A student was asked to find the arithmetic mean of the following 12 numbers :
3,11, 7,9, 15, 13, 8,19, 17, 21, 14 and x.
He found the mean to be 12. The value of x will be :

Options

A. 3
B. 7
C. 17
D. 31

Answer

To find the value of \( x \) in the given problem, we will follow these steps:

### Step 1: Understand the formula for the arithmetic mean
The arithmetic mean (average) of a set of numbers is calculated by dividing the sum of the numbers by the total count of the numbers.

### Step 2: Set up the equation
We know that the arithmetic mean of the 12 numbers is given as 12. Therefore, we can set up the equation as follows:

\[ \text{Mean} = \frac{\text{Sum of all numbers}}{\text{Total count of numbers}} \]

In this case, the total count of numbers is 12, and the mean is 12. Thus, we can write:

\[ 12 = \frac{3 + 11 + 7 + 9 + 15 + 13 + 8 + 19 + 17 + 21 + 14 + x}{12} \]

### Step 3: Calculate the sum of the known numbers
Now, let's calculate the sum of the known numbers:

\[ 3 + 11 + 7 + 9 + 15 + 13 + 8 + 19 + 17 + 21 + 14 \]

Calculating this step-by-step:

- \( 3 + 11 = 14 \)
- \( 14 + 7 = 21 \)
- \( 21 + 9 = 30 \)
- \( 30 + 15 = 45 \)
- \( 45 + 13 = 58 \)
- \( 58 + 8 = 66 \)
- \( 66 + 19 = 85 \)
- \( 85 + 17 = 102 \)
- \( 102 + 21 = 123 \)
- \( 123 + 14 = 137 \)

So, the sum of the known numbers is \( 137 \).

### Step 4: Substitute the sum into the equation
Now we can substitute this sum back into our equation:

\[ 12 = \frac{137 + x}{12} \]

### Step 5: Solve for \( x \)
To eliminate the fraction, multiply both sides by 12:

\[ 12 \times 12 = 137 + x \]

This simplifies to:

\[ 144 = 137 + x \]

Now, isolate \( x \) by subtracting 137 from both sides:

\[ x = 144 - 137 \]

Calculating this gives:

\[ x = 7 \]

### Final Answer
The value of \( x \) is \( 7 \).

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