A student was tasked with finding the arithmetic mean of the numbers 3, 11, 7, 9, 15, 13, 8, 19, 17, 21, 14, and $$ x $$. The calculated mean was found to be 12. Determine the value of $$ x $$.

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Accepted Answer

The formula for the arithmetic mean is:  
$$
	ext{Mean} = \frac{	ext{Sum of all numbers}}{	ext{Total count of numbers}}
$$

1. Total count of numbers = $$ 12 $$ (including $$ x $$).  
   The given mean = $$ 12 $$. Therefore, the total sum of all numbers is:  
   $$
   	ext{Total sum} = 	ext{Mean} 	imes 	ext{Total count} = 12 	imes 12 = 144
   $$

2. Calculate the sum of the known numbers (excluding $$ x $$):  
   $$
   3 + 11 + 7 + 9 + 15 + 13 + 8 + 19 + 17 + 21 + 14 = 137
   $$

3. Solve for $$ x $$:  
   $$
   x = 	ext{Total sum} - 	ext{Sum of known numbers} = 144 - 137 = 7
   $$

Thus, the value of $$ x $$ is **7**.

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