Question

The number of students in three classes is in the ratio \(3:13:6\). If 18 students are added to each class, the ratio changes to \(15:35:21\).
The total number of students in all the three classes in the beginning was:

• A. 22
• B. 66
• C. 88
• D. 110

Hint:

In ratio problems involving addition, always introduce a variable first and use the changed ratio to form equations.

Answer 1

The Correct Option is C

Step 1: Assume the initial numbers of students.
Let the number of students in the three classes initially be \[ 3x,\; 13x,\; 6x. \]

Step 2: Add 18 students to each class.
After adding 18 students to each class, the numbers become \[ 3x+18,\; 13x+18,\; 6x+18. \]

Step 3: Use the new ratio.
According to the question, \[ \frac{3x+18}{15} = \frac{13x+18}{35} = \frac{6x+18}{21}. \]

Step 4: Solve using the first two terms.
\[ \frac{3x+18}{15} = \frac{13x+18}{35} \] \[ 35(3x+18) = 15(13x+18) \] \[ 105x + 630 = 195x + 270 \] \[ 360 = 90x $\Rightarrow$ x = 4. \]

Step 5: Find the total number of students initially.
\[ 3x + 13x + 6x = 22x = 22 \times 4 = 88. \] % Final Answer

Final Answer: \[ \boxed{88} \]