The average of three numbers p, q and r is k. p is as much more than the average as q is less than the average. What is the value of r?

The correct option is k.

Step-by-Step Explanation:

Step 1: Express the given average mathematically
We are given that the average of three numbers, p, q, and r, is k.

$\frac{p+q+r}{3}=k$

Multiplying both sides by 3 gives the sum of the three numbers:

$p+q+r=3k$

Step 2: Use the relationship between p, q, and the average (k)
The question states that "p is as much more than the average as q is less than the average."

This means the difference between p and the average k is equal to the difference between the average k and q:

$p-k=k-q$

Rearranging this equation by moving q to the left side and k to the right side:

$p+q=2k$

Step 3: Substitute (p + q) back into the sum equation
Now, substitute p + q = 2k into the sum of three numbers equation from Step 1:

$(p+q)+r=3k$

$2k+r=3k$

Step 4: Solve for r
Subtract 2k from both sides:

$r=3k-2k$

$r=k$

Thus, the value of r is k.