The number of students in three classes are in the ratio 2 : 3:5. If 40 students are increased in each class, the ratio changes to 4:5:7. Originally, the total number of students was :

Options

A. 100
B. 180
C. 200
D. 400

Answer

To solve the problem step by step, let's denote the number of students in the three classes as follows:

Let the number of students in the three classes be represented as:
- Class A: 2x
- Class B: 3x
- Class C: 5x

### Step 1: Set up the original ratio
The original number of students in the three classes is in the ratio 2:3:5. Therefore, we can express the number of students in each class in terms of a variable \( x \):
- Number of students in Class A = \( 2x \)
- Number of students in Class B = \( 3x \)
- Number of students in Class C = \( 5x \)

### Step 2: Increase the number of students
According to the problem, 40 students are added to each class. Thus, the new number of students in each class becomes:
- Class A: \( 2x + 40 \)
- Class B: \( 3x + 40 \)
- Class C: \( 5x + 40 \)

### Step 3: Set up the new ratio
After adding 40 students to each class, the new ratio of students is given as 4:5:7. Therefore, we can set up the following equation based on the new ratio:
\[ \frac{2x + 40}{3x + 40} = \frac{4}{5} \]
\[ \frac{3x + 40}{5x + 40} = \frac{5}{7} \]

### Step 4: Solve the first equation
Cross-multiplying the first equation:
\[ 5(2x + 40) = 4(3x + 40) \]
Expanding both sides:
\[ 10x + 200 = 12x + 160 \]
Rearranging gives:
\[ 200 - 160 = 12x - 10x \]
\[ 40 = 2x \]
Thus, we find:
\[ x = 20 \]

### Step 5: Calculate the original number of students
Now that we have the value of \( x \), we can calculate the original number of students in each class:
- Class A: \( 2x = 2(20) = 40 \)
- Class B: \( 3x = 3(20) = 60 \)
- Class C: \( 5x = 5(20) = 100 \)

### Step 6: Find the total number of students
Now, we can find the total number of students originally:
\[ \text{Total} = 40 + 60 + 100 = 200 \]

### Final Answer
The original total number of students was **200**.

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