The average of six numbers is $x$ and the average of three of these is $y$. If the average of the remaining three is $z$, then

Concept & Formula The total sum of a set of observations is equal to the product of the number of observations and their average value. $$\text{Sum} = \text{Average} \times \text{Number of observations}$$ Step-by-Step Solution * The total sum of all 6 numbers with average $x$ is: $$\text{Sum of 6 numbers} = 6 \cdot x$$ * The sum of the first group of 3 numbers with average $y$ is: $$\text{Sum of first 3 numbers} = 3 \cdot y$$ * The sum of the remaining 3 numbers with average $z$ is: $$\text{Sum of remaining 3 numbers} = 3 \cdot z$$ * Since the total sum is equal to the sum of both individual subgroups: $$6x = 3y + 3z$$ * Divide the entire equation by 3: $$2x = y + z$$ Exam Strategy & Shortcut You can instantly pick values to verify algebraic relations. Let all 6 numbers be equal to 10. Then their average $x = 10$. The average of any 3 numbers $y = 10$, and the remaining 3 numbers $z = 10$. Test the equations: (a) $10 = 10 + 10$ (False) (b) $2(10) = 10 + 10$ (True) This verification takes less than 5 seconds. Common Pitfall Students often intuitively assume averages add up directly like variables, leading them to falsely choose $x = y + z$. Always convert averages to total sums before combining or comparing groups. Final Answer Therefore, the correct answer is $2x = y + z$.