If the median of the numbers 6, 14, 15, 17, x-1, x+2, 29, 32, 35, 45 written in ascending order is 21.5, then the value of x is

Options

A. 20
B. 21
C. 22
D. 21.5

Answer

To find the value of \( x \) given that the median of the numbers \( 6, 14, 15, 17, x-1, x+2, 29, 32, 35, 45 \) is \( 21.5 \), we will follow these steps:

### Step 1: Arrange the numbers in ascending order
The numbers we have are \( 6, 14, 15, 17, x-1, x+2, 29, 32, 35, 45 \). To find the median, we need to arrange these numbers in ascending order. The position of \( x-1 \) and \( x+2 \) will depend on the value of \( x \).

### Step 2: Determine the median formula for an even set of numbers
Since there are 10 numbers (which is even), the median is calculated as the average of the 5th and 6th terms in the ordered list.

### Step 3: Identify the 5th and 6th terms
The 5th term is \( x-1 \) and the 6th term is \( x+2 \).

### Step 4: Set up the median equation
According to the problem, the median is given as \( 21.5 \). Therefore, we can write the equation:
\[ \frac{(x-1) + (x+2)}{2} = 21.5 \]

### Step 5: Simplify the equation
Now, simplify the left side:
\[ \frac{(x-1) + (x+2)}{2} = \frac{2x + 1}{2} \]
Setting this equal to \( 21.5 \):
\[ \frac{2x + 1}{2} = 21.5 \]

### Step 6: Eliminate the fraction
Multiply both sides by 2 to eliminate the fraction:
\[ 2x + 1 = 43 \]

### Step 7: Solve for \( x \)
Subtract 1 from both sides:
\[ 2x = 42 \]
Now divide by 2:
\[ x = 21 \]

### Conclusion
Thus, the value of \( x \) is \( 21 \).