Kerala TET – Quantitative Aptitude
“Caselets”
(A concise guide for the Teacher Eligibility Test – an MIT‑level, first‑principles view)
Caselet: a short, self‑contained numerical problem (often 1–3 sentences) that mixes arithmetic, algebra, number theory, probability, or basic geometry.
The aim is to test quick reasoning, not mere plug‑and‑chug.
Why a separate section?
– They test applied reasoning rather than rote formulas.
– They often involve “hidden” constraints or non‑obvious constraints.
– Edge cases (trick wording) are frequent.
Table of Contents
| Section | Topic | Key Take‑aways |
|---|---|---|
| 1 | Introduction & Features | What makes a good caselet |
| 2 | Systematic Problem‑Solving Technique | Step‑by‑step framework |
| 3 | Core Mathematical Tools | Equations, inequalities, Diophantine, modular, combinatorics |
| 4 | Advanced Theory | From first principles to “deep” reasoning |
| 5 | Edge Cases & Common Traps | Ambiguity, rounding, invalid data |
| 6 | Challenging Practice Problem | Full solution |
| 7 | Quick‑Reference Cheat‑Sheet | One‑page summary |
| 8 | Mini‑Quiz | Test your grasp |
Tip: Always draw a diagram or table first. In 70 % of caselets, a visual representation turns the problem into a familiar pattern.
1. Introduction & Features
| Feature | Explanation | Example |
|---|---|---|
| Compactness | 1–3 sentences, ~20–35 words. | “A teacher has 7 pens and 9 pencils. If she gives 3 pens to a student, how many pens does she have left?” |
| Implicit Constraints | Numbers may hide relationships (e.g., “three times as many”). | “A class of 30 students with a 2:1 ratio of boys to girls.” |
| Multiple Steps | From extraction to algebraic manipulation. | “Find the sum of the first 10 even numbers that are multiples of 3.” |
| Trick Words | “At least”, “at most”, “exactly” change the solution space dramatically. | “At least 5 students passed” vs “exactly 5 students passed.” |
2. Systematic Problem‑Solving Technique
Three‑Phase Framework
- Parse & Diagram
- Mathematize
- Solve & Verify
2.1 Parse & Diagram
-
Identify the Question
What is being asked?
Is it a sum, difference, ratio, probability? -
Extract Quantities
List all numbers explicitly mentioned. -
Create a Representation
- Table (rows for categories, columns for quantities).
- Venn Diagram (for intersections).
- Equation System (for unknowns).
-
Spot Hidden Relations
- Ratio → Proportionality.
- “Three times more” → Additive vs multiplicative.
- “At least” → Inequality.
2.2 Mathematically Model
- Translate words into symbols.
- Write equations or inequalities.
- Reduce to simplest form.
2.3 Solve & Verify
- Solve algebraically or numerically.
- Check if all constraints satisfied.
- Beware of extraneous solutions (especially when squaring or multiplying by zero).
3. Core Mathematical Tools
| Tool | First Principles | Typical Use in Caselets | Edge Cases |
|---|---|---|---|
| Linear Equations | Solving (ax+by=c) from balancing two sides of a story | “Two friends buy 7 pens and 4 pencils for $30” | Division by zero, multiple solutions |
| Quadratic Identities | ( (x+y)^2 = x^2+2xy+y^2 ) | “Sum of squares of two consecutive numbers is 85” | Discriminant < 0 |
| Inequalities | (a<b) → strict, (a\le b) → non‑strict | “At least 3 students passed” | Misinterpreting “at least” vs “exactly” |
| Diophantine (Integer) Equations | Solve (ax+by=c) for integers | “Find integer ages adding to 30” | No integer solutions |
| Modular Arithmetic | (a \equiv b \pmod{m}) | “Number divisible by 4 and leaves remainder 3 when divided by 5” | Ambiguous modulus |
| Combinatorics | ( \binom{n}{k} ) or permutations | “Ways to seat 4 teachers in 4 chairs” | Overcounting due to symmetry |
| Probability | (P(E)=\frac{\text{favorable}}{\text{total}}) | “Probability of drawing a red ball” | Dependent vs independent events |
| Basic Geometry | Area/Perimeter formulas | “Area of a right triangle with legs 3 and 4” | Misreading “perimeter” vs “area” |
Key Idea: Each tool has a first‑principles justification; knowing this prevents misapplication.
4. Advanced Theory – From First Principles
4.1 Linear Systems
Fundamental Theorem: A system of (n) linear equations with (n\