Quantitative Aptitude – Caselets
(For Uttarakhand Patwari / Lekhpal Entrance Exam)
Caselets are mini‑problems that mimic real‑world scenarios.
They are short, self‑contained, and usually involve finding a single unknown after setting up an equation (or a few).
Mastering caselets means you can read a story, spot the key data, and convert it into a solvable math problem in seconds.
1. Simple Explanation
| Step | What to Do | Why |
|---|---|---|
| 1. Read the whole paragraph | Make sure you understand the situation. | Mis‑reading often leads to wrong assumptions. |
| 2. Identify the knowns & unknowns | Highlight numbers, units, and the quantity to find. | Sets up the variables. |
| 3. Choose the right formula | Ratio, proportion, profit‑loss, mixture, time‑work, etc. | Each caselet type has a “go‑to” equation. |
| 4. Write down the equation | Translate the story into algebra. | Keeps the math separate from the words. |
| 5. Solve & double‑check | Compute, round if needed, verify logic. | Avoids arithmetic slip‑ups. |
2. Key Points to Remember
| Topic | Common Formula | Typical Red Flags |
|---|---|---|
| Ratio / Proportion | ( \frac{a}{b} = \frac{c}{d} ) | Forgetting to cross‑multiply, sign errors |
| Profit–Loss | (\text{Profit} = \text{Selling Price} - \text{Cost Price}) | Wrong sign for loss, misreading “profit per unit” |
| Mixture / Weighted Average | ( \frac{m_1x_1 + m_2x_2}{m_1+m_2} = \text{Desired Avg} ) | Mixing units (kg vs. liters), wrong direction of equation |
| Time & Work | ( \frac{1}{T} = \frac{1}{T_A} + \frac{1}{T_B} ) | Forgetting that rates add, confusion between “time taken” and “rate” |
| Simple Interest | ( SI = \frac{P \times R \times T}{100} ) | Mixing up principal vs. amount, forgetting to convert years/months |
| Distance–Speed–Time | ( D = S \times T ) | Mixing up direction (e.g., “towards” vs. “away”) |
| Compound Interest | ( A = P(1+\frac{r}{n})^{nt} ) | Wrong exponent (use (nt)), rounding early |
3. Worked Example (Mixture Problem)
Caselet
“A tea shop sells 200 l of mixed tea every day. The shopkeeper mixes 3 parts of strong tea (price ₹12 per litre) with 5 parts of soft tea (price ₹8 per litre). What is the cost price per litre of the mixed tea?”
Step‑by‑Step Solution
-
Understand the problem
- We need the average cost per litre of the blend.
- The blend ratio: 3 : 5 (strong : soft).
- Prices: ₹12 (strong), ₹8 (soft).
-
Set up weighted average
[ \text{Cost per litre} = \frac{(3 \times 12) + (5 \times 8)}{3+5} ] -
Compute numerator
[ 3 \times 12 = 36,\quad 5 \times 8 = 40,\quad 36 + 40 = 76 ] -
Compute denominator
[ 3 + 5 = 8 ] -
Divide
[ \frac{76}{8} = 9.5 ] -
Answer
The cost price of the mixed tea is ₹9.50 per litre.
Quick Check
- The result lies between ₹8 and ₹12 – correct.
- If we bought 200 l at ₹9.50, total cost = ₹1900, which is reasonable.
4. Quick Revision Summary
| Concept | Quick Tip | Example |
|---|---|---|
| Ratio | Cross‑multiply, solve for the missing part | ( \frac{a}{b} = \frac{c}{d}) → (ad = bc) |
| Profit–Loss | Profit = (SP - CP); Loss = (CP - SP) | If SP = ₹150, CP = ₹120 → Profit = ₹30 |
| Mixture | Weighted average = (\frac{\sum m_i x_i}{\sum m_i}) | 3 L at ₹10 + 2 L at ₹5 → Avg = ₹8 |
| Time‑Work | Work rate = 1/Time → Sum rates → New Time | A=5 h, B=10 h → Combined Time = ( \frac{1}{\frac{1}{5}+\frac{1}{10}} = 3.33,h) |
| Simple Interest | (SI = \frac{PRT}{100}) | ₹1000 at 5% for 3 yrs → ₹150 |
| Compound Interest | (A = P(1+r/n)^{nt}) | ₹500 at 8%/yr, compounded quarterly for 2 yrs → ₹600.48 |
Final Advice
- Read fast but carefully – underline every number.
- Label variables – e.g., (x) = unknown cost, (y) = units, etc.
- Check units – litre vs. kg vs. rupees.
- **Do a