In the number analogy "12 : 72 :: 14 : ____", which number correctly completes the pattern using a square and division rule?

Introduction / Context:
This numerical analogy asks you to identify the mathematical operation that converts the first number of a pair into the second number. In the example "12 : 72", the number 72 is obtained from 12 by some rule. You must discover that rule and apply it to 14 to determine the missing number. Such questions often involve squaring, cubing, or combining operations like multiplication and division.

Given Data / Assumptions:

• First pair: 12 is related to 72.
• Second pair: 14 is related to an unknown number.
• Options: 98, 91, 42, 60, 84.
• We look for a clean arithmetic rule that works for both pairs.

Concept / Approach:
A helpful strategy is to try operations involving the square of the number. 12^2 equals 144. If we then divide this by 2, we get 144 / 2 = 72, which matches the second number in the first pair. So the rule appears to be: second number equals (first number squared) divided by 2. Applying this to 14, we get 14^2 = 196, and 196 / 2 = 98. Therefore, 98 is the number that completes the analogy correctly.

Step-by-Step Solution:

Step 1: Square 12 to see if that is related to 72. Compute 12^2 = 144. Step 2: Try dividing 144 by 2. 144 / 2 = 72, which matches the given second number. Step 3: Conclude that the transformation is "square the number and then divide by 2". Step 4: Apply this rule to 14. First compute 14^2 = 196. Step 5: Divide 196 by 2 to obtain 196 / 2 = 98. Step 6: Choose 98 from the options as the correct corresponding number for 14.

Verification / Alternative check:
Check that none of the other options can be reached from 14 by the same rule. If we try 14 * 7, we get 98 again, but that does not explain the original pair. 91 would be 14 * 6.5, and 42 would be 14 * 3, neither of which match the squared and halved pattern. 60 and 84 also cannot be obtained from 14 using the "square and divide by 2" rule. Thus, 98 is consistent with the pattern and the only valid choice.

Why Other Options Are Wrong:
All other numbers in the options require a different, more complicated, or arbitrary rule when starting from 14. Because analogies demand one simple rule that fits every pair involved, any option that needs a new rule is automatically incorrect. Only 98 satisfies the same transformation that takes 12 to 72.

Common Pitfalls:
A common error is to try direct multiplication factors first, such as 12 * 6, and become stuck when the same factor does not work with 14. Another pitfall is to ignore squaring because it seems too large at first glance. In many exam questions, squaring and then scaling up or down is exactly the intended pattern, so it should always be checked early.

Final Answer:
Using the rule "square the number and divide by 2", 12 maps to 72 and 14 maps to 98, completing the analogy as "12 : 72 :: 14 : 98".