Explanations:The concept of average is fundamental in arithmetic and is often used in various competitive exams. The average of a set of numbers is the sum of the numbers divided by the count of the numbers. In this problem, we are given the average of 22 numbers and the average of the last two numbers, and we need to find the average of the remaining 20 numbers.
Let's denote the sum of the 22 numbers as \( S \). The average of these 22 numbers is given as 11. Therefore, we can write the equation for the sum as:
\[
S = 22 \times 11 = 242
\]
Next, we are given that the average of the last two numbers is 16. Let's denote the sum of these last two numbers as \( S_2 \). The average of these two numbers is 16, so we can write:
\[
S_2 = 2 \times 16 = 32
\]
To find the sum of the remaining 20 numbers, we subtract the sum of the last two numbers from the total sum:
\[
S_{20} = S - S_2 = 242 - 32 = 210
\]
Now, we need to find the average of these 20 numbers. The average is the sum of these 20 numbers divided by 20:
\[
\text{Average of 20 numbers} = \frac{S_{20}}{20} = \frac{210}{20} = 10.5
\]
Thus, the average of the remaining 20 numbers is 10.5. This calculation demonstrates the importance of correctly applying the formula for average and adjusting the sum and count of the numbers as needed.
It is crucial to understand that the average of a subset of numbers can be different from the overall average, especially when the subset has a different sum and count. In this case, the last two numbers have a higher average than the overall average, which affects the average of the remaining numbers.
Common mistakes include not correctly adjusting the sum and count when calculating the average of a subset, misinterpreting the given averages, and incorrectly assuming that the average of the subset will be the same as the overall average. By carefully applying the formula and adjusting the sum and count, we can accurately find the average of the subset.