Average of ‘n’ observations is 38, average of ‘n’ other observations is 42 and average ofremaining ‘n’ observations is 55. Average of all the observations is:

Options

A. 45
B. 35
C. 55
D. 40

Answer

To find the average of all observations given the averages of three groups of observations, we can follow these steps:

### Step 1: Understand the given information
We have three groups of observations, each containing 'n' observations:
- The first group has an average of 38.
- The second group has an average of 42.
- The third group has an average of 55.

### Step 2: Calculate the total sum for each group
To find the total sum for each group, we can use the formula:
\[ \text{Total Sum} = \text{Average} \times \text{Number of Observations} \]
For the first group:
\[ \text{Total Sum}_1 = 38 \times n = 38n \]
For the second group:
\[ \text{Total Sum}_2 = 42 \times n = 42n \]
For the third group:
\[ \text{Total Sum}_3 = 55 \times n = 55n \]

### Step 3: Calculate the total sum of all observations
Now, we can find the total sum of all observations by adding the sums from all three groups:
\[ \text{Total Sum} = \text{Total Sum}_1 + \text{Total Sum}_2 + \text{Total Sum}_3 \]
\[ \text{Total Sum} = 38n + 42n + 55n = (38 + 42 + 55)n = 135n \]

### Step 4: Calculate the total number of observations
The total number of observations is the sum of the observations from all three groups:
\[ \text{Total Observations} = n + n + n = 3n \]

### Step 5: Calculate the average of all observations
The average of all observations can be calculated using the formula:
\[ \text{Average} = \frac{\text{Total Sum}}{\text{Total Observations}} \]
Substituting the values we calculated:
\[ \text{Average} = \frac{135n}{3n} \]
Here, \(n\) cancels out:
\[ \text{Average} = \frac{135}{3} = 45 \]

### Conclusion
The average of all observations is **45**.