In the series 3, 5/3, 1/3, ?, -7/3, -11/3, which fraction should replace the question mark to maintain the pattern?

Introduction / Context:
This problem examines your ability to work with fractions in a number series. Even though the values look complicated at first glance, they actually follow a simple arithmetic pattern when expressed with a common denominator. Recognising that structure is the key to solving the question quickly and accurately.

Given Data / Assumptions:
- The given series is: 3, 5/3, 1/3, ?, -7/3, -11/3.- All terms can be expressed with the same denominator to make comparison easy.- We assume a constant difference pattern, as is common in such questions.

Concept / Approach:
Whenever a series involves fractions, a good technique is to rewrite all terms using a common denominator. This converts fractional terms into numerators over that denominator, making it easier to observe any arithmetic progression. After finding the constant difference in this transformed form, we can return to the original fraction representation for the missing term.

Step-by-Step Solution:
- Express each term with denominator 3: 3 = 9/3, 5/3 = 5/3, 1/3 = 1/3.- So the series becomes: 9/3, 5/3, 1/3, ?, -7/3, -11/3.- Find the differences: (5/3 - 9/3) = -4/3, (1/3 - 5/3) = -4/3.- This suggests a constant difference of -4/3 between consecutive terms.- Apply the same difference from 1/3 to get the missing term: 1/3 - 4/3 = -3/3 = -1.- Continue to check: -1 - 4/3 = -3/3 - 4/3 = -7/3 and then -7/3 - 4/3 = -11/3.

Verification / Alternative check:
- The transformed series of numerators over 3 is: 9, 5, 1, -3, -7, -11.- This is an arithmetic progression with common difference -4.- Rewriting -3 as -3/3 confirms that the missing term is -1 in standard form.

Why Other Options Are Wrong:
- -2/3 and -4/3: These values do not maintain a constant difference of -4/3 when placed between 1/3 and -7/3.- -2: This corresponds to -6/3, which would break the arithmetic progression in the numerator sequence.- Only -1 gives a consistent arithmetic progression for all given terms in the series.

Common Pitfalls:
A typical mistake is to treat each fraction separately without converting to a common denominator, which hides the simple linear pattern. Another error is careless subtraction of negative fractions, leading to wrong intermediate values. Always standardise the denominators and compute differences carefully for fraction series questions.

Final Answer:
The fraction that correctly completes the series is -1, so the correct option is -1.