Percentages – Study Notes for Dadra & Nagar Haveli & Daman & Diu Police Constable (Quantitative Aptitude)
Why Percentages?
Percentages help you compare quantities, calculate discounts, tax, interest, and solve many real‑world problems. Mastering this topic gives you a strong foundation for all quantitative sections.
1. Simple Explanation
| Term | Meaning | Example |
|---|---|---|
| Percentage | A fraction out of 100. Represented by the symbol %. |
30% = ( \frac{30}{100} ) = 0.30 |
| Percent of a number | The part of a number expressed as a percentage. | 20% of 50 = ( \frac{20}{100} \times 50 = 10 ) |
| Percent change | Difference between two values expressed as a percentage of the original value. | Price goes from ₹200 to ₹240 → ( \frac{240-200}{200} \times 100 = 20% ) increase |
Conversion Rules
| From → To | Formula |
|---|---|
| Percentage → Decimal | Divide by 100 |
| Decimal → Percentage | Multiply by 100 |
| Fraction → Percentage | ( \frac{a}{b} \times 100 ) |
2. Key Points & Tricks
| # | Key Point | Quick Tip |
|---|---|---|
| 1 | Always convert percentages to decimals before multiplying. | 25% of 80 → (0.25 \times 80 = 20) |
| 2 | For percentage change, use difference/original × 100. | Price drops 10% → ( 1 - 0.10 = 0.90 ) of original |
| 3 | Percent of a percent = multiply the decimals. | 30% of 20% → (0.30 \times 0.20 = 0.06 = 6% ) |
| 4 | Complementary percentages add up to 100%. | If 35% is one part, the other part = 65% |
| 5 | Watch for double counting in word problems (e.g., “20% more than 80% of 100”). | Break into two steps: 80% of 100 = 80, then add 20% of 80 = 16 |
| 6 | Percent error in measurements = ( \frac{ | true - measured |
| 7 | Use the rule of three for simple percent problems: “X% of Y”. | ( \frac{X}{100} \times Y ) |
| 8 | Discount + Tax: Apply sequentially. | Price ₹1,000 → 10% discount → ₹900 → 5% tax → ₹945 |
| 9 | Quick mental math: 10% is one‑tenth, 25% is a quarter, 50% is half. | 25% of 120 = 30 (since 120/4 = 30) |
| 10 | Always check the units (₹, kg, L, etc.) – percentages are unit‑free. |
3. Worked Example (Police Constable Level)
Problem
A police station receives 1,200 complaints in a month.
- 30% are related to theft.
- 45% of the theft complaints are resolved.
- Among the unresolved theft complaints, 25% are due to lack of evidence.
Find:
a) Number of theft complaints.
b) Number of resolved theft complaints.
c) Number of unresolved theft complaints due to lack of evidence.
d) Percentage of total complaints that were resolved.
Solution
- Total complaints = 1,200
a) Theft complaints
( 30% ) of 1,200
( = 0.30 \times 1,200 = 360 )
b) Resolved theft complaints
( 45% ) of 360
( = 0.45 \times 360 = 162 )
c) Unresolved theft complaints
Unresolved = Total theft – Resolved = (360 - 162 = 198)
Among these, 25% are due to lack of evidence
( 25% ) of 198
( = 0.25 \times 198 = 49.5 )
Since complaints are whole numbers, round appropriately → 50 complaints.
d) Percentage of total complaints resolved
Resolved theft complaints = 162 (only theft complaints are resolved).
If the station resolved only theft complaints, then
( \frac{162}{1,200} \times 100 = 13.5% )
Answer Summary
| Item | Value |
|---|---|
| a) Theft complaints | 360 |
| b) Resolved theft complaints | 162 |
| c) Unresolved theft complaints due to lack of evidence | 50 |
| d) % of total complaints resolved | 13.5 % |
4. Quick Revision Summary
- Percentage = Fraction × 100
- Percent of a number: ( \frac{p}{100} \times N )
- Percent change: ( \frac{New - Old}{Old} \times 100 )
- Double percent: Multiply decimals (e.g.,
30% of 20%= 0.30 × 0.20 = 0.06 = 6%) - Complementary: ( p + q = 100% ) → ( q = 100 - p )
- Sequential operations: Apply one after another (discount → tax).
- Rule of three: `x% of y = (x × y)/100