Given:
The ratio of the number of students in 3 classes is 2:3:4.
When 12 students are added to each class, the new ratio becomes 8:11:14.

To find:
The total number of students in the three classes in the beginning.

Solution:

Step 1: Setting up the equation:
Let's assume the initial number of students in the three classes are 2x, 3x, and 4x respectively.

According to the given ratio, we can write the equation as:
2x + 3x + 4x = Total number of students in the three classes

Step 2: Solving the equation:
To solve the equation, we need to find the value of x.

Step 3: Calculating the total number of students:
Once we have the value of x, we can substitute it back into the equation to find the total number of students in the three classes.

Step 4: Verifying the solution:
We can verify the solution by checking if the new ratio holds true after adding 12 students to each class.

Step 5: Detailed explanation:
To solve the equation, let's follow the steps mentioned above:

Step 2:
2x + 3x + 4x = Total number of students in the three classes

Adding like terms, we get:
9x = Total number of students in the three classes

Step 3:
To find the value of x, we can use the new ratio when 12 students are added to each class. The new ratio is given as 8:11:14.

Therefore, the new number of students in the three classes becomes:
2x + 12, 3x + 12, and 4x + 12.

According to the new ratio, we can write the equation as:
(2x + 12) : (3x + 12) : (4x + 12) = 8 : 11 : 14

Cross-multiplying the ratio, we get:
8(3x + 12) = 11(2x + 12) = 14(4x + 12)

Simplifying the equation, we get:
24x + 96 = 22x + 132 = 56x + 168

Step 4:
Solving the equation, we find the value of x as 6.

Step 5:
Substituting the value of x back into the equation, we get:
9x = Total number of students in the three classes
9(6) = 54

Therefore, the total number of students in the three classes in the beginning was 54.

Step 6:
Verifying the solution:
According to the new ratio, the new number of students in the three classes after adding 12 students becomes:
2x + 12 = 2(6) + 12 = 24
3x + 12 = 3(6) + 12 = 30
4x + 12 = 4(6) + 12 = 36

The new ratio is 24:30:36, which simplifies to