Correct Option

The problem provides two key pieces of information:

The average of three numbers p, q, and r is k. This can be mathematically represented as:
(p + q + r) / 3 = k
Which implies: p + q + r = 3k (Equation 1)
p is as much more than the average (k) as q is less than the average (k). This condition can be written as:
p - k = k - q
Rearranging this equation to isolate p + q:
p + q = k + k
p + q = 2k (Equation 2)

Now, substitute the value of (p + q) from Equation 2 into Equation 1:

(2k) + r = 3k

To find the value of r, subtract 2k from both sides of the equation:

r = 3k - 2k

r = k

Therefore, the value of r is k.

Incorrect Options

Options 2 (k - 1), 3 (k + 1), and 4 (k/2) are incorrect because they contradict the derived value of r. Based on the given conditions that the average of p, q, and r is k, and that p and q deviate equally from k (one above, one below), the only value for r that maintains the average at k is k itself. Any other value for r would alter the sum p + q + r such that the average would no longer be k, or it would violate the relationship between p, q, and k.