The average of six numbers is x and the average of three of these numbers is y. If the average of the remaining three numbers is z, which of the following relations between x, y and z is correct?

Introduction:
This question tests your understanding of how averages of subsets relate to the average of the entire set. We are given the average of all six numbers and the averages of two groups of three numbers each. From this, we can derive a formula connecting x, y and z.

Given Data / Assumptions:
- There are six numbers in total. - Average of all six numbers = x. - Average of three of these numbers = y. - Average of the remaining three numbers = z. - We must find the correct relation among x, y and z.

Concept / Approach:
Average is defined as total sum divided by number of items. The total sum of all six numbers can be expressed in two different ways: using x, and using y and z. Equating these expressions allows us to derive a relation between x, y and z. This is a classic weighted average situation where each subgroup has the same size.

Step-by-Step Solution:
Step 1: Let the six numbers be a1, a2, a3, a4, a5 and a6. Step 2: Average of all six numbers is x, so total sum S = 6x. Step 3: Suppose the first group of three numbers has average y. Then the sum of those three numbers is 3y. Step 4: The remaining three numbers have average z, so their sum is 3z. Step 5: The total sum S can also be written as sum of first three numbers plus sum of last three numbers, that is S = 3y + 3z. Step 6: From Step 2 and Step 5, we have 6x = 3y + 3z. Step 7: Divide both sides by 3: 2x = y + z. Step 8: Therefore, the correct relation is 2x = y + z.

Verification / Alternative Check:
Consider an example where the three numbers in the first group are 2, 4 and 6 (average y = 4) and the three numbers in the second group are 8, 10 and 12 (average z = 10). The overall average x is (2 + 4 + 6 + 8 + 10 + 12) / 6 = 42 / 6 = 7. Now compute y + z and 2x. We have y + z = 4 + 10 = 14 and 2x = 2 * 7 = 14. Hence 2x = y + z holds for this example.

Why Other Options Are Wrong:
Option x = y + z is incorrect because it ignores the fact that each of y and z is an average, not a total. Option 3x = 2y - 2z does not follow from any correct manipulation of the sums. Option x = (y + z) / 2 would only be true if the subgroup sizes were equal and x was the average of y and z directly, but here we have already used the sizes in the derivation and obtained 2x = y + z instead. Option 3x = y + z also fails dimensional reasoning, since multiplying x by 3 changes the scale incorrectly.

Common Pitfalls:
Some learners forget that averages must be multiplied by the count to recover total sums. Others confuse the expression for the total and try to combine x, y and z directly without using the number of items in each group. Always go back to total sums, equate them and then simplify.

Final Answer:
The correct relationship between x, y and z is 2x = y + z.