The average of N numbers is 10. If a number 6 is removed, then average becomes 12. What is the value of N?

Options

A. 5
B. 4
C. 6
D. 3

Answer

To solve the problem step by step, let's break it down:

### Step 1: Understand the average formula
The average of N numbers is given by the formula:
\[ \text{Average} = \frac{\text{Sum of numbers}}{\text{Total numbers}} \]
From the problem, we know that the average of N numbers is 10.

### Step 2: Calculate the sum of N numbers
Using the average, we can express the sum of N numbers:
\[ \text{Sum of N numbers} = \text{Average} \times N = 10N \]

### Step 3: Remove the number 6 and find the new average
When we remove the number 6, the new average of the remaining (N-1) numbers becomes 12. Therefore, we can express the sum of the remaining numbers:
\[ \text{Sum of (N-1) numbers} = \text{Average} \times \text{Total numbers} = 12 \times (N - 1) \]

### Step 4: Set up the equation
The sum of the (N-1) numbers can also be expressed in terms of the original sum:
\[ \text{Sum of (N-1) numbers} = \text{Sum of N numbers} - 6 = 10N - 6 \]
Now, we can set the two expressions for the sum of (N-1) numbers equal to each other:
\[ 10N - 6 = 12(N - 1) \]

### Step 5: Simplify the equation
Now, let's simplify the equation:
\[ 10N - 6 = 12N - 12 \]
Rearranging gives:
\[ 10N - 12N = -12 + 6 \]
\[ -2N = -6 \]

### Step 6: Solve for N
Dividing both sides by -2:
\[ N = 3 \]

### Conclusion
The value of N is 3.