The numbers 25, 34, 46, 48, 2 x + 1, 4 x + 3, 105, 110, 114, 122 are written in ascending order and their median is 77. The value of x is

Options

A. 22
B. 24
C. 28
D. 25

Answer

To solve the problem, we need to find the value of \( x \) such that the median of the given numbers is 77. The numbers are:

\[ 25, 34, 46, 48, 2x + 1, 4x + 3, 105, 110, 114, 122 \]

### Step 1: Arrange the numbers in ascending order
Since we need to find the median, we first need to arrange the numbers in ascending order. The numbers can be expressed as:

1. \( 25 \)
2. \( 34 \)
3. \( 46 \)
4. \( 48 \)
5. \( 2x + 1 \)
6. \( 4x + 3 \)
7. \( 105 \)
8. \( 110 \)
9. \( 114 \)
10. \( 122 \)

### Step 2: Identify the position of the median
Since there are 10 numbers, the median will be the average of the 5th and 6th numbers in the ordered list.

### Step 3: Set up the equation for the median
According to the problem, the median is given as 77. Therefore, we can set up the equation:

\[ \frac{(2x + 1) + (4x + 3)}{2} = 77 \]

### Step 4: Simplify the equation
Now, simplify the left side of the equation:

\[ \frac{(2x + 1 + 4x + 3)}{2} = 77 \]

This simplifies to:

\[ \frac{(6x + 4)}{2} = 77 \]

### Step 5: Multiply both sides by 2
To eliminate the fraction, multiply both sides by 2:

\[ 6x + 4 = 154 \]

### Step 6: Solve for \( x \)
Now, subtract 4 from both sides:

\[ 6x = 150 \]

Next, divide both sides by 6:

\[ x = 25 \]

### Conclusion
The value of \( x \) is \( 25 \).