The average of 9 numbers is 18. If the average of first five numbers is 19 and the average of last 5 numbers is 17, find the 5th number.

Options

A. 16
B. 20
C. 18
D. 22

Answer

To find the 5th number in the given scenario, we can follow these steps:

### Step 1: Calculate the total sum of the 9 numbers.
Given that the average of 9 numbers is 18, we can find the total sum of these numbers using the formula:

\[ \text{Total Sum} = \text{Average} \times \text{Number of Terms} \]

So,

\[ \text{Total Sum} = 18 \times 9 = 162 \]

### Step 2: Calculate the total sum of the first 5 numbers.
The average of the first 5 numbers is given as 19. Therefore, the total sum of the first 5 numbers can be calculated as:

\[ \text{Sum of First 5 Numbers} = 19 \times 5 = 95 \]

### Step 3: Calculate the total sum of the last 5 numbers.
The average of the last 5 numbers is given as 17. Thus, the total sum of the last 5 numbers is:

\[ \text{Sum of Last 5 Numbers} = 17 \times 5 = 85 \]

### Step 4: Set up the equation to find the 5th number.
Let the 5th number be \( x_5 \). The sum of the first 5 numbers includes \( x_5 \) and the sum of the last 5 numbers also includes \( x_5 \). Therefore, we can express the total sum of the 9 numbers as:

\[ \text{Sum of First 5 Numbers} + \text{Sum of Last 5 Numbers} - x_5 = \text{Total Sum} \]

Substituting the values we calculated:

\[ 95 + 85 - x_5 = 162 \]

### Step 5: Solve for \( x_5 \).
Now, we can simplify the equation:

\[ 180 - x_5 = 162 \]

Rearranging gives:

\[ x_5 = 180 - 162 = 18 \]

### Conclusion
The 5th number is \( 18 \).

---