📊 Percentages – Study Notes for RPF Constable
(Quantitative Aptitude – RPF Constable Examination)
1️⃣ Simple Explanation
A percentage is a way of expressing a number as a fraction of 100.
- Percent sign (%) = “per hundred”.
- If x is 20% of y, then
[ x = \frac{20}{100}\times y = 0.20 \times y ] - Conversely, if you know x and want the percentage of y that it represents:
[ \text{Percentage} = \frac{x}{y}\times100 ]
2️⃣ Key Points & Formulas
| Concept | Formula | What it Means |
|---|---|---|
| Simple Percentage | ( \displaystyle P = \frac{\text{Part}}{\text{Whole}}\times 100 ) | Percent of whole |
| Finding Part from Percentage | ( \displaystyle \text{Part} = \frac{P}{100}\times \text{Whole} ) | How much is P% of whole |
| Finding Whole from Part | ( \displaystyle \text{Whole} = \frac{\text{Part}\times 100}{P} ) | Whole that gives part as P% |
| Percentage Increase | ( \displaystyle \text{Increase} = \frac{\text{New} - \text{Old}}{\text{Old}}\times100 ) | % increase from old to new |
| Percentage Decrease | ( \displaystyle \text{Decrease} = \frac{\text{Old} - \text{New}}{\text{Old}}\times100 ) | % decrease from old to new |
| Final Value after % increase/decrease | ( \displaystyle \text{Final} = \text{Old}\times\left(1\pm\frac{P}{100}\right) ) | Add or subtract % |
| Common Ratio / Comparison | ( \displaystyle \frac{A}{B} = \frac{P_A}{P_B} ) | If A is (P_A%) and B is (P_B%) of the same whole |
Tips for Quick Calculation
- 10% of a number = divide by 10.
- 5% = half of 10%.
- 25% = divide by 4.
- 50% = half.
- 20% = one‑fifth.
- 30% = ( \frac{1}{3}) of 10% (i.e., divide by 3 and then by 10).
Use these shortcuts for mental math.
3️⃣ Worked Example
Problem
A police station had 80 constables last year. This year the number increased by 15%.
- How many constables are there now?
- What is the percentage increase in the total strength of the police department if the department has 400 constables in total?
Solution
1. New number of constables
- Part = 15% increase of 80
[ \text{Increase} = \frac{15}{100}\times 80 = 0.15\times 80 = 12 ] - Final number
[ \text{New} = 80 + 12 = 92 ]
2. Percentage increase of the department
- Old department strength = 400
- New strength = 400 + 12 = 412
- Increase = 12
[ \text{Percentage Increase} = \frac{12}{400}\times100 = 3% ]
Answer
- 92 constables now.
- Department strength increased by 3%.
4️⃣ Quick Revision Summary
| Topic | Formula | Quick Tip |
|---|---|---|
| Percent of a number | ( \displaystyle \frac{P}{100}\times N ) | 50% = half, 25% = quarter |
| Reverse (find whole) | ( \displaystyle \frac{N\times100}{P} ) | Use 100/P as multiplier |
| Increase/Decrease | ( \displaystyle \frac{\text{Change}}{\text{Old}}\times100 ) | Signs (+/-) matter |
| Final after % change | ( \displaystyle \text{Old}\times(1\pm P/100) ) | Add for increase, subtract for decrease |
| Mental calc shortcuts | 10% → /10, 5% → /20, 25% → /4 | Saves time in exams |
Practice Tip
- Convert every word problem into a formula statement.
- Always keep the units consistent (e.g., money, people).
- Double‑check whether the question asks for the part, whole, or percentage.
Happy studying! 🚓📚