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Let's break down the problem:

We have 11 observations with an average of 50, meaning the sum of all 11 observations is:

\[
\text{Sum of 11 observations} = 50 \times 11 = 550
\]

The average of the first 6 observations is 40, so the sum of the first 6 observations is:

\[
\text{Sum of first 6 observations} = 40 \times 6 = 240
\]

The average of the last 6 observations is 55, so the sum of the last 6 observations is:

\[
\text{Sum of last 6 observations} = 55 \times 6 = 330
\]

Now, notice that the 6th observation is included in both the first and the last 6 observations. So, the sum of the 11 observations can also be written as:

\[
\text{Sum of 11 observations} = (\text{Sum of first 5 observations}) + (\text{6th observation}) + (\text{Sum of last 5 observations})
\]

From the given information, we know that the total sum of all observations is 550, the sum of the first 6 is 240, and the sum of the last 6 is 330. Therefore, the 6th observation is:

\[
\text{6th observation} = \text{Sum of 11 observations} - (\text{Sum of first 5 observations} + \text{Sum of last 5 observations})
\]

\[
\text{6th observation} = 240 + 330 - 550 = 20
\]

So, the 6th observation is **20**, which corresponds to option (D).

Would you like further clarification on any steps, or details?

Here are 5 related questions to think about:
1. What would happen if the average of the last 6 observations was different?
2. How does changing the average of the first 6 observations affect the result?
3. Can you calculate the average if you know the sum but not the number of observations?
4. How can this concept be applied to larger data sets?
5. How would you determine the sum of observations if only partial averages were given?

**Tip:** To solve problems involving averages, always remember that the average is just the total sum divided by the number of observations. Try to express everything in terms of the sum to simplify calculations.

The average of 11 observations is 50. If the average of the first six observations is 40 and the average of the last six observations is 55, then the sixth observation is:

Solve the problem of determining the sixth observation given the averages of 11 observations, the first six, and the last six. Learn the concept of averages and how to apply the arithmetic mean theorem to calculate individual data points. Suitable for students in Grades 8-10.

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