On mixing two classes A and B of students having average marks 25 and 40 respectively, the overall average obtained is 30. Find the ratio of the students in the classes A and B.

Options

A. `2:1`
B. `5:8`
C. `5:6`
D. `3:4`

Answer

To solve the problem, we need to find the ratio of the number of students in classes A and B based on their average marks and the overall average. Let's break it down step by step.

### Step 1: Define Variables
Let:
- \( x \) = number of students in class A
- \( y \) = number of students in class B

### Step 2: Calculate Total Marks for Each Class
The total marks for each class can be calculated using the average marks:
- Total marks in class A = \( 25x \) (since the average is 25)
- Total marks in class B = \( 40y \) (since the average is 40)

### Step 3: Calculate Overall Total Marks
The overall average is given as 30. The total number of students is \( x + y \), and the overall total marks can be expressed as:
- Overall total marks = \( 30(x + y) \)

### Step 4: Set Up the Equation
We can set up the equation based on the total marks:
\[ 25x + 40y = 30(x + y) \]

### Step 5: Expand and Simplify the Equation
Expanding the right side:
\[ 25x + 40y = 30x + 30y \]
Now, rearranging the equation:
\[ 25x + 40y - 30x - 30y = 0 \]
This simplifies to:
\[ -5x + 10y = 0 \]

### Step 6: Solve for the Ratio
Rearranging gives:
\[ 10y = 5x \]
Dividing both sides by 5:
\[ 2y = x \]
Thus, the ratio of students in class A to class B is:
\[ \frac{x}{y} = \frac{2}{1} \]

### Final Answer
The ratio of the number of students in class A to class B is \( 2:1 \).

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