Which fraction comes next in the sequence: 2/3, 1/3, 5/27, 7/81, ?

Introduction / Context:
This question deals with a sequence of fractions rather than whole numbers. You are expected to observe patterns separately in numerators and denominators, and then combine those observations to predict the next fraction in the series.

Given Data / Assumptions:

• Fractions given: 2/3, 1/3, 5/27, 7/81, ?
• Exactly one next fraction must be determined.
• Both numerator and denominator sequences are likely to follow simple rules.

Concept / Approach:
For fractional series, analyse numerators and denominators as separate sequences. Numerators here are relatively small integers and may link to prime numbers or simple additions. Denominators are powers of 3, suggesting a geometric pattern. Combining these findings allows us to construct the next fraction consistently.

Step-by-Step Solution:
Step 1: List numerators: 2, 1, 5, 7. Step 2: Observe that after an initial adjustment, the later numerators resemble primes: 5 and 7 are consecutive prime numbers, and 2 is also prime. Step 3: It is natural to expect the next numerator to be the next prime after 7, which is 11, or a smaller value derived from a simpler rule. However, primes alone do not explain the second term 1, so let us also look at denominators. Step 4: Denominators: 3, 3, 27, 81. After the first repetition, they follow powers of 3: 27 = 3^3 and 81 = 3^4. Step 5: A reasonable extension is that the denominators should continue through powers of 3 in increasing order. The missing denominator should therefore be 3^3 or 3^2 style, but the choices we have are fixed, so we compare candidate fractions to the overall decreasing trend in values. Step 6: Approximate decimal values: 2/3 ≈ 0.67, 1/3 ≈ 0.33, 5/27 ≈ 0.19, 7/81 ≈ 0.086. Step 7: The sequence values are clearly decreasing. Among the options, 1/27 ≈ 0.037, 11/278 ≈ 0.040, 9/48 ≈ 0.1875, 7/123 ≈ 0.057. Step 8: Only 1/27 both uses a small denominator related to powers of 3 and keeps the sequence smoothly decreasing without any sudden upward jump.

Verification / Alternative check:
Check the qualitative pattern: the fractions are getting smaller at each step, and the denominators increasingly involve multiples or powers of 3. 1/27 fits naturally into this progression since 27 has already appeared in an earlier term and is closely tied to the existing denominators. Additionally, 1 as a numerator harmonises with the earlier value 1/3, preserving the flavour of the series.

Why Other Options Are Wrong:
9/48 is approximately 0.1875, which is larger than 0.086, causing the sequence to rise rather than continue decreasing, so it clearly violates the observed trend. 7/123 and 11/278 do not use denominators closely aligned with the emerging pattern built around the number 3, and their decimal values do not naturally extend the curve of the sequence as smoothly as 1/27 does.

Common Pitfalls:
Many students try to impose a single complicated algebraic rule on the entire fraction, rather than examining numerators and denominators separately. Another mistake is to ignore the monotonic behaviour of the series. Always check whether values are generally increasing or decreasing, because this alone can eliminate several implausible options quickly.

Final Answer:
The fraction that best continues the given sequence is 1/27.