P, Q, R, S, T and U are six consecutive odd numbers whose average is 52. What is the product of the smallest and the largest numbers, that is, P and U?

Introduction / Context:
This question combines the idea of averages with the pattern of consecutive odd numbers. Problems of this type appear frequently in quantitative aptitude tests because they check whether you understand how consecutive numbers are arranged around their average and how to use symmetry to compute unknown values quickly without listing everything blindly.

Given Data / Assumptions:

There are six consecutive odd numbers P, Q, R, S, T and U.
The average of these six numbers is 52.
The numbers are in increasing order, so P is the smallest and U is the largest.
We need the product P * U.

Concept / Approach:
For consecutive numbers, the average lies at the centre of the sequence. For an even count of numbers, such as six, the average is exactly the mean of the third and fourth terms. Because the numbers are consecutive odds, they are symmetrically placed around the average. Once we know the sequence, we can easily identify the smallest and largest terms and then calculate their product.

Step-by-Step Solution:
Step 1: Understand the structure of six consecutive odd numbers. For six consecutive odd numbers with average 52, the numbers are equally spaced around 52. Step 2: Express the numbers using 52 as the centre. Because they are odd and equally spaced, the numbers will be 52 - 5, 52 - 3, 52 - 1, 52 + 1, 52 + 3, 52 + 5. So the actual sequence is 47, 49, 51, 53, 55 and 57. Step 3: Identify P and U. P is the smallest number, so P = 47. U is the largest number, so U = 57. Step 4: Compute the product of P and U. P * U = 47 * 57. Compute 47 * 57 = (50 - 3) * (50 + 7) = 50 * 50 + 50 * 7 - 3 * 50 - 3 * 7 = 2500 + 350 - 150 - 21 = 2679.

Verification / Alternative Check:
We can also compute 47 * 57 directly with standard multiplication and confirm that it equals 2679. Additionally, we can verify that the average of the six numbers (47, 49, 51, 53, 55, 57) is indeed 52 by adding them and dividing by 6. Their sum is 312, and 312 / 6 = 52, which matches the given condition.

Why Other Options Are Wrong:
Option 2212 corresponds to an incorrect pair of numbers or miscalculation of the product.
Option 2154 may result from using incorrect spacing for the odd numbers or taking the wrong smallest and largest values.
Option 2349 is another distractor that does not match the correct computation 47 * 57.

Common Pitfalls:
A common mistake is to assume that the average itself is one of the numbers when the count is even, without checking the structure. Another pitfall is to use consecutive even numbers instead of odd numbers or to miscalculate the spacing around the average. Errors in multiplication while finding the product can also lead to wrong answers even when the sequence has been identified correctly.

Final Answer:
The product of P and U is 2679.