Lakshadweep PSC – Quantitative Aptitude
Topic: Percentages
Goal – Master the fundamentals of percentages, solve typical PSC-style problems in 5–10 minutes, and avoid common pitfalls.
1. What is a Percentage?
A percentage is a way of expressing a number as a fraction of 100.
- Formula
[ \text{Percent} = \frac{\text{Part}}{\text{Whole}} \times 100 ] - Example
25 % of 200 = ( \frac{25}{100} \times 200 = 50 ).
Tip – Whenever you see “percent of”, treat it as “(given number) × (percent/100)”.
2. Key Concepts & Formulas
| Concept | Formula | Notes |
|---|---|---|
| Percentage of a number | ( P% \text{ of } N = \frac{P}{100} \times N ) | Replace (P) and (N) with actual numbers. |
| Percentage increase | New Value = Original × ( (1 + \frac{I}{100}) ) | (I) = increase percent. |
| Percentage decrease | New Value = Original × ( (1 - \frac{D}{100}) ) | (D) = decrease percent. |
| Find original when decreased/increased | Original = New Value ÷ ( (1 \pm \frac{p}{100}) ) | Use “+” for increase, “–” for decrease. |
| Rate of change (from old to new) | ( \frac{New - Old}{Old} \times 100% ) | Gives increase or decrease? |
| Compound percentage | After (n) successive changes: ( \text{Final} = \text{Initial} \times \prod_{i=1}^{n} (1 \pm \frac{p_i}{100}) ) | Useful for multiple tax or discount layers. |
| Reciprocal percentage | If (x%) of (y) equals (z), then (y = \frac{z \times 100}{x}) | Classic “part–whole” reversal. |
3. Common Pitfalls
| Mistake | Why it Happens | Correct Approach |
|---|---|---|
| Treating percent as fraction without multiplying by 100 | Forgetting that 1 % = 0.01 | Always divide by 100 first. |
| Adding percentages to numbers directly (e.g., 100 + 10 %) | Misreading “increase by 10 % of 100” | Compute (100 \times 1.10 = 110). |
| Using “+” for a decrease or “–” for an increase | Sign confusion | Remember: increase → multiply by (>1); decrease → multiply by (<1). |
| Ignoring the order of operations in compound problems | Mistaking 10 % + 20 % → 30 % | Treat each change sequentially: ( \times 1.10 \times 1.20 ). |
| Assuming “percentage change” is the same as “percent of” | Different contexts | Clarify whether the question asks for a part of a whole or a change in value. |
4. Worked Example (Lakshadweep PSC Style)
Problem
A shopkeeper sells a watch for ₹2500. The selling price includes a 12 % profit on the cost price.
- Find the cost price.
- If the shopkeeper next offers a 15 % discount on the selling price, what will be the final price?
Step 1 – Find Cost Price
Let CP be the cost price.
Selling Price (SP) = CP + 12 % of CP
[
2500 = CP \times (1 + 0.12) = CP \times 1.12
]
[
CP = \frac{2500}{1.12} = 2232.14 \text{ (approx.)}
]
Answer: CP ≈ ₹2232.14
Step 2 – Apply 15 % Discount on SP
Discount = 15 % of SP = (0.15 \times 2500 = 375)
Final Price = SP – Discount = (2500 - 375 = 2125)
Answer: Final price = ₹2125
Quick Check
- Profit margin = 12 % of ₹2232.14 ≈ ₹267.86 → SP = 2232.14 + 267.86 = 2500 ✔
- Discount on SP = 15 % of 2500 = 375 ✔
5. Quick Revision Summary
| Topic | Formula | Quick Tip |
|---|---|---|
| Percent of a number | ( P% \times N = \frac{P}{100} \times N ) | Convert % → decimal first. |
| Increase/Decrease | New = Old × (1 ± (p/100)) | “+” for increase, “–” for decrease. |
| Reverse (find whole) | Whole = Part × 100 ÷ Percent | Example: 40 % of X = 80 → X = 200. |
| Compound changes | Final = Initial × Π(1 ± (p_i/100)) | Multiply each factor sequentially. |
| Percentage change between two values | (\frac{New - Old}{Old} × 100%) | Positive → increase, negative → decrease. |
| **Common |