📚 Study Notes – Percentages
Program: Nagaland Forest Guard
Subject: Quantitative Aptitude
1️⃣ Simple Explanation
A percentage is a way of expressing a number as a fraction of 100.
- 1% = 1/100 = 0.01
- 25% = 25/100 = 0.25
Percentages are used to compare, find parts of a whole, calculate increases/decreases, and solve many common aptitude problems.
2️⃣ Key Points & Formulas
| Concept | Symbol | Formula | Example |
|---|---|---|---|
| Percent of a number | (P%) of (X) | (\displaystyle \frac{P}{100}\times X) | 20% of 150 = 0.2×150 = 30 |
| Finding Whole from a part | If (P%) of whole = part | (\displaystyle \text{Whole} = \frac{\text{Part}\times 100}{P}) | 40% of area = 200 km² → Whole = 200×100/40 = 500 km² |
| Percentage increase | New = Old + increase | (\displaystyle \text{New} = \text{Old}\times(1 + \frac{I}{100})) | 10% increase: 100 → 110 |
| Percentage decrease | New = Old – decrease | (\displaystyle \text{New} = \text{Old}\times(1 - \frac{D}{100})) | 5% decrease: 100 → 95 |
| Compound percentage | Successive changes | Multiply the factors: ( (1+\frac{I_1}{100})(1-\frac{D_1}{100})\dots ) | Increase 10% then decrease 5%: 100×1.10×0.95 = 104.5 |
| Percentage change (net) | (\displaystyle \frac{\text{New} - \text{Old}}{\text{Old}}\times 100) | 100 → 104.5 → ((104.5-100)/100×100 = 4.5%) | |
| Part of a part | (P%) of (Q%) of (X) | (\displaystyle \frac{P}{100}\times\frac{Q}{100}\times X = \frac{P\cdot Q}{10,000}\times X) | 20% of 30% of 200 = 0.2×0.3×200 = 12 |
| Multiple parts of a whole | (P_1%+P_2%+…+P_n% = 100% ) | Check if fractions add up to 1 | 40% + 30% + 30% = 100% |
| Finding percent of a percent | ( \frac{\text{Part of part}}{\text{Whole}}\times 100 ) | 12 / 200 × 100 = 6% |
3️⃣ Worked Example
Problem
- A forest guard’s salary was ₹60,000 in 2021.
- It increased by 10% in 2022.
- Then it decreased by 5% in 2023.
- The forest covers 500 km² in total.
- 40% of it is densely forested (trees).
- The remaining area is cleared land.
- In 2023, 25% of the cleared land was re‑planted with saplings.
Find:
- a) Salary in 2023.
- b) Net percentage change in salary from 2021 to 2023.
- c) Area of forested land.
- d) Area of cleared land.
- e) Area of re‑planted land in 2023.
Solution
| Step | Calculation | Result |
|---|---|---|
| a) 2022 salary | (60,000 \times 1.10 = 66,000) | ₹66,000 |
| a) 2023 salary | (66,000 \times 0.95 = 62,700) | ₹62,700 |
| b) Net % change | (\frac{62,700-60,000}{60,000}\times100 = 4.5%) | +4.5 % |
| c) Forested area | (500 \times 0.40 = 200) km² | 200 km² |
| d) Cleared land | (500 - 200 = 300) km² | 300 km² |
| e) Re‑planted land | (300 \times 0.25 = 75) km² | 75 km² |
4️⃣ Quick Revision Summary
- 1% = 1/100 = 0.01
- **P