Students of two schools appeared for a common test carrying 100 marks. The arthmetic means of their marks for school I and II are 82 and 86 respectively, If the number of students of school II is 1.5 times the number of students of school I, what is the arithmetic mean of the marks of all the students, of both the school ?

Options

A. 84.5
B. 84.2
C. 84.4
D. This cannot be calculated with the given data

Explanation

To find the arithmetic mean of the marks of all the students from both schools, we can follow these steps:

### Step 1: Define the number of students in each school
Let the number of students in School I be \( x \).
Since the number of students in School II is 1.5 times the number of students in School I, we can express the number of students in School II as:
\[ \text{Number of students in School II} = 1.5x \]

### Step 2: Calculate the total marks for each school
The arithmetic mean of the marks for School I is 82, so the total marks for School I can be calculated as:
\[ \text{Total marks for School I} = \text{Mean} \times \text{Number of students} = 82 \times x \]

For School II, the arithmetic mean of the marks is 86, so the total marks for School II can be calculated as:
\[ \text{Total marks for School II} = \text{Mean} \times \text{Number of students} = 86 \times (1.5x) = 129x \]

### Step 3: Calculate the combined total marks
Now, we can find the combined total marks for both schools:
\[ \text{Combined total marks} = \text{Total marks for School I} + \text{Total marks for School II} = 82x + 129x = 211x \]

### Step 4: Calculate the total number of students
The total number of students from both schools is:
\[ \text{Total students} = x + 1.5x = 2.5x \]

### Step 5: Calculate the overall arithmetic mean
The overall arithmetic mean of the marks of all the students can be calculated by dividing the combined total marks by the total number of students:
\[ \text{Overall mean} = \frac{\text{Combined total marks}}{\text{Total students}} = \frac{211x}{2.5x} \]

### Step 6: Simplify the expression
The \( x \) cancels out in the numerator and denominator:
\[ \text{Overall mean} = \frac{211}{2.5} = 84.4 \]

Thus, the arithmetic mean of the marks of all the students from both schools is \( 84.4 \).

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