What is the average of first 11 multiples of 11?

Options

A. 22
B. 44
C. 55
D. 66

Answer

To find the average of the first 11 multiples of 11, we can follow these steps:

### Step 1: Identify the first 11 multiples of 11
The first 11 multiples of 11 are:
1. \( 11 \times 1 = 11 \)
2. \( 11 \times 2 = 22 \)
3. \( 11 \times 3 = 33 \)
4. \( 11 \times 4 = 44 \)
5. \( 11 \times 5 = 55 \)
6. \( 11 \times 6 = 66 \)
7. \( 11 \times 7 = 77 \)
8. \( 11 \times 8 = 88 \)
9. \( 11 \times 9 = 99 \)
10. \( 11 \times 10 = 110 \)
11. \( 11 \times 11 = 121 \)

So, the first 11 multiples of 11 are:
\[ 11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 121 \]

### Step 2: Calculate the sum of these multiples
Now, we will sum these multiples:
\[ 11 + 22 + 33 + 44 + 55 + 66 + 77 + 88 + 99 + 110 + 121 \]

We can group them for easier calculation:
\[ (11 + 121) + (22 + 110) + (33 + 99) + (44 + 88) + (55 + 77) + 66 \]
Calculating each pair:
- \( 11 + 121 = 132 \)
- \( 22 + 110 = 132 \)
- \( 33 + 99 = 132 \)
- \( 44 + 88 = 132 \)
- \( 55 + 77 = 132 \)
- Plus the middle term \( 66 \)

So, the total sum is:
\[ 5 \times 132 + 66 = 660 + 66 = 726 \]

### Step 3: Calculate the average
The average is calculated by dividing the total sum by the number of terms:
\[ \text{Average} = \frac{\text{Total Sum}}{\text{Number of Terms}} = \frac{726}{11} \]

Now, performing the division:
\[ 726 \div 11 = 66 \]

### Conclusion
Thus, the average of the first 11 multiples of 11 is \( \boxed{66} \).

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