The average of 2, 7, 6 and $x$ is 5 and the average of 18, 1, 6, $x$ and $y$ is 10. What is the value of $y$?

### Concept & Logic The formula for the average is the total sum of observations divided by the number of observations. By solving the first equation for the unknown variable, we can substitute it into the second equation to find the final target value. $$Average = \frac{Sum\ of\ Observations}{Total\ Number\ of\ Observations}$$ ### Step-by-Step Solution * **Given:** The average of 2, 7, 6, and $x$ is 5. * The average of 18, 1, 6, $x$, and $y$ is 10. * **Calculation:** Set up the first equation to find $x$. * $\frac{2 + 7 + 6 + x}{4} = 5$ * $15 + x = 20$ * $x = 5$ * Now, substitute $x = 5$ into the second equation to find $y$. * $\frac{18 + 1 + 6 + 5 + y}{5} = 10$ * $\frac{30 + y}{5} = 10$ * $30 + y = 50$ * $y = 20$ ### Exam Strategy & Shortcut Think in terms of "Total Sum" immediately rather than writing out the fraction. For the first group: 4 numbers averaging 5 means the total sum is 20. The knowns sum to 15, so $x$ is 5. For the second group: 5 numbers averaging 10 means the total sum is 50. The knowns (including $x=5$) sum to 30. Therefore, $y = 50 - 30 = 20$. This mental math saves crucial seconds. ### Common Pitfall A common mistake is forgetting to count $y$ as one of the numbers in the second set, leading to dividing by 4 instead of 5, which fundamentally ruins the algebraic equation. Always verify your total count $n$. ### Final Answer **Therefore, the correct answer is 20.**