The median of 8 numbers is 40. If 7 of these numbers are 22,58,31,44,25,45 and 50. find the eighth number.

Answer

To find the eighth number given that the median of eight numbers is 40 and seven of those numbers are 22, 58, 31, 44, 25, 45, and 50, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Median**: The median of a set of numbers is the middle value when the numbers are arranged in ascending order. For an even number of observations (like 8), the median is the average of the two middle numbers.

2. **Arrange the Given Numbers**: First, we need to arrange the seven given numbers in ascending order:
- The numbers are: 22, 58, 31, 44, 25, 45, and 50.
- Arranging these in ascending order gives us: 22, 25, 31, 44, 45, 50, 58.

3. **Identify the Position of the Eighth Number**: Since we need to find the eighth number (let's call it \( x \)), we will insert \( x \) into the sorted list. The median of these eight numbers is given as 40.

4. **Calculate the Sum of the Middle Two Terms**: Since there are 8 numbers, the median is the average of the 4th and 5th numbers in the sorted list. Therefore:
\[ \text{Median} = \frac{\text{4th term} + \text{5th term}}{2} = 40 \]
This means:
\[ \text{4th term} + \text{5th term} = 80 \]

5. **Determine the Middle Terms**: The 4th and 5th terms in the sorted list are 44 and 45. If we assume \( x \) is less than 44, then the 4th term will be 44 and the 5th term will be \( x \). If \( x \) is greater than 45, then the 4th term will be \( x \) and the 5th term will be 45.

6. **Finding the Eighth Number**:
- If \( x \) is the 4th term, then:
\[ 44 + x = 80 \implies x = 80 - 44 = 36 \]
- Now, we check if 36 fits in the sorted order. If we insert 36 into the list, we get:
- 22, 25, 31, 36, 44, 45, 50, 58.
- The 4th term is now 36 and the 5th term is 44. The median is:
\[ \text{Median} = \frac{36 + 44}{2} = \frac{80}{2} = 40 \]
- This confirms that our assumption is correct.

7. **Conclusion**: The eighth number is \( x = 36 \).

### Final Answer:
The eighth number is **36**.