In the number series 3240, 540, 108, 27, ?, 4.5, what value should replace the question mark so that the series follows a consistent pattern?

Introduction / Context:
This number series problem focuses on division by successively changing divisors. To solve such questions, it is important to examine how each term transforms into the next, especially when the numbers decrease significantly. The series 3240, 540, 108, 27, ?, 4.5 can be cracked by comparing ratios of consecutive terms.

Given Data / Assumptions:
- The series is 3240, 540, 108, 27, ?, 4.5. - There is one missing term between 27 and 4.5. - We assume a simple multiplicative or divisive pattern.

Concept / Approach:
- Compute the ratio of each term to the previous term. - If the ratios form a simple pattern, such as division by consecutive integers, extend that pattern to find the missing term. - Confirm that the same pattern works consistently through the entire series.

Step-by-Step Solution:
Step 1: Find the ratio from 3240 to 540. 3240 / 540 = 6, so 3240 has been divided by 6 to get 540. Step 2: Find the ratio from 540 to 108. 540 / 108 = 5, so 540 is divided by 5 to give 108. Step 3: Find the ratio from 108 to 27. 108 / 27 = 4, so 108 is divided by 4 to get 27. Step 4: We can see a pattern: the series divides successively by 6, 5, and 4. Step 5: Following this pattern, the next divisor should be 3, and after that 2. Step 6: Compute the unknown term after 27 by dividing 27 by 3. This gives 27 / 3 = 9. Step 7: Now check the final step: 9 divided by 2 should be 4.5, which matches the given last term.

Verification / Alternative check:
We can summarize the pattern as: 3240 / 6 = 540, 540 / 5 = 108, 108 / 4 = 27, 27 / 3 = 9, and 9 / 2 = 4.5. This shows a clear and consistent sequence of divisors: 6, 5, 4, 3, 2. Since the pattern works perfectly across all transitions, the missing term must be 9.

Why Other Options Are Wrong:
Option A (7): 27 / 3 is not 7, and 7 / 2 does not give 4.5, so it destroys the clean divisor pattern. Option B (10): Similarly, 27 / 3 is not 10 and 10 / 2 is 5, which does not match 4.5. Option D (8): 27 / 3 is 9, not 8, and 8 / 2 is 4, not 4.5. Option E (6): This would lead to 6 / 2 = 3 as the last term, which is inconsistent with the given 4.5.

Common Pitfalls:
- Looking at differences instead of ratios in a sequence where numbers shrink quickly. - Assuming a random or non systematic pattern without testing a simple descending divisor pattern. - Forgetting to verify that the pattern holds both before and after the missing term.

Final Answer:
The missing term that keeps the descending divisor pattern intact is 9.