The average of 13 numbers is 8. If each number is multiplied by 7, then what will the new average be:53

The correct option is 56.

Step-by-Step Explanation:

Step 1: Understanding the Concept
A basic property of averages states that if every number in a given set is multiplied by a constant factor $k$, the average of the set is also multiplied by that same factor $k$.

Let the original 13 numbers be $x_1,x_2,...,x_{13}$.

The original average is given as:

$\text{Original Average}=\frac{x_1+x_2+...+x_{13}}{13}=8$

From this, the sum of the original 13 numbers is:

$x_1+x_2+...+x_{13}=13\times 8=104$

Step 2: Calculating the New Average
If each of the 13 numbers is multiplied by 7, the new numbers become $7x_1,7x_2,...,7x_{13}$.

The new average is calculated as:

$\text{New Average}=\frac{7x_1+7x_2+...+7x_{13}}{13}$

Factoring out 7 from the numerator:

$\text{New Average}=7\times \left(\frac{x_1+x_2+...+x_{13}}{13}\right)$

Substitute the value of the original average (8):

$\text{New Average}=7\times 8=56$

Thus, the new average is 56.