricks like digit reversal unless clearly suggested.

Concept / Approach:
The approach is to test simple, natural operations on 42 and see which one leads to 20. A good starting point is to try halving or dividing by small numbers and then adding or subtracting a constant. Once a suitable rule is found, that same formula is applied to 64. The pattern should be simple, consistent, and easy to explain, because exam setters typically avoid overly complicated constructions for analogies.

Step-by-Step Solution:
Step 1: Try halving 42. We get 42 / 2 = 21, which is close to 20 but not equal.Step 2: To adjust 21 to 20, subtract 1. This suggests the rule result = (number / 2) - 1.Step 3: Check this rule thoroughly for the first pair: (42 / 2) - 1 = 21 - 1 = 20, which matches the given second number.Step 4: Now apply the same rule to 64: (64 / 2) - 1 = 32 - 1 = 31.Step 5: Among the options, 31 appears and therefore is the only number consistent with the discovered pattern.

Verification / Alternative check:
Test whether any other simple formula could plausibly generate 20 from 42 and then give a different value for 64 that also appears among the options. For example, adding or subtracting a fixed number directly does not work: 42 - 22 = 20, but then 64 - 22 = 42, which is not an option. Similarly, multiplying by a fraction like 0 point 5 and then adding or subtracting large numbers is less natural. The formula (number / 2) - 1 is simple, elegant, and consistent, so it is the most reasonable pattern.

Why Other Options Are Wrong:
- 32: This would come from 64 / 2 with no adjustment, which does not fit the given pair because 42 / 2 equals 21, not 20.
- 33: There is no simple symmetric rule that transforms 42 to 20 and 64 to 33 in one clean step, so this value is inconsistent.
- 34: Similar to 33, selecting 34 would require an artificial rule that is not supported by the first pair.

Common Pitfalls:
Many candidates guess that the pattern is simply halving and therefore choose 32 directly, overlooking the small subtract one adjustment that is necessary for 42 to become 20. Others may try more complicated combinations and become confused. To avoid this, always verify that the same operation works smoothly on both numbers. Focusing on simple divide then add or subtract patterns keeps your reasoning clear and reduces errors.

Final Answer:
Using the consistent pattern result = (number / 2) - 1, 42 maps to 20 and 64 maps to 31, so 31 is the correct answer.