A series of ordered pairs is given with one pair missing. Choose the correct alternative from the given options that will complete the series: (3, 5), (5, 7), ( ? ), (11, 13).

Introduction / Context:
This question uses ordered pairs of numbers arranged in a sequence. The task is to identify the pattern that connects consecutive pairs and then determine which missing pair fits that pattern. Here the pattern is based on prime numbers and their positions in the sequence, testing number sense and recognition of prime progressions.

Given Data / Assumptions:
Given pairs: (3, 5), (5, 7), ( ? ), (11, 13).We must find the missing third pair.We assume that the pairs follow a logical progression involving prime numbers.

Concept / Approach:
The key idea is to examine the first and second elements of each pair separately. We observe that all visible numbers 3, 5, 7, 11, and 13 are primes. The pairs appear to link consecutive primes in an overlapping way: the second element of one pair often becomes the first element of the next. By listing primes in order, we can map how the sequence of pairs is constructed and insert the missing pair accordingly.

Step-by-Step Solution:
Step 1: List primes in order: 2, 3, 5, 7, 11, 13, and so on.Step 2: Observe that (3, 5) contains the primes 3 and 5, and (5, 7) contains 5 and 7. These are consecutive primes: (3rd, 4th) and (4th, 5th) primes if we index from 2.Step 3: The last given pair (11, 13) contains 11 and 13, which are again consecutive primes later in the list.Step 4: To continue the overlapping pattern, after (5, 7) we expect a pair formed by the next two consecutive primes after 7, which are 7 and 11.Step 5: Therefore, the missing pair is (7, 11).

Verification / Alternative check:
Write the sequence of pairs using consecutive primes in a sliding window: (3, 5), (5, 7), (7, 11), (11, 13). Each pair consists of two consecutive primes and the second prime of one pair becomes the first of the next pair, except where the sequence jumps from 7 to 11 due to the absence of 9, which is composite. This structure is consistent and natural, confirming that (7, 11) is the correct missing pair.

Why Other Options Are Wrong:
Option (7, 9) introduces 9, which is not a prime number, breaking the prime-based pattern.Option (9, 11) again includes 9, a composite number, so it does not fit the strict prime sequence.Option (9, 13) contains 9 and also skips 11 in the second element, disrupting both primality and the idea of consecutive primes.

Common Pitfalls:
Some learners focus only on numerical differences between elements of each pair and overlook the key property that all valid numbers must be prime. Another common mistake is to treat 9 as a prime simply because it is odd. Always verify primality by checking divisibility, especially for smaller numbers like 9 which are frequently used as traps in exam questions.

Final Answer:
The missing ordered pair that completes the prime-based sequence is (7, 11).