The average of 19 numbers is 22.8. The average of the first ten numbers is 18.4 and that of the last ten numbers is 28.6. If the `10^(th)`number is excluded from the given numbers, then what is the average of the remaining numbers? (Your answer should be nearest to an integer.)

Options

A. 21
B. 23
C. 22
D. 20

Answer

To solve the problem step by step, we will follow these calculations:

### Step 1: Calculate the total sum of the 19 numbers
Given that the average of 19 numbers is 22.8, we can find the total sum of these numbers using the formula:

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

\[ \text{Total Sum} = 22.8 \times 19 = 433.2 \]

### Step 2: Calculate the sum of the first 10 numbers
The average of the first 10 numbers is given as 18.4. Therefore, the sum of the first 10 numbers can be calculated as:

\[ \text{Sum of First 10 Numbers} = 18.4 \times 10 = 184 \]

### Step 3: Calculate the sum of the last 10 numbers
The average of the last 10 numbers is given as 28.6. Thus, the sum of the last 10 numbers is:

\[ \text{Sum of Last 10 Numbers} = 28.6 \times 10 = 286 \]

### Step 4: Calculate the 10th number
The 10th number is included in both the first 10 and the last 10 numbers. Therefore, we can express the total sum of the 19 numbers as:

\[ \text{Total Sum} = \text{Sum of First 10 Numbers} + \text{Sum of Last 10 Numbers} - \text{10th Number} \]

Substituting the values we have:

\[ 433.2 = 184 + 286 - \text{10th Number} \]

\[ 433.2 = 470 - \text{10th Number} \]

Rearranging gives us:

\[ \text{10th Number} = 470 - 433.2 = 36.8 \]

### Step 5: Calculate the new sum after excluding the 10th number
Now, we will calculate the new sum of the remaining 18 numbers by excluding the 10th number:

\[ \text{New Sum} = \text{Total Sum} - \text{10th Number} \]

\[ \text{New Sum} = 433.2 - 36.8 = 396.4 \]

### Step 6: Calculate the average of the remaining 18 numbers
The average of the remaining 18 numbers can be calculated as:

\[ \text{Average} = \frac{\text{New Sum}}{\text{Number of Remaining Terms}} = \frac{396.4}{18} \]

Calculating this gives:

\[ \text{Average} = 22.0222 \]

### Step 7: Round to the nearest integer
Finally, rounding 22.0222 to the nearest integer gives us:

\[ \text{Final Answer} = 22 \]

### Summary of Steps
1. Calculate the total sum of 19 numbers using the average.
2. Calculate the sum of the first 10 numbers using their average.
3. Calculate the sum of the last 10 numbers using their average.
4. Determine the 10th number using the total sum equation.
5. Calculate the new sum after excluding the 10th number.
6. Calculate the average of the remaining numbers.
7. Round the average to the nearest integer.