In the following number analogy, 16 is related to 50 in a particular way. Using the same pattern, which number should replace the question mark in 16 : 50 :: 24 : ?

Introduction / Context:
This is a number analogy problem where the relation between 16 and 50 must be discovered and then applied to 24 to obtain the missing number. Such questions typically use simple arithmetic operations like multiplication and addition, and they test your ability to detect and extend numeric patterns quickly and accurately.

Given Data / Assumptions:
We are given the pair 16 : 50. We must find the number that completes 16 : 50 :: 24 : ?. All values are positive integers. The same arithmetic transformation must apply in both pairs.

Concept / Approach:
The first step is to look for a direct arithmetic relation between 16 and 50. We may test multiplication by a constant followed by the addition or subtraction of a small number. Once a simple and consistent operation is identified for the first pair, we use that identical rule for 24. The pattern should be one that is natural for exam questions and does not require complicated formulas.

Step-by-Step Solution:
Step 1: Check possible multipliers. 16 * 2 = 32, which is too small. 16 * 3 = 48, which is close to 50. Step 2: Bridge from 48 to 50. We notice that 48 + 2 = 50. So, 50 can be written as 16 * 3 + 2. Step 3: Treat this as the rule. Transformation rule: multiply the given number by 3 and then add 2. Step 4: Apply the same rule to 24. 24 * 3 = 72. 72 + 2 = 74. Therefore, the required number is 74.

Verification / Alternative check:
To verify, we can restate the pattern as: output = 3 * input + 2. For the first pair, 3 * 16 + 2 = 48 + 2 = 50, which is correct. For the second pair, 3 * 24 + 2 = 72 + 2 = 74, which matches our derived value. No other option can be expressed using exactly the same operation applied to 24, so the pattern is consistent and unique.

Why Other Options Are Wrong:
75 is 24 * 3 + 3, so it fails to follow the +2 part of the pattern. 70 would require subtracting 2 after tripling 24, which contradicts the rule established by 16 and 50. 79 does not correspond to 24 multiplied by any simple integer followed by a constant addition in the same way as the first pair. 68 is less than 3 * 24 and does not fit the add 2 rule either.

Common Pitfalls:
A common mistake is to jump directly to guesswork with squares, cubes, or digit sums when a simple linear relation is present. Another pitfall is to assume that the same difference will always be added, ignoring the possibility of multiplication first. The safest strategy is to test small integer multipliers and then check whether adding or subtracting a small constant yields the second number. Once the right operation is found, apply it consistently to the second pair without modification.

Final Answer:
The number that correctly completes the analogy is 74.