The average of 11 numbers is 7. If every number is doubled, what will be the new average of the numbers?

Options

A. 3.5
B. 7
C. 10.5
D. 14

Answer

To solve the problem, we will follow these steps:

1. **Understand the Given Information**: We know that the average of 11 numbers is 7. The average is calculated by dividing the sum of the numbers by the total count of numbers.

2. **Calculate the Total Sum of the Numbers**:
\[ \text{Average} = \frac{\text{Sum of numbers}}{\text{Count of numbers}} \]
Given that the average is 7 and there are 11 numbers, we can find the total sum (S) of the numbers:
\[ S = \text{Average} \times \text{Count} = 7 \times 11 = 77 \]

3. **Determine the New Numbers After Doubling**: If every number is doubled, the new numbers will be:
\[ 2A_1, 2A_2, 2A_3, \ldots, 2A_{11} \]

4. **Calculate the New Total Sum**: The new total sum (S') after doubling each number will be:
\[ S' = 2A_1 + 2A_2 + 2A_3 + \ldots + 2A_{11} = 2(A_1 + A_2 + A_3 + \ldots + A_{11}) = 2S \]
Substituting the value of S we found earlier:
\[ S' = 2 \times 77 = 154 \]

5. **Calculate the New Average**: The new average (A') will be:
\[ A' = \frac{S'}{\text{Count}} = \frac{154}{11} \]
Simplifying this gives:
\[ A' = 14 \]

Thus, the new average of the numbers after doubling is **14**.