The average of 11 numbers is 32 .If the average of the first six numbers is 34 and that of the last six is 33 , then the sixth number is

Options

A. 34
B. 25
C. 50
D. 36

Answer

To solve the problem step by step, we will follow these calculations:

### Step 1: Find the sum of the 11 numbers
The average of 11 numbers is given as 32. To find the sum of these numbers, we use the formula for average:

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

So, we can rearrange this to find the sum:

\[ \text{Sum of 11 numbers} = \text{Average} \times \text{Total numbers} = 32 \times 11 = 352 \]

### Step 2: Find the sum of the first 6 numbers
The average of the first 6 numbers is given as 34. Using the same formula:

\[ \text{Sum of first 6 numbers} = \text{Average} \times \text{Total numbers} = 34 \times 6 = 204 \]

### Step 3: Find the sum of the last 6 numbers
The average of the last 6 numbers is given as 33. Again, using the average formula:

\[ \text{Sum of last 6 numbers} = \text{Average} \times \text{Total numbers} = 33 \times 6 = 198 \]

### Step 4: Set up the equation for the sixth number
Let \( x \) be the sixth number. The sum of the last 6 numbers includes the sixth number, so:

\[ \text{Sum of last 6 numbers} = x + \text{(sum of numbers 7 to 11)} \]

From the previous steps, we know:

\[ \text{Sum of last 6 numbers} = 198 \]

The sum of the last 6 numbers can also be expressed as:

\[ x + (204 - x) = 198 \]

Where \( 204 \) is the sum of the first 6 numbers, and \( x \) is included in both sums.

### Step 5: Combine the equations
Now, we can express the total sum of the 11 numbers as:

\[ \text{Sum of first 6 numbers} + \text{Sum of last 6 numbers} - x = \text{Sum of 11 numbers} \]

Substituting the values we found:

\[ 204 + 198 - x = 352 \]

### Step 6: Solve for \( x \)
Now we can solve for \( x \):

\[ 402 - x = 352 \]

Subtracting 352 from both sides gives:

\[ 402 - 352 = x \]

\[ x = 50 \]

### Conclusion
The sixth number is \( 50 \).

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