The ratio of students in three class of school is `4:5:6` .If 60 students increase in each class ,the ratio becomes `6:7:8` , then what was the total no of students in the three classes of school initially ?

Options

A. A)630
B. B)450
C. C)540
D. D)350

Answer

To solve the problem, we need to find the total number of students in three classes initially, given their ratios and the changes after adding students. Let's break it down step by step.

### Step 1: Define the initial number of students
Let the number of students in the three classes be represented as:
- Class A: \(4x\)
- Class B: \(5x\)
- Class C: \(6x\)

Here, \(x\) is a common multiplier.

### Step 2: Set up the equation after the increase
According to the problem, if 60 students are added to each class, the new number of students becomes:
- Class A: \(4x + 60\)
- Class B: \(5x + 60\)
- Class C: \(6x + 60\)

The new ratio of students in the classes is given as \(6:7:8\).

### Step 3: Write the new ratio as an equation
From the new ratio, we can set up the following equations:
\[ \frac{4x + 60}{6} = \frac{5x + 60}{7} = \frac{6x + 60}{8} \]

### Step 4: Solve the first two parts of the ratio
Let's solve the first part of the ratio:
\[ \frac{4x + 60}{6} = \frac{5x + 60}{7} \]
Cross-multiplying gives:
\[ 7(4x + 60) = 6(5x + 60) \]
Expanding both sides:
\[ 28x + 420 = 30x + 360 \]
Rearranging gives:
\[ 28x - 30x = 360 - 420 \]
\[ -2x = -60 \]
\[ x = 30 \]

### Step 5: Find the number of students in each class
Now that we have \(x\), we can find the number of students in each class:
- Class A: \(4x = 4 \times 30 = 120\)
- Class B: \(5x = 5 \times 30 = 150\)
- Class C: \(6x = 6 \times 30 = 180\)

### Step 6: Calculate the total number of students initially
Now, we can find the total number of students initially:
\[ \text{Total} = 120 + 150 + 180 = 450 \]

### Final Answer
The total number of students in the three classes initially is **450**.

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