The average of 9 numbers is 11. If each of these 9 numbers is multiplied by 5 and then 5 is added to each of these resultant numbers, then the new average is :

Options

A. 20
B. 30
C. 60
D. 50

Answer

To solve the problem step by step, we need to find the new average after performing the operations on the original numbers.

### Step 1: Calculate the total sum of the original numbers
Given that the average of 9 numbers is 11, we can find the total sum of these numbers.

\[ \text{Total Sum} = \text{Average} \times \text{Number of elements} = 11 \times 9 = 99 \]

**Hint:** Remember that the sum of the numbers can be calculated by multiplying the average by the number of elements.

### Step 2: Apply the operations to each number
Each of the 9 numbers is multiplied by 5 and then 5 is added to each resultant number. If we denote the original numbers as \( x_1, x_2, \ldots, x_9 \), the new numbers will be:

\[ \text{New Number} = 5x_i + 5 \quad \text{for each } i = 1, 2, \ldots, 9 \]

### Step 3: Calculate the new total sum
Now, we can express the total sum of the new numbers:

\[ \text{New Total Sum} = (5x_1 + 5) + (5x_2 + 5) + \ldots + (5x_9 + 5) \]

This can be simplified as:

\[ \text{New Total Sum} = 5(x_1 + x_2 + \ldots + x_9) + 5 \times 9 \]

Substituting the total sum we calculated in Step 1:

\[ \text{New Total Sum} = 5(99) + 45 = 495 + 45 = 540 \]

### Step 4: Calculate the new average
Now, we can find the new average by dividing the new total sum by the number of elements:

\[ \text{New Average} = \frac{\text{New Total Sum}}{\text{Number of elements}} = \frac{540}{9} = 60 \]

### Final Answer
The new average is **60**.

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