In this number analogy, “66 is to 36 as 99 is to ______”. Select the number that completes the analogy by following the same pattern of taking the product of the digits.

Introduction / Context:
This analogy links two digit numbers to other numbers using a digit based operation. The pair 66 : 36 suggests that the transformation may be related to the digits themselves rather than the whole number only. One natural idea is to multiply the digits of the two digit number and see if that gives the second number. After identifying the rule in the first pair, we apply the same to 99 and select the correct result from the options. This type of question is common in mental ability tests.

Given Data / Assumptions:

• First pair: 66 and 36. • Second pair: 99 and an unknown number. • Options: 18, 27, 81, 108. • We consider operations such as product or square of digits for two digit numbers.

Concept / Approach:
For a two digit number like 66, the digits are 6 and 6. The product 6 * 6 is 36, which exactly matches the second number. This gives a clear rule: the second number is the product of the two digits of the first number. We then apply the same idea to 99, whose digits are 9 and 9. The product 9 * 9 gives the related number. We finally match the result to the options.

Step-by-Step Solution:
Step 1: Analyse the first pair. The digits of 66 are 6 and 6. Compute the product: 6 * 6 = 36. So the second number 36 is equal to the product of the digits of 66. Step 2: Apply the same rule to 99. Digits of 99 are 9 and 9. Compute the product: 9 * 9 = 81. Step 3: Compare with options. Among 18, 27, 81, and 108, only 81 is the product of 9 and 9.

Verification / Alternative check:
We can also check other plausible operations. For instance, the sum of digits of 66 is 6 + 6 = 12, not 36. The square of the number 66 is much larger than 36. Therefore, the only simple rule that fits 66 : 36 is digit product. Applying the same logic to 99 gives 81, which fits neatly with the pattern. This coherence across both pairs confirms the rule.

Why Other Options Are Wrong:
• 18: This is 2 * 9, not related to the digits of 99 directly. • 27: Could be 3³, but it does not arise from 9 and 9 as digits in the given pattern. • 108: This is 12 * 9 or 9 * 12, but no such operation is indicated by the first pair.

Common Pitfalls:
Many candidates try to connect 66 and 36 by operations on the whole number, such as subtracting 30. However, this does not give a clear way to handle 99. When two digit numbers have repeated digits, you should always consider operations involving those digits, especially multiplication, since repeated digit products often lead to familiar squares like 36 and 81.

Final Answer:
The number that correctly completes the analogy is 81.