Explanations:The concept of average, also known as the arithmetic mean, is a fundamental concept in mathematics. The average of a set of numbers is calculated by summing all the numbers in the set and then dividing by the count of the numbers. Mathematically, if we have a set of \( n \) numbers \( x_1, x_2, \ldots, x_n \), the average \( \bar{x} \) is given by:
\[
\bar{x} = \frac{x_1 + x_2 + \ldots + x_n}{n}
\]
In this problem, we are given that the average of 10 numbers is 7. This means:
\[
\bar{x} = 7
\]
Now, if each number in the set is multiplied by 12, the new set of numbers will be \( 12x_1, 12x_2, \ldots, 12x_n \). The average of the new set of numbers can be calculated as follows:
\[
\text{New Average} = \frac{12x_1 + 12x_2 + \ldots + 12x_n}{n}
\]
We can factor out the 12 from the numerator:
\[
\text{New Average} = \frac{12(x_1 + x_2 + \ldots + x_n)}{n}
\]
Since the sum of the original numbers divided by \( n \) is the original average, we can substitute \( \bar{x} \) for \( \frac{x_1 + x_2 + \ldots + x_n}{n} \):
\[
\text{New Average} = 12 \times \bar{x}
\]
Given that the original average \( \bar{x} \) is 7, we can substitute this value into the equation:
\[
\text{New Average} = 12 \times 7 = 84
\]
Therefore, the average of the new set of numbers, after each number is multiplied by 12, is 84. This demonstrates that when each number in a set is multiplied by a constant, the average of the new set is also multiplied by that constant.