CAT 2025

Slot 1

QA

Question & SolutionQues & Sol

The ratio of the number of students in the morning shift and afternoon shift of a school was 13 : 9. After 21 students moved from the morning shift to the afternoon shift, this ratio became 19 : 14. Next, some new students joined the morning and afternoon shifts in the ratio 3 : 8 and then the ratio of the umber of students in the morning shift and the afternoon shift became 5 : 4.

The number of new students who joined is:

1. Concept Used

• Topic: Ratio and Proportion — Setting up algebraic equations from ratio conditions across multiple stages
• Formula: $\frac{a + x}{b + y} = \frac{p}{q} \Rightarrow q(a+x) = p(b+y)$

2. Calculation

Let the initial number of students in the morning and afternoon shifts be $13k$ and $9k$ respectively, based on the given ratio $13:9$.

Stage 1 — Finding k: After 21 students move from morning to afternoon, the new counts become $13k - 21$ (morning) and $9k + 21$ (afternoon), and the ratio becomes $19:14$.

$\frac{13k - 21}{9k + 21} = \frac{19}{14}$

Cross-multiplying: $14(13k - 21) = 19(9k + 21)$

$182k - 294 = 171k + 399$

$11k = 693 \Rightarrow k = 63$

Stage 2 — Actual student counts after transfer:

Morning shift: $13(63) - 21 = 819 - 21 = 798$

Afternoon shift: $9(63) + 21 = 567 + 21 = 588$

Stage 3 — Finding new students: Let $3t$ and $8t$ be the number of new students joining morning and afternoon shifts respectively (ratio $3:8$).

After joining, the new ratio becomes $5:4$:

$\frac{798 + 3t}{588 + 8t} = \frac{5}{4}$

Cross-multiplying: $4(798 + 3t) = 5(588 + 8t)$

$3192 + 12t = 2940 + 40t$

$252 = 28t \Rightarrow t = 9$

Total new students joined $= 3t + 8t = 11t = 11 \times 9 = 99$

3. Solution

Answer = Option 4 ✅

The total number of new students who joined is 99.