The following numbers are written in descending order of their values:
`68,60,42,x-3,x-8,x-11,30,25,22` and 20.
If their median is 39, find the value of x.

Answer

To solve the problem step by step, we will find the value of \( x \) given that the median of the numbers is 39.

### Step 1: Understand the Median
The median of a set of numbers is the middle value when the numbers are arranged in order. If there is an even number of observations, the median is the average of the two middle numbers.

### Step 2: Identify the Number of Terms
We have the following numbers in descending order:
\[ 68, 60, 42, x-3, x-8, x-11, 30, 25, 22, 20 \]
This gives us a total of \( n = 10 \) terms.

### Step 3: Find the Positions of the Median
Since \( n = 10 \) (an even number), the median will be the average of the 5th and 6th terms.
- 5th term: \( x - 8 \)
- 6th term: \( x - 11 \)

### Step 4: Set Up the Equation for the Median
According to the problem, the median is given as 39. Therefore, we can set up the equation:
\[ \frac{(x - 8) + (x - 11)}{2} = 39 \]

### Step 5: Simplify the Equation
Multiply both sides by 2 to eliminate the fraction:
\[ (x - 8) + (x - 11) = 78 \]
Combine like terms:
\[ 2x - 19 = 78 \]

### Step 6: Solve for \( x \)
Add 19 to both sides:
\[ 2x = 78 + 19 \]
\[ 2x = 97 \]
Now divide by 2:
\[ x = \frac{97}{2} = 48.5 \]

### Final Answer
The value of \( x \) is \( 48.5 \).

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