Caselets – Quantitative Aptitude
(MIT‑Level Study Notes)
A “caselet” is a very short, self‑contained problem (often 1–3 sentences) that tests quick reasoning, arithmetic, combinatorics, or algebraic manipulation. The goal is to answer in 30–60 seconds, so the solution must be systematic, not ad‑hoc.
1. What Makes a Caselet Hard?
| Feature | Why It Matters | Typical Trick |
|---|---|---|
| Ambiguity | Words like “at least”, “more than”, “exactly” change the answer. | Treat each wording as an interval or a strict inequality. |
| Implicit Constraints | Not all numbers are given; you must infer them. | Look for hidden relationships (e.g., “total” vs “difference”). |
| Large Numbers | Prevents brute‑force enumeration. | Reduce to modular arithmetic or use combinatorial identities. |
| Multiple Sub‑Questions | Each sub‑question may build on the previous. | Solve sequentially, storing intermediate results. |
| Edge Cases | “What if” scenarios (e.g., zero, negative, parity). | Consider parity, divisibility, or extremal cases early. |
2. Fundamental Tools (First Principles)
| Tool | Core Idea | Typical Use‑Case in Caselets |
|---|---|---|
| Direct Counting | Count objects that satisfy a property directly. | “How many ways can 3 friends sit?” |
| Inclusion–Exclusion | Add counts of individual events, subtract overlaps. | Counting arrangements with forbidden positions. |
| Pigeonhole Principle | If more items than boxes, at least one box holds ≥ ⌈n/k⌉. | “At least 2 girls must sit together” |
| Probability (Basic) | Probability = favorable / total. | “If 3 balls are drawn, probability of at least one red.” |
| Modular Arithmetic | Work with remainders to simplify large numbers. | “Sum of first 100 primes mod 5.” |
| Algebraic Manipulation | Solve linear/ quadratic equations quickly. | “If 2x+3y=12 and x−y=4, find x.” |
| Generating Functions | Encode combinatorial counts as power series. | “Number of ways to distribute 5 identical apples to 3 kids.” |
| Integer Programming (light) | Set up inequalities and solve by bounds. | “Minimize cost subject to constraints.” |
3. Common Caselet Types
| Type | Typical Question | Key Insight |
|---|---|---|
| Number Sequences | “What is the 7th term of 2, 4, 8, 16, …?” | Identify pattern (geometric, arithmetic, factorial). |
| Time & Speed | “Two trains cover 300 km in 5 hrs and 3 hrs respectively; what is their relative speed?” | Use (v = d/t) and relative velocity formula. |
| Ratios & Proportions | “If A:B = 3:5 and B:C = 4:7, find A:C.” | Multiply ratios: A:C = 3×4 : 5×7. |
| Combinatorial Arrangements | “In how many ways can 5 people sit with 2 adjacent?” | Treat adjacent pair as a single block. |
| Probability with Replacement | “Draw 2 balls from a bag of 10 red, 5 blue. Probability both are red?” | P = (10/15) * (9/14). |
| Logical Deduction | “Only two of the statements are true. Which?” | Exhaustive check of combinations. |
| Graph/Network | “Shortest path between A and C via B takes 8 min.” | Basic shortest‑path reasoning. |
| Financial/Interest | “$1000 grows to $1100 in 2 yrs at compound interest. Find r.” | Solve (1100 = 1000(1+r)^2). |
4. Systematic Problem‑Solving Approach
-
Read Carefully
- Highlight exactly, at least, maximum, minimum.
- Identify all variables and constraints.
-
Translate Words to Equations / Inequalities
- Write a short algebraic representation.
- Example: “At least 3 of the 6 students are girls” → (G \ge 3).
-
Choose the Right Tool
- Direct counting → simple “count” question.
- Inclusion–Exclusion → overlapping restrictions.
- Pigeonhole → “at least” or “at most” large groups.
-
Simplify Early
- Reduce fractions, cancel common factors.
- Use parity or modular checks to rule out impossible cases.
-
Compute
- Keep numbers small, use mental arithmetic wherever possible.
- Keep track of units (e.g., minutes, dollars).
-
Verify
- Check if the answer satisfies all constraints.
- Quick sanity check (e.g., answer cannot exceed total possibilities).
5. Edge Cases & Common Pitfalls
| Pitfall | What It Looks Like | How to Avoid |
|---|---|---|
| Misreading “at most” vs “at least” | Assuming “at most 2” means “≤ 2” but forgetting “0” is allowed. | Map to inequalities: “≤” or “≥”. |
| Overcounting in Inclusion–Exclusion | Adding too many overlap terms. | Count how many items satisfy each combination; stop when higher‑order overlaps are impossible. |
| Ignoring Zero Cases | Forgetting that choosing 0 objects is a valid option. | Include the “0” term in combin |