Caselets, Sets and Conditional Data Reasoning Notes for CAT: Concepts, Formulas, Worked Examples & Practice

Caselets, Sets and Conditional Data Reasoning notes for CAT: 32 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice questions.

Chapter Roadmap: Caselets, Sets and Conditional Data Reasoning

Chapter Journey

This chapter tests your ability to extract hidden constraints from dense paragraphs and organize them logically.

Puzzle Solving and Draw Competitions

Decode random scores, moving averages, and sequential progress.

Operator and Registration Set Classification

Classify entities across overlapping sets and binary conditions.

Subscriber and Course Enrollment Sets

Analyze multi-category enrollments and conditional exclusions.

By the end, you will master translating complex text into clean, solvable grids.

Puzzle Solving and Draw Competitions

Mastering the art of extracting hidden numerical constraints from seemingly random events and sequential tasks.

What you will learn here:

01 Decode random draw scores and bounded sums

02 Handle moving averages and consecutive day links

03 Track sequential puzzle progress and time bottlenecks

04 Optimize logic grids and eliminate impossible cases

Anatomy of a Draw Competition

Core Structure

Element	Description	Example
Participants	Entities competing	Akhil, Bimal, Chatur
Timeframe	Duration of event	5 days
Score Bounds	Min and max daily score	1 to 9
Total Score	Sum of all daily scores	Sum of Day 1 to Day 5

The Power of Bounds

The most critical feature is the bounded score range.

If the maximum score is 9, and a participant's total score over 2 days is 17, the only possible daily scores are 8 and 9.

Rule: Bounded sums drastically reduce the number of valid combinations. Always write down the min and max possible sums first.

The 2-Day Average Trap

Definition

The 2-day average on Day $i$ (for $i > 1$ ) is defined as:

2-Day Average_{i} = \frac{Score _{i} + Score _{i - 1}}{2}

Why It Matters

This condition strictly links two consecutive days. It does not average all days up to that point.

Immediate Deductions

If the 2-day average on Day 3 is 6, then:

Score_{3} + Score_{2} = 12

This creates a chain of constraints:

Day 1 and Day 2 are linked.
Day 2 and Day 3 are linked.
Day 3 and Day 4 are linked.

Key Insight: A known score on Day 1 instantly ripples through the entire timeline, fixing or restricting all subsequent days.

Caselets, Sets and Conditional Data Reasoning: Solved Questions with Step-by-Step Explanations (2 Problems)

Question 1 · Data Interpretation and Logical Reasoning MCQ

A business incubator has 250 startups, classified into two sectors: 'DeepTech' (150 startups) and 'E-Commerce' (100 startups). Each startup is registered for at least one of three government grant schemes: Alpha, Beta, and Gamma.

For the E-Commerce sector:

The total number of registrations across the three schemes is 220 (with Alpha: 90, Beta: 80, Gamma: 50).
The number of startups registered for exactly two schemes is equal to the number registered for all three schemes.

For the DeepTech sector:

The total number of registrations across the three schemes is 230 (with Alpha: 100, Beta: 80, Gamma: 50).

Global Condition:

The total number of startups registered for ALL THREE schemes across both sectors combined is exactly 65.

What is the maximum possible number of DeepTech startups registered for ONLY scheme Alpha?

A.
65
B.
70
C.
75
D.
80

Correct Answer:

Step-by-Step Solution

Key idea: This is a multi-set optimization problem linked by a global constraint. We must use the Conservation of Sums identity ( $S - N = E_{2} + 2 E_{3}$ ) to find the exact parameters of each sector before applying bounding techniques to maximize a specific region.

Step 1: Analyze the E-Commerce sector.

Total startups $N_{E C} = 100$ . Sum of registrations $S_{E C} = 220$ .

Using the identity: $S_{E C} - N_{E C} = E_{2, E C} + 2 E_{3, E C} ⟹ 220 - 100 = 120$ .

We are given $E_{2, E C} = E_{3, E C}$ . Substituting this gives $3 E_{3, E C} = 120 ⟹ E_{3, E C} = 40$ .

Step 2: Use the global condition to find DeepTech parameters.

Total ALL THREE across both sectors = $E_{3, E C} + E_{3, D T} = 65$ .

Since $E_{3, E C} = 40$ , we have $E_{3, D T} = 65 - 40 = 25$ .

For DeepTech, $N_{D T} = 150$ and $S_{D T} = 230$ .

$S_{D T} - N_{D T} = E_{2, D T} + 2 E_{3, D T} ⟹ 230 - 150 = 80$ .

Substituting $E_{3, D T} = 25$ : $E_{2, D T} + 50 = 80 ⟹ E_{2, D T} = 30$ .

The number of DeepTech startups in exactly one scheme is $E_{1, D T} = N_{D T} - E_{2, D T} - E_{3, D T} = 150 - 30 - 25 = 95$ .

Step 3: Maximize DeepTech ONLY Alpha ( $X$ ).

Let $A, B, C$ be the exactly-two regions for DeepTech (Alpha-Beta, Beta-Gamma, Gamma-Alpha).

We know $A + B + C = E_{2, D T} = 30$ .

The total for Alpha is $X + A + C + E_{3, D T} = 100 ⟹ X + A + C = 75$ .

To maximize $X$ , we must minimize $(A + C)$ . Since $A + C = 30 - B$ , we must maximize $B$ .

The total for Gamma is $Z + B + C + E_{3, D T} = 50 ⟹ Z + B + C = 25$ .

Since $Z \geq 0$ and $C \geq 0$ , the maximum possible value for $B$ is 25.

If $B = 25$ , then $C = 0$ and $Z = 0$ .

This gives $A + C = 30 - 25 = 5$ .

Substituting back into the Alpha equation: $X + 5 = 75 ⟹ X = 70$ .

(Checking Beta: $Y + A + B = 80 - 25 = 55 ⟹ Y + 5 + 25 = 55 ⟹ Y = 25$ . Total exactly one: $X + Y + Z = 70 + 25 + 0 = 95$ , which matches $E_{1, D T}$ . All constraints are satisfied).

Answer: 70

Question 2 · Data Interpretation and Logical Reasoning NAT

A tech company has three specialized departments: Cloud (C), Security (S), and AI (A). Every employee is assigned to at least one department. The following data is known about the workforce:

The total number of employees in Cloud is $120$ .
The total number of employees in Security is $100$ .
The total number of employees in AI is $80$ .
The number of employees working in exactly two departments is $x$ .
The number of employees working in all three departments is $y$ .

If it is known that $x + y = 60$ and $x \geq 2 y$ , what is the sum of all possible integer values for the total number of employees in the company?

Correct Answer:

4830

Step-by-Step Solution

Key idea: This is a reverse_engineering problem combining Venn algebra with inequality constraints and arithmetic progression sums.

Step 1: Establish the Sum of Sets and Union relationship.

Sum of sets = $120 + 100 + 80 = 300$ .

Let $N$ be the total number of employees. Since everyone is in at least one department, Union = $N$ .

Formula: $\sum ∣ S e t s ∣ = E_{1} + 2 E_{2} + 3 E_{3}$ .

Also, Union = $E_{1} + E_{2} + E_{3} ⟹ E_{1} = N - E_{2} - E_{3}$ .

Substitute $E_{1}$ : $300 = (N - E_{2} - E_{3}) + 2 E_{2} + 3 E_{3} ⟹ 300 = N + E_{2} + 2 E_{3}$ .

So, $N = 300 - E_{2} - 2 E_{3}$ .

Step 2: Substitute the given constraints.

We are given $E_{2} = x$ and $E_{3} = y$ .

$N = 300 - x - 2 y$ .

We are also given $x + y = 60 ⟹ x = 60 - y$ .

Substitute $x$ : $N = 300 - (60 - y) - 2 y = 240 - y$ .

Step 3: Determine the valid range for $y$ .

Constraint 1: $x \geq 2 y ⟹ (60 - y) \geq 2 y ⟹ 60 \geq 3 y ⟹ y \leq 20$ .

Constraint 2: $y \geq 0$ (Region counts cannot be negative).

Constraint 3: $x \geq 0 ⟹ 60 - y \geq 0 ⟹ y \leq 60$ .

Combining these, the valid integer range for $y$ is $0 \leq y \leq 20$ .

Step 4: Calculate the sum of all possible values for $N$ .

Since $N = 240 - y$ , as $y$ ranges from $0$ to $20$ , $N$ ranges from $240$ down to $220$ .

This is an arithmetic progression with $21$ terms: $240, 239, 238, \dots, 220$ .

Sum = $\frac{Number of terms}{2} \times (First term + Last term)$

Sum = $\frac{21}{2} \times (240 + 220) = \frac{21}{2} \times 460 = 21 \times 230 = 4830$ .

Answer: 4830

More notes in this unit

chapter

Caselets, Sets and Conditional Data Reasoning Notes for CAT: Concepts, Formulas, Worked Examples & Practice

Caselets, Sets and Conditional Data Reasoning notes for CAT: 32 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice questi

A question from this chapter

Question 1

A business incubator has 250 startups, classified into two sectors: 'DeepTech' (150 startups) and 'E-Commerce' (100 startups). Each startup is registered for at least one of three government grant schemes: Alpha, Beta, and Gamma. For the E-Commerce sector: The total number of registrations across the three schemes is 220 (with Alpha: 90, Beta: 80, Gamma: 50). The number of startups registered for exactly two schemes is equal to the number registered for all three schemes. For the DeepTech sector: The total number of registrations across the three schemes is 230 (with Alpha: 100, Beta: 80, Gamma: 50). Global Condition: The total number of startups registered for ALL THREE schemes across both sectors combined is exactly 65. What is the maximum possible number of DeepTech startups registered for ONLY scheme Alpha?

Question 2

A tech company has three specialized departments: Cloud (C), Security (S), and AI (A). Every employee is assigned to at least one department. The following data is known about the workforce: The total number of employees in Cloud is 120 . The total number of employees in Security is 100 . The total number of employees in AI is 80 . The number of employees working in exactly two departments is x . The number of employees working in all three departments is y . If it is known that x + y = 60 and x \geq 2 y, what is the sum of all possible integer values for the total number of employees in the company?

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