A student was asked to find the arithmetic mean of the numbers 3, 11, 7, 9, 15, 13, 8, 19, 17, 21, 14 and $x$. He found the mean to be 12. What should be the number in place of $x$?

### Concept & Logic The mean relates the total sum of all elements to the number of elements. By using the given mean and total count, we can find the required total sum and isolate the unknown variable. $$Total\ Sum = Mean \times Total\ Count$$ ### Step-by-Step Solution * **Given:** The numbers are 3, 11, 7, 9, 15, 13, 8, 19, 17, 21, 14, and $x$. * The given mean is 12. * **Deduction:** Count the total number of terms. There are 11 known numbers plus $x$, making exactly 12 terms in total. * Calculate the required total sum for the mean to be 12. * $Total\ Sum = 12 \times 12 = 144$. * **Calculation:** Sum the 11 known numbers. * $Known\ Sum = 3 + 11 + 7 + 9 + 15 + 13 + 8 + 19 + 17 + 21 + 14 = 137$. * Set up the algebraic equation to find $x$. * $137 + x = 144$ * $x = 144 - 137 = 7$. ### Exam Strategy & Shortcut You can use the "Deviation Method". Compare each number to the mean (12). Deviations: -9, -1, -5, -3, +3, +1, -4, +7, +5, +9, +2. Sum of deviations = $(-9+9) + (-1+1) + (-5+5) + (-3+3) + (-4+7+2) = 5$. Since the net deviation of known numbers is +5, $x$ must be 5 *below* the mean to balance it out to 0. $x = 12 - 5 = 7$. This avoids adding up large numbers! ### Common Pitfall A common error is miscounting the total number of terms as 11 instead of 12, leading to an incorrect total sum of $(11 \times 12 = 132)$. Always double-check your $n$ value when a variable is mixed into a list. ### Final Answer **Therefore, the correct answer is 7.**