Percentage — Study Notes for Maharashtra Police Bharti
Overview
Percentage is one of the most scoring and frequently tested topics in the Mathematics section of Maharashtra Police Bharti exam. The word "percent" means "per hundred" — so 25% simply means 25 out of 100. This concept forms the foundation for many other topics like Profit & Loss, Simple & Compound Interest, and Data Interpretation.
Mastering percentage means mastering quick mental calculations — the exam rewards speed, and most percentage problems can be solved in 30–60 seconds with the right approach.
Students must be fluent in converting between fractions, decimals, and percentages, and should memorize key fraction-percentage equivalents to save precious time during the exam.
Key Concepts
- Basic Definition: x% = x/100. To find x% of a number N, calculate (x × N)/100.
- Percentage to Fraction: Divide by 100 and simplify. Example: 25% = 25/100 = 1/4.
- Fraction to Percentage: Multiply by 100. Example: 3/5 = (3/5) × 100 = 60%.
- Percentage Increase/Decrease: When a value changes from A to B, the percentage change = [(B − A)/A] × 100. Positive result means increase, negative means decrease.
- Successive Percentage Change: When two successive changes of a% and b% occur, the net effect = a + b + (ab/100)%. This is crucial for discount-on-discount and population growth problems.
- Base Value Matters: "A is 20% more than B" means A = 1.2B. But "A is 20% less than B" means A = 0.8B. The base is always what comes after "than" or "of".
- Reverse Percentage: If a number after x% increase becomes N, original = N × (100/(100 + x)). If after x% decrease, original = N × (100/(100 − x)).
Formulas / Key Facts
| Concept | Formula |
|---|---|
| x% of N | (x × N)/100 |
| Percentage change | [(New − Old)/Old] × 100 |
| A is x% more than B | A = B × (100 + x)/100 |
| A is x% less than B | A = B × (100 − x)/100 |
| Successive change (a%, b%) | Net = a + b + (ab/100)% |
| Find original after x% increase | Original = Final × 100/(100 + x) |
| Find original after x% decrease | Original = Final × 100/(100 − x) |
Must-Memorize Fraction-Percentage Equivalents:
| Fraction | Percentage | Fraction | Percentage |
|---|---|---|---|
| 1/2 | 50% | 1/8 | 12.5% |
| 1/3 | 33.33% | 1/9 | 11.11% |
| 1/4 | 25% | 1/10 | 10% |
| 1/5 | 20% | 1/11 | 9.09% |
| 1/6 | 16.67% | 1/12 | 8.33% |
| 1/7 | 14.28% | 2/3 | 66.67% |
Worked Examples
Example 1: Basic Percentage Calculation
Question: Find 35% of 240.
Solution:
- 35% of 240 = (35 × 240)/100
- = 8400/100 = 84
- Answer: 84
Shortcut: 35% = 30% + 5%. Calculate 30% of 240 = 72, and 5% of 240 = 12. Total = 72 + 12 = 84.
Example 2: Percentage Change
Question: The population of a village increased from 25,000 to 28,000. Find the percentage increase.
Solution:
- Change = 28,000 − 25,000 = 3,000
- Percentage increase = (3,000/25,000) × 100
- = (3/25) × 100 = 12%
- Answer: 12%
Example 3: Successive Discounts
Question: A shopkeeper gives two successive discounts of 20% and 10% on a shirt marked at ₹500. Find the selling price.
Solution:
- Method 1 (Step-by-step):
- After 20% discount: 500 − (20% of 500) = 500 − 100 = ₹400
- After 10% discount on ₹400: 400 − (10% of 400) = 400 − 40 = ₹360
- Method 2 (Formula for net discount):
- Net discount = 20 + 10 − (20 × 10)/100 = 30 − 2 = 28%
- SP = 500 × (100 − 28)/100 = 500 × 0.72 = ₹360
- Answer: ₹360
Example 4: Reverse Percentage
Question: After a 15% increase, the price of rice became ₹46 per kg. What was the original price?
Solution:
- Let original price = x
- x + 15% of x = 46
- x × (115/100) = 46
- x = 46 × (100/115) = 4600/115 = 40
- Answer: ₹40 per kg
Common Mistakes
- Wrong base in comparison: "A is 25% more than B" — students calculate 25% of A instead of B. Fix: The base is always what comes after "than" or "of".
- Adding successive percentages directly: Two successive 10% increases ≠ 20% increase. Fix: Use the formula a + b + (ab/100). Two 10% increases = 10 + 10 + 1 = 21%.
- Confusing increase and decrease reversal: If something increases by 20%, it does NOT decrease by 20% to return to original. Fix: A 20% increase needs a 16.67% decrease (1/6) to reverse.
- Calculating percentage OF vs percentage MORE/LESS: "A is 20% of B" means A = 0.2B. "A is 20% more than B" means A = 1.2B. These are completely different.
- Forgetting to convert percentage to decimal: Writing 25% as 25 instead of 0.25 in calculations leads to answers 100 times larger or smaller.
Quick Reference
- x% of N = (x × N)/100 — the most basic formula.
- Percentage change = (Difference/Original) × 100 — always divide by the OLD value.
- Successive changes of a% and b% = a + b + (ab/100)% — negative sign for decrease.
- To reverse x% increase, decrease by (100x)/(100 + x)%.
- 25% = 1/4, 20% = 1/5, 12.5% = 1/8 — memorize these for speed.
- "More than" adds to 100%, "Less than" subtracts from 100% — A is 30% less than B means A = 70% of B.