A number series involving fractions and decimals is given with one term missing. Observe the pattern in 1.5, 9/4, 3, ?, 4.5, 21/4 and choose the correct alternative that completes the series.

Introduction / Context:
This question presents a number series that mixes fractions and decimals. The sequence is 1.5, 9/4, 3, ?, 4.5, 21/4. The goal is to identify the underlying pattern and determine the missing term. Mixed-format series like this check whether a candidate can comfortably move between decimal and fractional representations while still spotting a simple arithmetic relationship.

Given Data / Assumptions:
Given series: 1.5, 9/4, 3, ?, 4.5, 21/4.
We can freely convert between decimals and fractions for analysis.
There is one consistent pattern governing the entire sequence.
Exactly one option will fit the pattern for the missing position.

Concept / Approach:
A good strategy is to convert all terms to either pure decimals or pure fractions. Doing so often reveals a straightforward arithmetic progression. Converting the given numbers to decimals: 1.5, 2.25, 3.0, ?, 4.5, 5.25. This suggests that each term might be increasing by a constant difference. Checking quickly shows that the difference between successive terms is 0.75, which is equal to 3/4. Therefore the series is an arithmetic progression with common difference 0.75.

Step-by-Step Solution:
Convert 1.5 to decimal: 1.5. 9/4 as decimal is 2.25. 3 is already 3.00. Difference from 1.5 to 2.25 is 0.75, and from 2.25 to 3.00 is again 0.75. Add 0.75 to 3.00 to get the missing term: 3.00 + 0.75 = 3.75. 3.75 expressed as a fraction is 15/4. Next: 3.75 + 0.75 = 4.50, and 4.50 + 0.75 = 5.25, which matches 4.5 and 21/4. So the missing term must be 15/4.

Verification / Alternative check:
We verify the arithmetic progression: each step increases by 0.75. All known terms fit this rule perfectly when expressed as decimals. In fractional form, the common difference is 3/4. Converting 15/4 gives 3.75, which keeps the progression intact between 3 and 4.5. This shows that the found value is consistent with both the earlier and later terms in the sequence.

Why Other Options Are Wrong:
Option B: 7/2 is 3.5, which would use a step of only 0.5 from 3 and then a different step to reach 4.5, breaking the constant difference rule.
Option C: 8 is far away from 3 and would destroy the smooth progression of small increments.
Option D: 9 is even more unsuitable and does not reflect the gentle growth of the series.
Option E: 13/4 gives 3.25, which does not maintain the 0.75 step size either.

Common Pitfalls:
One common mistake is to avoid converting fractions to decimals or vice versa and instead try to see a complicated pattern. Another pitfall is to assume that mixed-format series are always advanced, while here the relationship is just a simple arithmetic progression. Candidates should also be careful with fractional conversions, since calculation errors can lead to incorrect conclusions about the common difference.

Final Answer:
The missing term that preserves a constant increment of 0.75 is 15/4, so the correct option is 15/4.