To solve the problem step by step, we will follow the information given in the question and derive the necessary values.
### Step 1: Calculate the sum of all eleven numbers.
The average of eleven numbers is given as 68. We can find the sum of these numbers using the formula:
\[
\text{Sum} = \text{Average} \times \text{Number of Observations}
\]
So, the sum of the eleven numbers is:
\[
\text{Sum} = 68 \times 11 = 748
\]
### Step 2: Calculate the sum of the first four numbers.
The average of the first four numbers is given as 78. Therefore, we can find their sum as follows:
\[
\text{Sum of first four numbers} = 78 \times 4 = 312
\]
### Step 3: Calculate the sum of the next four numbers.
The average of the next four numbers is given as 63. Thus, their sum is:
\[
\text{Sum of next four numbers} = 63 \times 4 = 252
\]
### Step 4: Calculate the sum of the ninth, tenth, and eleventh numbers.
Now, we can find the sum of the ninth, tenth, and eleventh numbers by subtracting the sums of the first eight numbers from the total sum:
\[
\text{Sum of 9th, 10th, and 11th numbers} = \text{Total Sum} - \text{Sum of first four numbers} - \text{Sum of next four numbers}
\]
Calculating this gives:
\[
\text{Sum of 9th, 10th, and 11th numbers} = 748 - 312 - 252 = 184
\]
### Step 5: Set up equations for the ninth, tenth, and eleventh numbers.
Let the 11th number be \( x \). Then, according to the problem:
- The 9th number is \( 2x \).
- The 10th number is \( \frac{x}{2} \).
Now, we can express the sum of the ninth, tenth, and eleventh numbers in terms of \( x \):
\[
\text{Sum} = 2x + \frac{x}{2} + x = 2x + x + \frac{x}{2} = 3.5x
\]
### Step 6: Set up the equation using the sum of the ninth, tenth, and eleventh numbers.
From Step 4, we know that:
\[
3.5x = 184
\]
### Step 7: Solve for \( x \).
To find \( x \):
\[
x = \frac{184}{3.5} = \frac{1840}{35} = 52.57 \text{ (approximately)}
\]
### Step 8: Calculate the ninth and eleventh numbers.
Now we can find the values of the 9th and 11th numbers:
- The 11th number \( x \) is approximately \( 52.57 \).
- The 9th number \( 2x \) is approximately \( 2 \times 52.57 = 105.14 \).
### Step 9: Calculate the average of the ninth and eleventh numbers.
The average of the 9th and 11th numbers is:
\[
\text{Average} = \frac{\text{9th number} + \text{11th number}}{2} = \frac{105.14 + 52.57}{2} = \frac{157.71}{2} = 78.85
\]
### Final Answer:
The average of the 9th and 11th numbers is approximately **78.85**.
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