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

Caselets, Sets and Conditional Data Reasoning short notes for CAT: 3 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice questions.

Quick Revision: Key Formulas and Insights

Quick Revision

Core Formulas

Concept	Formula
2-Day Average	$Avg_{i} = \frac{Score _{i} + Score _{i - 1}}{2}$
Cumulative Time	$Cum_{k} = \sum_{i = 1}^{k} Time_{i}$
Individual Time	$Time_{k} = Cum_{k} - Cum_{k - 1}$

Key Insights

Bounds dictate possibilities
- A sum near the maximum possible value leaves very few valid combinations.
Moving averages create chains
- A 2-day average links Day $i$ to Day $i - 1$ . One known value fixes the rest.
Sequential means no parallel work
- You cannot start puzzle $k$ until puzzle $k - 1$ is complete.
Time gaps reveal categories
- If categories have fixed time costs, the individual time gap instantly identifies the category.

Common Pitfalls

Confusing 2-day average with overall average.
Forgetting score bounds when listing combinations.
Reading cumulative time as individual time.
Assuming puzzles can be solved out of order.

The Set Classification Master Checklist

1. Verify & Sum (The Foundation)

Are the sets mutually exclusive? (Exactly one category per entity).
Write the equation: $Total = Set_{A} + Set_{B}$ .

2. Decode Aggregates (The Bounds)

Map the Median to its exact positional index (e.g., 3rd out of 5).
Write the inequalities: Left of median $\leq M$ ; Right of median $\geq M$ .
Check for the word "distinct". If missing, allow repetitions.

3. Build the Matrix (The Interactions)

Rows = Source, Columns = Destination.
Diagonal = Same-category, Off-diagonal = Cross-category.
Fill in knowns and mark variables for unknowns.

4. Apply Conservation (The Solver)

$\sum Row Totals = \sum Column Totals$ (Outgoing = Incoming).
Use part-whole sums to find missing cells.

5. Optimize (The Final Push)

To maximize a variable $\to$ minimize all other unconstrained variables.
Ensure no directional double-counting when summing totals.

The Enrollment Set Master Checklist

1. Identify Set Type (The Foundation)

Are sets disjoint (mutually exclusive) or overlapping?
If overlapping, use Inclusion-Exclusion.

2. Translate Conditions (The Constraints)

"Must take both" $\to$ Group is in Intersection.
"Take only one" $\to$ Group is in Exclusive Regions.
"Cannot take both" $\to$ Intersection is zero for that group.

3. Algebraic Venn Method (The Solver)

Define variables for each distinct region (Only A, Only B, Both, etc.).
Write equations for Row/Column totals.
Include a variable for "Neither" if not explicitly excluded.

4. Precision Checks (The Traps)

Distinguish "At Least One" (Union) from "Exactly One".
Convert percentages to absolute numbers before comparing years.
Ensure $\sum Regions + Neither = Grand Total$ .

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 short notes in this unit

chapter

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

Caselets, Sets and Conditional Data Reasoning short notes for CAT: 3 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice q

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?

Free preview ends here

Login to view the complete short notes

Creating an account is free. You get the rest of this chapter, step-by-step solutions, and a study plan built around the topics you are actually weak at.

Why MastersUp

Personalised first. High quality throughout.

Most platforms hand everyone the same content. Here the content moves with your performance, topic by topic.

Built around you, not around a syllabus PDF

Every answer you give moves your topic-level intelligence rate. The next question, the next revision card and tomorrow's plan all change with it.

Revision that hits your weak spots

We only revise topics you have actually attempted and are still below the safe bar on — never the same chapter on repeat.

Questions calibrated to the real exam

Each question carries a measured toughness. You are served a rung above your current level, so practice keeps stretching you.

Notes written for recall, not for volume

Full lesson cards for first study, curated short-note cards for the last mile — with derivations, traps and exam patterns marked.

One place for everything

Notes, chapter practice, previous-year questions, test series and full-length papers — all feeding one picture of your preparation.

Honest progress

No vanity streaks. Progress here means chapters mastered and accuracy that held up on harder questions.

Unlock the whole course

Full notes and short notes, the complete question bank with worked solutions, mock tests, full-length papers, and an adaptive plan that rebuilds itself as you improve.

View all plans Start free