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 number of students in the morning shift and the afternoon shift became 5 : 4. The number of new students who joined is

CAT 2025 Slot 1 QA Question 1

Multiple choice (+3 / −1) · Arithmetic · Ratio, Proportion & Variation · Try it, then check the answer and solution below.

CAT 2025 Slot 1QAArithmetic • Ratio, Proportion & VariationModerate

The ratio of the number of students in the morning shift and afternoon shift of a school was 13 :

After 21 students moved from the morning shift to the afternoon shift, this ratio became 19 :

Next, some new students joined the morning and afternoon shifts in the ratio 3 : 8 and then the ratio of the number of students in the morning shift and the afternoon shift became 5 :

The number of new students who joined is

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Official Convenor Answer Key: C

Answer and solution

Answer: C) 99

📌 Core Concept

The problem involves ratios and requires setting up equations based on given ratios and solving for variables. The key is to use the concept of ratios and proportions to find the number of students.

🔢 Step-by-Step Solution

Initial Ratio:

Let the number of students in the morning shift be

13x

and in the afternoon shift be

9x

After 21 Students Move:

Morning students:

13x - 21

Afternoon students:

9x + 21

New ratio:

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

Cross-multiplying:

14(13x - 21) = 19(9x + 21)

Simplifying:

182x - 294 = 171x + 399

Solving for

x

11x = 693 \Rightarrow x = 63

Number of Students After Transfer:

Morning:

13x - 21 = 13(63) - 21 = 798

Afternoon:

9x + 21 = 9(63) + 21 = 588

New Students Join in Ratio 3:8:

Let new students be

3y

in morning and

8y

in afternoon.

New ratio:

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

Cross-multiplying:

4(798 + 3y) = 5(588 + 8y)

Simplifying:

3192 + 12y = 2940 + 40y

Solving for

y

252 = 28y \Rightarrow y = 9

Total New Students:

3y + 8y = 11y = 11(9) = 99

⚡ 30-Second Shortcut

Use the initial ratio to find the multiplier \(x

CAT 2025 Slot 1 QA Question 1

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