The mean of marks obtained in an examination by a group of 100 students was found to be 49.96. The mean of the marks obtained in the same examination by another group of 200 students was 52.32, then the mean of the marks obtained by both the groups of students taken together is

Options

A. 51.5
B. 52
C. 52.5
D. 53

Answer

To find the mean of the marks obtained by both groups of students together, we can follow these steps:

### Step 1: Calculate the total marks for the first group of students.
The mean marks of the first group of 100 students is given as 49.96.

\[ \text{Total marks of first group} = \text{Number of students} \times \text{Mean} = 100 \times 49.96 = 4996 \]

### Step 2: Calculate the total marks for the second group of students.
The mean marks of the second group of 200 students is given as 52.32.

\[ \text{Total marks of second group} = \text{Number of students} \times \text{Mean} = 200 \times 52.32 = 10464 \]

### Step 3: Calculate the total marks for both groups combined.
Now, we add the total marks of both groups to find the overall total marks.

\[ \text{Total marks of both groups} = 4996 + 10464 = 15460 \]

### Step 4: Calculate the total number of students.
The total number of students from both groups is:

\[ \text{Total number of students} = 100 + 200 = 300 \]

### Step 5: Calculate the mean marks for all students combined.
Finally, we can find the mean marks for all 300 students by dividing the total marks by the total number of students.

\[ \text{Mean of both groups} = \frac{\text{Total marks of both groups}}{\text{Total number of students}} = \frac{15460}{300} = 51.53 \]

### Final Answer:
The mean of the marks obtained by both groups of students taken together is **51.53**.

---