📊 Percentages – XAT Quant Study Notes
Percentages are a way of expressing a part of a whole as a fraction of 100.
They are ubiquitous in XAT – from basic percent calculations to complex percent‑change, percent‑on‑percent, compound interest, and “percent of a percent” problems.
1. Simple Explanation
| Concept | What it means | Example |
|---|---|---|
| Percent | “Per hundred” – a number divided by 100. | 25 % = 25/100 = 0.25 |
| Percent of a number | value × (percent/100) |
30 % of 200 = 200 × 0.30 = 60 |
| Percent change | (new – old) / old × 100 |
Price rises from ₹50 to ₹60 → (60‑50)/50×100 = 20 % increase |
| Reverse percent | Find the base when the percent and the final value are known. | 20 % of X = 12 → X = 12 / 0.20 = 60 |
| Percent on percent | percent1 × percent2 / 100 |
20 % of 25 % of 500 = 0.20 × 0.25 × 500 = 25 |
2. Key Points & Formulae
| Category | Formula | Notes |
|---|---|---|
| Percent of a number | P% of N = N × (P/100) |
If P is negative → subtraction. |
| Percent change (increase/decrease) | Δ% = ((New – Old)/Old) × 100 |
Positive → increase, negative → decrease. |
| Reverse percent | Base = Value / (P/100) |
Useful for “Find the original amount”. |
| Percent on percent | P1% of P2% of N = N × (P1/100) × (P2/100) |
Equivalent to N × P1 × P2 / 10,000. |
| Two‑step percent change | Final = Initial × (1 + a%/100) × (1 + b%/100) |
For successive increases/decreases. |
| Compound interest (annual) | A = P (1 + r/100)^t |
r = rate per annum, t = years. |
| Continuous compounding | A = P e^{rt} |
e ≈ 2.71828. |
| Percent of a percent of a percent | P1% of P2% of P3% of N = N × P1 × P2 × P3 / 10^6 |
Each percent multiplies the base. |
3. Worked Example (XAT‑style)
Problem
A shop sells a T‑shirt for ₹400. The shopkeeper applies a 15 % discount, then gives an additional 10 % off on the discounted price. What is the final price?
Step‑by‑step solution
-
First discount (15 %)
[ \text{Discounted price}_1 = 400 \times (1 - 0.15) = 400 \times 0.85 = ₹340 ] -
Second discount (10 %) on the new price
[ \text{Final price} = 340 \times (1 - 0.10) = 340 \times 0.90 = ₹306 ]
Answer: ₹306
Alternative one‑step formula
[
\text{Final price} = 400 \times (1 - 0.15) \times (1 - 0.10) = 400 \times 0.85 \times 0.90 = ₹306
]
Quick check
- 15 % of 400 = 60 → ₹340
- 10 % of 340 = 34 → ₹306
✓ All steps are integer‑friendly – handy for XAT mental math.
4. Quick Revision Summary
| Topic | Formula | Quick Tip |
|---|---|---|
| Percent of a number | N × P/100 |
Memorise 10 % = 1/10, 25 % = 1/4. |
| Percent change | (Δ/Old) × 100 |
Positive → ↑, Negative → ↓. |
| Reverse percent | Base = Value / (P/100) |
Think “What base gives this percent value?” |
| Percent on percent | N × P1 × P2 / 10,000 |
Two percents → divide by 10,000. |
| Two‑step change | Initial × (1 + a%) × (1 + b%) |
Multiply successive multipliers. |
| Compound interest | P (1 + r/100)^t |
Quick mental power: 10 % → ×1.1, 20 % → ×1.2. |
| Continuous compounding | P e^{rt} |
Use e ≈ 2.718; often not needed for XAT. |
Final Tips for XAT
- Convert to decimals early – saves time.
- Use “reverse” trick for “What percent of X is Y?” →
X = Y / (percent/100). - Break down compound problems into successive percentages.
- Check sanity: if a percent exceeds 100 %, the result might be negative or zero.
- Practice mental math: remember simple fractions: 25 % = 1/4, 50 % = 1/2, 75 % = 3/4.
Good luck! 🚀