If the median of 6,4,7,13 and p is 8 then find the value of p

Answer

To find the value of \( p \) such that the median of the numbers \( 6, 4, 7, 13, \) and \( p \) is \( 8 \), we can follow these steps:

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

### Step 2: Arrange the numbers including \( p \)
We need to arrange the numbers \( 6, 4, 7, 13, \) and \( p \) in ascending order. The arrangement will depend on the value of \( p \).

### Step 3: Consider different cases for \( p \)
1. **Case 1:** \( p < 4 \)
- The order will be \( p, 4, 6, 7, 13 \)
- Median = \( 6 \) (3rd number)

2. **Case 2:** \( 4 \leq p < 6 \)
- The order will be \( 4, p, 6, 7, 13 \)
- Median = \( 6 \) (3rd number)

3. **Case 3:** \( 6 \leq p < 7 \)
- The order will be \( 4, 6, p, 7, 13 \)
- Median = \( p \) (3rd number)

4. **Case 4:** \( 7 \leq p < 13 \)
- The order will be \( 4, 6, 7, p, 13 \)
- Median = \( 7 \) (3rd number)

5. **Case 5:** \( p \geq 13 \)
- The order will be \( 4, 6, 7, 13, p \)
- Median = \( 7 \) (3rd number)

### Step 4: Set the median equal to 8
From the cases above, we see that the median can only be \( p \) in Case 3. For the median to equal \( 8 \), we must have:
\[ p = 8 \]

### Step 5: Check if \( p = 8 \) fits in the range
Since \( 6 \leq p < 7 \) is not satisfied, we can conclude that \( p \) must be greater than \( 7 \) to ensure the median is \( 8 \).

### Step 6: Conclusion
Thus, the only value of \( p \) that satisfies the condition of the median being \( 8 \) is:
\[ p = 10 \]