Quantitative Aptitude – Caselets
Advanced, MIT‑Level Study Guide (First Principles → Edge Cases)
What is a “Caselet”?
A caselet is a short, self‑contained problem (typically 4–10 lines) that tests a single quantitative concept. Unlike long‑form word problems, caselets focus on precision and speed. They often appear in competitive exams (e.g., JEE‑Advanced, GRE, GMAT, UPSC, and aptitude tests for certifications like RHCSA).
1. Foundations: Why Caselets Matter
| Property | Why it matters |
|---|---|
| Conciseness | Requires you to parse information quickly. |
| Concept‑centric | Each caselet focuses on one mathematical skill (ratio, probability, algebra, etc.). |
| Precision | A single mis‑read can lead to an incorrect answer. |
| Time pressure | Typically 3–5 minutes per caselet (even less in timed exams). |
2. Classification of Caselets
| Category | Typical Topics | Sample Question |
|---|---|---|
| Ratios & Proportions | Direct & inverse proportionality, compound ratios | “The ratio of boys to girls in a class is 3:4. If 6 more girls join, what is the new ratio?” |
| Algebra & Equations | Linear/quadratic equations, simultaneous equations | “If x + y = 10 and xy = 21, find x and y.” |
| Arithmetic Sequences | First term, common difference, nth term, sum | “The sequence 2, 5, 8… has a sum of 275. What is the last term?” |
| Geometry (Plane) | Area, perimeter, Pythagoras, circles | “Triangle ABC has AB = 10, BC = 6, ∠ABC = 90°. Find area of ABC.” |
| Geometry (Solid) | Volume, surface area, sphere properties | “A cube’s diagonal is 12 cm. What is its surface area?” |
| Probability | Simple, conditional, independent events | “Two dice are rolled. What is the probability that the sum is 7?” |
| Percentages & Discount | Increase/decrease, compound interest | “A book costs Rs. 200 and is sold at a 25 % discount. What is the selling price?” |
| Time & Speed | Work, motion, relative speed | “Two workers start at the same time. One completes a task in 8 hrs, the other in 12 hrs. How long will they take together?” |
| Data Interpretation | Tables, graphs, averages | “From the table below, find the mean of the data set.” |
Tip: 70 % of caselets revolve around ratio & proportion and simple algebra.
3. General Strategy for Solving Caselets
| Step | Action | Rationale |
|---|---|---|
| 1. Read Carefully | Read whole problem once, underline numbers & keywords. | Prevents misinterpretation. |
| 2. Identify the Concept | Match keywords (ratio, sum, product, probability, etc.). | Directs you to the right formula. |
| 3. Extract Variables | Assign variables to unknowns (e.g., let x be the missing quantity). | Keeps reasoning linear. |
| 4. Write the Equation(s) | Translate words into mathematical expressions. | First‑principle approach. |
| 5. Solve Algebraically | Simplify, isolate, compute. | Avoids mistakes due to mental arithmetic. |
| 6. Check Units & Reasonableness | Verify result makes sense (e.g., probability ∈ [0,1]). | Detects algebraic errors. |
| 7. Double‑Check | Quick mental check or plug back into the problem. | Final safety net. |
Edge‑Case Awareness
- Zero / Negative Values: Some problems set constraints (e.g., “positive integers only”).
- Fractional/Decimal Handling: Convert to common denominators early.
- Equivalence of Ratios: Simplify before substituting.
4. Deep Theoretical Insights
4.1 Ratios & Proportions
- Theorem (Proportionality):
If (A : B = C : D), then (\frac{A}{B} = \frac{C}{D}). - Compound Ratio:
((a:b)(c:d) = (ac):(bd)). - Inverse Proportion:
If (x \propto \frac{1}{y}), then (xy = k) (constant).
Proof Sketch:
From (x = \frac{k}{y}) → (xy = k).
Therefore, if (y) doubles, (x) halves, preserving the product.
4.2 Algebraic Equations
- Simultaneous Linear Equations
Solve via substitution, elimination, or matrix inversion.
Matrix Approach:
(\begin{bmatrix} a & b \ c & d \end{bmatrix} \begin{bmatrix} x \ y \end{bmatrix} = \begin{bmatrix} e \ f \end{bmatrix})
→ ( \begin{bmatrix} x \ y \end{bmatrix} = \begin{bmatrix} a & b \ c & d \end{bmatrix}^{-1} \begin{bmatrix} e \ f \end{bmatrix})
4.3 Geometric Formulas
- Pythagorean Theorem: (a^2 + b^2 = c^2).
- Circle Area/Perimeter: