In the following question, select the related number pair from the given alternatives: 31 is to 42 as which of the following pairs?

Introduction / Context:
This is a simple number pair analogy. The pair 31 : 42 is given, and you must select the option in which the two numbers are related in the same way. A common and natural pattern in such questions is a fixed difference between the numbers. Here, the difference between 31 and 42 is constant and suggests the same difference should hold in the correct answer pair.

Given Data / Assumptions:

• The base pair is 31 on the left and 42 on the right.
• The candidate pairs are 53 : 64, 47 : 59, 61 : 73, and 44 : 34.
• We assume the relationship is likely to be arithmetic, such as adding a constant number.
• We need to find the pair where the difference between second and first number equals the difference in the base pair.

Concept / Approach:
First compute the difference between 31 and 42. Then compute the difference for each candidate pair. The correct option will have the same difference, indicating the same rule second = first + 11, or a similar consistent relation. This straightforward approach is efficient and aligns with typical patterns used in exam questions.

Step-by-Step Solution:
Step 1: Compute the difference in the base pair: 42 - 31 = 11. Step 2: Check option a, 53 : 64. Compute 64 - 53 = 11. Step 3: Check option b, 47 : 59. Compute 59 - 47 = 12, not 11. Step 4: Check option c, 61 : 73. Compute 73 - 61 = 12, again not 11. Step 5: Check option d, 44 : 34. Compute 34 - 44 = -10, which is not 11 and also reverses the order. Step 6: Conclude that only 53 : 64 preserves the same difference of 11 as in the base pair 31 : 42.

Verification / Alternative check:
If we tried to impose a more complex rule, such as multiplying by a factor or combining digits, we would not match both the example pair and one of the options as neatly. The pair 31 and 42 do not share any special digit manipulation pattern, but the constant difference of 11 is obvious. Applying 11 to 53 gives 64 exactly, making 53 : 64 a natural parallel. The other pairs do not reproduce this difference, so the constant difference interpretation is well supported.

Why Other Options Are Wrong:
The pair 47 : 59 differs by 12, not 11. The pair 61 : 73 also differs by 12. The pair 44 : 34 actually decreases by 10, which contradicts both the magnitude and direction of the change in the base pair. Since analogy demands a uniform rule, these pairs fail to follow the same pattern as 31 : 42 and cannot be correct.

Common Pitfalls:
Test takers sometimes overcomplicate patterns when a simple difference is sufficient. Another pitfall is to miscalculate differences, especially when working quickly. Carefully subtracting the numbers in each pair and comparing the results avoids these errors and leads directly to the correct answer.

Final Answer:
The pair that mirrors the +11 difference in 31 : 42 is 53 : 64.