Explanation: The concept of averages is fundamental in mathematics and is often used in various competitive examinations. The average (or arithmetic mean) of a set of numbers is calculated by summing all the numbers and then dividing by the count of the numbers. Mathematically, if we have a set of numbers \( x_1, x_2, \ldots, x_n \), the average \( A \) is given by:
\[ A = \frac{x_1 + x_2 + \ldots + x_n}{n} \]
In this problem, we are given that the average of 10 numbers is 7. This means:
\[ A = \frac{x_1 + x_2 + \ldots + x_{10}}{10} = 7 \]
If each number in the set is multiplied by 12, the new set of numbers becomes \( 12x_1, 12x_2, \ldots, 12x_{10} \). The average of the new set of numbers is:
\[ A_{\text{new}} = \frac{12x_1 + 12x_2 + \ldots + 12x_{10}}{10} \]
We can factor out the 12 from the numerator:
\[ A_{\text{new}} = \frac{12(x_1 + x_2 + \ldots + x_{10})}{10} \]
Since we know from the original average that \( x_1 + x_2 + \ldots + x_{10} = 70 \) (because \( 7 \times 10 = 70 \)), we can substitute this into the equation:
\[ A_{\text{new}} = \frac{12 \times 70}{10} = \frac{840}{10} = 84 \]
Thus, the average of the new set of numbers is 84. This demonstrates the property that when each number in a set is multiplied by a constant, the average of the new set is the original average multiplied by that constant. This property simplifies the calculation of the new average without needing to recalculate the sum of the new set of numbers.
Understanding this concept is crucial for solving similar problems efficiently, especially in competitive exams where time management is critical. It is important to remember that the average scales linearly with the multiplication factor applied to each number in the set.