The average of five numbers is 7. When three new numbers are included, the average of the eight numbers becomes 8.5. The average of the three new numbers is:

Options

A. 9
B. 10.5
C. 11
D. 11.5

Answer

To solve the problem step by step, we will follow the logical flow of calculations based on the averages given.

### Step 1: Calculate the sum of the first five numbers.
The average of the first five numbers is given as 7. The formula for average is:

\[ \text{Average} = \frac{\text{Sum of numbers}}{\text{Number of numbers}} \]

So, we can rearrange this to find the sum:

\[ \text{Sum of first 5 numbers} = \text{Average} \times \text{Number of numbers} = 7 \times 5 = 35 \]

### Step 2: Calculate the sum of all eight numbers.
The average of the eight numbers (which includes the three new numbers) is given as 8.5. Using the same formula:

\[ \text{Sum of all 8 numbers} = \text{Average} \times \text{Number of numbers} = 8.5 \times 8 = 68 \]

### Step 3: Calculate the sum of the three new numbers.
To find the sum of the three new numbers, we subtract the sum of the first five numbers from the sum of all eight numbers:

\[ \text{Sum of 3 new numbers} = \text{Sum of all 8 numbers} - \text{Sum of first 5 numbers} = 68 - 35 = 33 \]

### Step 4: Calculate the average of the three new numbers.
Now that we have the sum of the three new numbers, we can find their average:

\[ \text{Average of 3 new numbers} = \frac{\text{Sum of 3 new numbers}}{\text{Number of new numbers}} = \frac{33}{3} = 11 \]

### Final Answer:
The average of the three new numbers is **11**.

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