GATE CS 2021 Set 2 Q8 — General Aptitude

Let the initial number of students in the three classes be $3x$, $13x$, and $6x$, respectively, where $x$ is a positive constant.

The total number of students in the beginning is: $$\text{Total} = 3x + 13x + 6x = 22x$$ Since $x$ must be an integer, the total number of students must be a multiple of $22$.

When $18$ students are added to each class, the new strengths become $(3x + 18)$, $(13x + 18)$, and $(6x + 18)$. The new ratio is given as $15 : 35 : 21$.

Using the ratio of the first two classes: $$\frac{3x + 18}{13x + 18} = \frac{15}{35}$$ Simplifying the fraction $\frac{15}{35}$ to $\frac{3}{7}$: $$\frac{3x + 18}{13x + 18} = \frac{3}{7}$$ Cross-multiplying to solve for $x$: $$7(3x + 18) = 3(13x + 18)$$ $$21x + 126 = 39x + 54$$ $$39x - 21x = 126 - 54$$ $$18x = 72 \implies x = 4$$

We can verify this value of $x$ with the ratio of the second and third classes: $$\frac{13(4) + 18}{6(4) + 18} = \frac{52 + 18}{24 + 18} = \frac{70}{42} = \frac{35}{21}$$ This matches the given ratio perfectly.

Therefore, the total number of students in the beginning was: $$\text{Total} = 22x = 22 \times 4 = 88$$