CAT 2025Slot 1QAQuestion & Solution
Question
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
Options
110
88
121
99
Solution
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.