Caselets: Quantitative Aptitude Notes for Karnataka Police Sub-Inspector (SI)

Karnataka Police Sub‑Inspector (SI) – Quantitative Aptitude

“Caselets” – A Comprehensive Guide

(MIT‑Level, First Principles, Advanced Reasoning)

What is a “Caselet”?
A caselet is a short, real‑world narrative that embeds one or more mathematical questions. It requires the examinee to parse the story, extract data, formulate equations, and arrive at a numerical answer. Caselets test reading comprehension, logical deduction, and mathematical fluency simultaneously.

1. Why Caselets Matter for the SI Exam

Feature	Significance
Real‑world context	Mimics the decision‑making environment of a police officer.
Time‑pressure	Requires quick extraction of relevant facts.
Interdisciplinary	Combines arithmetic, algebra, probability, and geometry.
Ambiguity	Sharpens interpretation skills—critical for investigative work.

2. Fundamental Principles

2.1 Textual Analysis

Identify the Problem Statement: The last sentence often contains the question; earlier sentences provide data.
Highlight Key Quantities: Numbers, ratios, units, and qualifiers (“at most”, “more than”, “exactly”).
Determine Variables: Assign symbols (e.g., x, y) to unknowns.
- Example: “The vehicle travelled 60 km at 30 km/h and the rest at 50 km/h.” →
  Let t₁ and t₂ be times.

2.2 Mathematical Modeling

Equations: Translate textual relationships into algebraic equations.
- Distance = speed × time.
- Proportion: a:b = c:d → a/b = c/d.
System of Equations: Often, two or more equations are needed.
Constraints: Non‑negativity, integer conditions, bounds.

2.3 Logical Deduction & Edge Cases

Uniqueness: Check if the solution is unique.
Multiple Solutions: Ask for “minimum” or “maximum” to disambiguate.
Rounding: Keep intermediate results exact until the final answer.
Units: Consistency is paramount.

3. Core Topics in Caselets

Topic	Representative Concepts	Typical Question Type
Arithmetic & Ratios	Sum, difference, product, common ratios	“If the ratio of boys to girls is 3:2…”
Algebra	Linear equations, simultaneous equations, inequalities	“Solve for `x` given `2x+3y=12` and `x-y=1`.”
Probability	Conditional probability, Bayes theorem, expected value	“Probability that a suspect is guilty given evidence.”
Combinatorics	Permutations, combinations, inclusion–exclusion	“Number of ways to form a committee.”
Geometry	Perimeter, area, volume, similarity	“Area of a trapezoid given bases and height.”
Data Interpretation	Tables, bar charts, pie charts	“Percentage of crimes solved in a month.”

4. Advanced Problem‑Solving Strategies

Strategy	When to Use	Example
Backward Reasoning	Start from the answer form and work backwards.	“If the final answer is an integer, what must the intermediate quantities be?”
Dimensional Analysis	Verify unit consistency.	“Speed × time = distance”
Graphical Interpretation	Plot variable relationships to spot patterns.	“Linear relationship between cost and time.”
Bounding Technique	Estimate upper/lower limits when exact values are hard.	“Maximum number of vehicles that can be inspected in 8 hrs.”
Paradox Elimination	Exclude impossible scenarios early.	“A case where a suspect has 0% probability of being guilty is impossible if evidence exists.”
Symmetry & Substitution	Reduce variables by exploiting symmetry.	“Let `x` be the number of days; then `y = total – x`.”

5. Edge Cases & Common Pitfalls

Pitfall	Remedy
Misreading “at most” vs “exactly”	Highlight qualifiers; treat “at most” as an inequality.
Ignoring Units	Write units next to each number; convert before calculation.
Rounding Early	Keep results in fractions or decimals until the final step.
Over‑counting in Combinations	Use inclusion–exclusion if objects are not distinct.
Assuming Independence in Probability	Verify independence; if not, use conditional probability.
Assuming Integer Solutions	Check if the problem states “whole numbers” or allows fractional values.

6. Sample Practice Problem Set

Problem 1 – Arithmetic & Ratios
A patrol car covers a distance of 120 km in 3 hrs. During the first 2 hrs, it travels at a constant speed v₁. In the last hour, it speeds up to v₂ such that the ratio v₁ : v₂ = 5 : 7.
Find the speeds v₁ and v₂.

Problem 2 – Probability
In a city, 70% of all crimes are thefts, 20% are assaults, and 10% are frauds. A forensic report shows a fingerprint that matches a suspect with probability 0.8 if the crime is a theft, 0.4 if assault, and 0.2 if fraud. The witness identifies the suspect with 0.9 confidence.
What is the probability that the suspect is guilty?