CAT 2025 Slot1 Quant: The ratio of the number of students in the morning shift and afternoon shift

Step 1: Find Initial Number of Students
Let Morning $(M) = 13x$ and Afternoon $(A) = 9x$.
After 21 move: $\frac{13x - 21}{9x + 21} = \frac{19}{14}$
$14(13x - 21) = 19(9x + 21) \implies 182x - 294 = 171x + 399$
$11x = 693 \implies x = 63$.
Students after move: $M = 13(63)-21 = 798$, $A = 9(63)+21 = 588$.

Step 2: Add New Students
Let new students be $3k$ and $8k$.
$\frac{798 + 3k}{588 + 8k} = \frac{5}{4}$
$4(798 + 3k) = 5(588 + 8k) \implies 3192 + 12k = 2940 + 40k$
$28k = 252 \implies k = 9$.

Step 3: Calculate Total New Students
Total new students $= 3k + 8k = 11k$.
Total $= 11 \times 9 = \mathbf{99}$.