The mean of 11 numbers is 35. If the mean of the first 6 numbers is 32 and that of the last 6 numbers is 37, then the sixth number is

Options

A. 27
B. 29
C. 30
D. 32

Answer

To solve the problem step by step, we will use the given information about the means of the numbers.

### Step 1: Calculate the total sum of the 11 numbers.
The mean of 11 numbers is given as 35. We can use the formula for mean:

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

Let the sum of the 11 numbers be \( S \).

\[ 35 = \frac{S}{11} \]

To find \( S \), we multiply both sides by 11:

\[ S = 35 \times 11 = 385 \]

### Step 2: Calculate the total sum of the first 6 numbers.
The mean of the first 6 numbers is given as 32. Let the sum of the first 6 numbers be \( S_1 \).

Using the mean formula again:

\[ 32 = \frac{S_1}{6} \]

To find \( S_1 \), we multiply both sides by 6:

\[ S_1 = 32 \times 6 = 192 \]

### Step 3: Calculate the total sum of the last 6 numbers.
The mean of the last 6 numbers is given as 37. Let the sum of the last 6 numbers be \( S_2 \).

Using the mean formula:

\[ 37 = \frac{S_2}{6} \]

To find \( S_2 \), we multiply both sides by 6:

\[ S_2 = 37 \times 6 = 222 \]

### Step 4: Relate the sums to find the sixth number.
The sixth number is included in both the sum of the first 6 numbers and the sum of the last 6 numbers. Let the sixth number be \( x \).

From the sums we have:

\[ S_1 + S_2 = 192 + 222 = 414 \]

However, the sixth number \( x \) is counted twice (once in \( S_1 \) and once in \( S_2 \)), so we need to subtract it once to find the total sum of the 11 numbers:

\[ S = S_1 + S_2 - x \]

Substituting the values we found:

\[ 385 = 414 - x \]

### Step 5: Solve for \( x \).
Now, we can solve for \( x \):

\[ x = 414 - 385 = 29 \]

Thus, the sixth number is \( 29 \).

### Final Answer:
The sixth number is **29**.

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