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

Introduction / Context:
This question is about the relationship between the overall average of a set of numbers and the averages of two equal sized subgroups within that set. It is a simple algebraic problem that tests whether you can convert verbal statements about averages into equations using sums and then derive the correct relationship between x, y and z.

Given Data / Assumptions:

• There are 6 numbers in total.
• The average of all 6 numbers is x.
• The average of 3 of these numbers is y.
• The average of the remaining 3 numbers is z.
• We must find the correct equation relating x, y and z.

Concept / Approach:
Average is defined as total sum divided by number of items. If the average of 6 numbers is x, then their total sum is 6x. If the average of one group of 3 numbers is y, then the sum of those 3 numbers is 3y. Similarly, the sum of the remaining 3 numbers is 3z. Since the 6 numbers consist exactly of these two subgroups, the total sum 6x must equal 3y + 3z. We then simplify this equation to get a direct relation between x, y and z.

Step-by-Step Solution:
Step 1: Let S be the total sum of all 6 numbers. Step 2: Given that the average of 6 numbers is x, we have S = 6x. Step 3: Sum of the first group of 3 numbers = 3y because their average is y. Step 4: Sum of the second group of 3 numbers = 3z because their average is z. Step 5: These two groups together make up all 6 numbers, so S = 3y + 3z. Step 6: Equate the two expressions for S: 6x = 3y + 3z. Step 7: Divide both sides by 3 to simplify: 2x = y + z. Step 8: Therefore, the correct relationship is 2x = y + z.

Verification / Alternative check:
Take a numerical example. Suppose the three numbers in the first group are 1, 2 and 3, and the three numbers in the second group are 7, 8 and 9. Then y = (1 + 2 + 3) / 3 = 2 and z = (7 + 8 + 9) / 3 = 8. The overall average x is (1 + 2 + 3 + 7 + 8 + 9) / 6 = 30 / 6 = 5. Now compute y + z = 2 + 8 = 10, and 2x = 2 * 5 = 10. Since 2x equals y + z for this example, this supports the derived relationship.

Why Other Options Are Wrong:
The equation x = y + z would suggest the overall average is larger than each subgroup average, which conflicts with the numeric example and the derived formula. The expression 3x = 2y - 2z has the wrong structure and can be rejected by substitution into a simple numeric case. The option x = (y + z) / 3 treats the overall average as a third of the average of the subgroup averages, which makes no sense dimensionally. The option 'none of these' is incorrect because we have found a specific correct relationship.

Common Pitfalls:
Some students mistakenly average y and z directly, writing x = (y + z) / 2, forgetting that x is the average over all 6 numbers, not just a simple mean of the subgroup averages. Others confuse sum and average and write equations like 6x = y + z, which ignore the fact that y and z are averages, not totals. Always remember that sum = average * number of items, and ensure that you use sums when combining groups.

Final Answer:
The correct relationship between the averages is 2x = y + z.