If the mean of x, y and z is K, then the mean of x/y, 1 and z/y is:

Concept: The arithmetic mean of a list of quantities is their sum divided by how many quantities there are. If every term of a list is divided by the same non-zero constant, the mean of the new list equals the original mean divided by that same constant.

Application

1. The mean of x, y and z is K, so (x + y + z)/3 = K, which gives x + y + z = 3K.

2. Rewrite the middle term of the new list as 1 = y/y, so the list x/y, 1, z/y is the same as x/y, y/y, z/y — each of the three original quantities divided by y.

3. Mean of the new list = (x/y + y/y + z/y)/3 = (x + y + z)/(3y).

4. Substitute x + y + z = 3K to get (3K)/(3y) = K/y.

Cross-check: Take x = 1, y = 2, z = 3. Then K = (1 + 2 + 3)/3 = 2. The new list is 1/2, 1, 3/2, whose sum is 3 and whose mean is 1. And K/y = 2/2 = 1, which agrees.

Hence the required mean is K/y.