Percentages – Study Notes for West Bengal PSC
(Quantitative Aptitude – Part 1)
Why you need these notes
Percentages are the most frequently asked topic in the PSC exam. Mastering the basic formulas, practising a variety of “percentage” problems, and remembering quick tricks will boost both speed and accuracy.
1. Simple Explanation
- Percentage = “per hundred”; it is a way to express a part of a whole as a fraction of 100.
- Formula
[ \text{Percentage} = \frac{\text{Part}}{\text{Whole}}\times100 ] - Common variations:
- Part from a given percentage and whole.
- Whole from a given part and percentage.
- Percentage change (increase/decrease) between two values.
2. Key Points & Formulas
| Case | Formula | Example |
|---|---|---|
| 1. Find % of a number | (\text{% of } X = \dfrac{Y}{100}\times X) | 15 % of 200 = (0.15\times200=30) |
| 2. Find part given % and whole | (\text{Part} = \dfrac{\text{Percentage}}{100}\times\text{Whole}) | 30 % of 120 = (0.30\times120=36) |
| 3. Find whole given part and % | (\text{Whole} = \dfrac{\text{Part}\times100}{\text{Percentage}}) | A 25 % of a number is 75 ⇒ Whole = (75\times4=300) |
| 4. Percentage increase | (\text{New %} = \dfrac{\text{Old}+\Delta}{\text{Old}}\times100) | 120 → 140: (\dfrac{140}{120}\times100=116.67%) |
| 5. Percentage decrease | (\text{New %} = \dfrac{\text{Old}-\Delta}{\text{Old}}\times100) | 150 → 120: (\dfrac{120}{150}\times100=80%) |
| 6. Net change after successive increases/decreases | Multiply the individual factors: ((1+I/100)(1-D/100)-1) | +10 % then –10 % → ((1.10)(0.90)-1=0) (no net change) |
| 7. “Part of a part” | Multiply percentages: (a%) of (b%) of (X = \dfrac{a,b}{10000}\times X) | 20 % of 30 % of 200 = (0.20\times0.30\times200=12) |
Mnemonic:
Part Of Whole =>P = (W%/100) × Whole
Whole Part Percentage =>W = (Part × 100) / P%
3. Worked Example (Step‑by‑Step)
Problem
If 75 % of 80 % of a number is 48, find the number.
Step‑1 – Convert to a Single Fraction
(75% = 0.75) and (80% = 0.80).
(0.75 \times 0.80 = 0.60).
Step‑2 – Set Up the Equation
(0.60 \times X = 48).
Step‑3 – Solve for (X)
(X = \dfrac{48}{0.60} = 80).
Answer: The number is 80.
Quick Tip: When the problem involves “% of % of X”, first multiply the percentages (as decimals) to get a single factor.
4. Sample Problems (with Solutions)
| # | Problem | Solution |
|---|---|---|
| 1 | 18 % of 250? | (0.18\times250 = 45) |