Average of 17 observations is 28. The average of first 8 observations is 25 and last 8 observation is 30. find the 9th observation.

Options

A. 36
B. 20
C. 29
D. none of the above

Answer

To find the 9th observation given the averages of the observations, we can follow these steps:

### Step 1: Calculate the total sum of 17 observations
The average of 17 observations is given as 28. The total sum can be calculated using the formula:
\[ \text{Total Sum} = \text{Average} \times \text{Number of Observations} \]
So,
\[ \text{Total Sum} = 28 \times 17 = 476 \]

### Step 2: Calculate the total sum of the first 8 observations
The average of the first 8 observations is given as 25. Similarly, we can calculate the total sum:
\[ \text{Sum of First 8 Observations} = 25 \times 8 = 200 \]

### Step 3: Calculate the total sum of the last 8 observations
The average of the last 8 observations is given as 30. We calculate the total sum:
\[ \text{Sum of Last 8 Observations} = 30 \times 8 = 240 \]

### Step 4: Set up the equation for the 9th observation
We know that the total sum of all 17 observations is equal to the sum of the first 8 observations, the last 8 observations, and the 9th observation. Let the 9th observation be \( x \). Thus, we can write:
\[ \text{Total Sum} = \text{Sum of First 8} + \text{Sum of Last 8} + x \]
Substituting the values we found:
\[ 476 = 200 + 240 + x \]

### Step 5: Solve for the 9th observation
Now, we simplify the equation:
\[ 476 = 440 + x \]
Subtracting 440 from both sides gives:
\[ x = 476 - 440 = 36 \]

### Conclusion
The 9th observation is \( 36 \).

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