The numbers `5, 7, 8, 10, 12, 13 and N` are arranged in ascending order. If the mean of the numbers is equal to the median. the value of N is:

Options

A. 16
B. 25
C. 15
D. 14

Answer

To solve the problem step by step, we need to find the value of \( N \) such that the mean of the numbers \( 5, 7, 8, 10, 12, 13, N \) is equal to the median.

### Step 1: Arrange the numbers in ascending order
The numbers given are \( 5, 7, 8, 10, 12, 13, N \). We need to consider the value of \( N \) and how it affects the order.

### Step 2: Determine the number of terms
There are 7 numbers in total: \( 5, 7, 8, 10, 12, 13, N \).

### Step 3: Calculate the median
Since there are 7 numbers, the median will be the 4th number when arranged in ascending order.

- If \( N \) is less than or equal to 10, the 4th number will be 10.
- If \( N \) is greater than 10, the 4th number will still be 10 because the first three numbers will still be \( 5, 7, 8 \).

Thus, in either case, the median is \( 10 \).

### Step 4: Calculate the mean
The mean is calculated using the formula:
\[ \text{Mean} = \frac{\text{Sum of all terms}}{\text{Number of terms}} \]

The sum of the numbers is:
\[ 5 + 7 + 8 + 10 + 12 + 13 + N = 55 + N \]

The number of terms is \( 7 \), so the mean becomes:
\[ \text{Mean} = \frac{55 + N}{7} \]

### Step 5: Set the mean equal to the median
Since we know the median is \( 10 \), we set up the equation:
\[ \frac{55 + N}{7} = 10 \]

### Step 6: Solve for \( N \)
To eliminate the fraction, multiply both sides by \( 7 \):
\[ 55 + N = 70 \]
Now, subtract \( 55 \) from both sides:
\[ N = 70 - 55 \]
\[ N = 15 \]

### Conclusion
The value of \( N \) is \( 15 \).

---