The average of 15 numbers is 18. The average of first 8 is 19 and that last 8 is 17, then the 8th number is

Options

A. 15
B. 16
C. 18
D. 20

Answer

To solve the problem step by step, we will follow the information provided in the question regarding averages and sums.

### Step-by-Step Solution:

1. **Calculate the sum of 15 numbers:**
- Given that the average of 15 numbers is 18, we can find the sum of these numbers using the formula:
\[ \text{Sum} = \text{Average} \times \text{Number of items} \]
- Therefore, the sum of 15 numbers is:
\[ \text{Sum} = 18 \times 15 = 270 \]

2. **Calculate the sum of the first 8 numbers:**
- The average of the first 8 numbers is given as 19. Thus, the sum of the first 8 numbers can be calculated as:
\[ \text{Sum of first 8} = 19 \times 8 = 152 \]

3. **Calculate the sum of the last 8 numbers:**
- The average of the last 8 numbers is given as 17. Therefore, the sum of the last 8 numbers is:
\[ \text{Sum of last 8} = 17 \times 8 = 136 \]

4. **Set up the equation to find the 8th number:**
- We note that the 8th number is included in both the sum of the first 8 numbers and the sum of the last 8 numbers. Therefore, if we denote the 8th number as \( x \), we can express the total sum of the numbers as:
\[ \text{Sum of first 8} + \text{Sum of last 8} - \text{8th number} = \text{Total sum of 15 numbers} \]
- This can be written as:
\[ 152 + 136 - x = 270 \]

5. **Solve for the 8th number \( x \):**
- Combine the sums of the first and last 8 numbers:
\[ 288 - x = 270 \]
- Rearranging gives:
\[ x = 288 - 270 = 18 \]

### Final Answer:
The 8th number is **18**.