The total number of students in three classes of an Engineering Institute is 885. The strength of the students in the first two classes are in the ratio of 4: 9. The ratio of the number of students in the second and the third classes is 6: 11. How many students are there in the class that has the maximum number of students?

Options

A. 594
B. 495
C. 954
D. 459

Answer

To solve the problem step by step, we will first define the variables based on the information given in the question.

### Step 1: Define Variables
Let the number of students in the first class be \(4x\) and in the second class be \(9x\) (since the ratio of the first two classes is 4:9).

Let the number of students in the third class be \(y\).

### Step 2: Set Up the Equation
According to the problem, the total number of students in all three classes is 885. Therefore, we can write the equation:
\[ 4x + 9x + y = 885 \]
This simplifies to:
\[ 13x + y = 885 \quad \text{(1)} \]

### Step 3: Use the Second Ratio
We are also given that the ratio of the number of students in the second and third classes is 6:11. This can be expressed as:
\[ \frac{9x}{y} = \frac{6}{11} \]
Cross-multiplying gives us:
\[ 11 \cdot 9x = 6y \]
This simplifies to:
\[ 99x = 6y \quad \text{(2)} \]

### Step 4: Solve for y
From equation (2), we can express \(y\) in terms of \(x\):
\[ y = \frac{99x}{6} = 16.5x \]

### Step 5: Substitute y in Equation (1)
Now we substitute \(y\) in equation (1):
\[ 13x + 16.5x = 885 \]
This simplifies to:
\[ 29.5x = 885 \]

### Step 6: Solve for x
Now we can solve for \(x\):
\[ x = \frac{885}{29.5} = 30 \]

### Step 7: Find the Number of Students in Each Class
Now that we have \(x\), we can find the number of students in each class:
- First class: \(4x = 4 \times 30 = 120\)
- Second class: \(9x = 9 \times 30 = 270\)
- Third class: \(y = 16.5x = 16.5 \times 30 = 495\)

### Step 8: Identify the Class with Maximum Students
Now we compare the number of students in each class:
- First class: 120 students
- Second class: 270 students
- Third class: 495 students

The class with the maximum number of students is the third class with **495 students**.

### Final Answer
The number of students in the class that has the maximum number of students is **495**.

---