Percentages – RRB ALP Study Notes
(Quantitative Aptitude – Percentages)
1. Simple Explanation
| Term | Definition | Example |
|---|---|---|
| Percentage | “Per cent” means “per hundred.” It’s a way to express a part of a whole as a fraction of 100. | 25 % of 200 = 0.25 × 200 = 50 |
| Part | The quantity whose percentage of the whole we want to find. | In “25 % of 200”, 50 is the part. |
| Whole | The total quantity that the part comes from. | In the same example, 200 is the whole. |
Basic Formula
[ \text{Percentage} = \frac{\text{Part}}{\text{Whole}} \times 100 ]
2. Key Points (Quick Reference)
| What to Find | Formula (solve for the unknown) |
|---|---|
| Part | (\text{Part} = \frac{\text{Percentage} \times \text{Whole}}{100}) |
| Whole | (\text{Whole} = \frac{\text{Part} \times 100}{\text{Percentage}}) |
| Percentage | (\text{Percentage} = \frac{\text{Part}}{\text{Whole}} \times 100) |
| Percentage Increase / Decrease | (\Delta % = \frac{\text{Change}}{\text{Original}} \times 100) |
| New Value After Increase | (\text{New} = \text{Original} \times \left(1 + \frac{\Delta %}{100}\right)) |
| New Value After Decrease | (\text{New} = \text{Original} \times \left(1 - \frac{\Delta %}{100}\right)) |
Conversion Tips
- Decimal ↔ Percentage:
- Decimal to %: multiply by 100.
- % to Decimal: divide by 100.
| Decimal | % |
|---|---|
| 0.25 | 25 % |
| 0.10 | 10 % |
3. Worked Examples
Example 1 – Finding the Whole
Question: 20 % of a number is 15. Find the number.
Solution:
Let the number be (x).
(20% \text{ of } x = 15)
[ \frac{20}{100} \times x = 15 ;\Rightarrow; 0.20x = 15 ;\Rightarrow; x = \frac{15}{0.20} = 75 ]
Answer: The number is 75.
Example 2 – Percentage Increase
Question: A computer’s price increased from Rs. 18 000 to Rs. 21 600. What is the percentage increase?
Solution:
Change (= 21,600 - 18,000 = 3,600)
Original price (= 18,000)
[ \Delta% = \frac{3,600}{18,000} \times 100 = 20% ]
Answer: 20 % increase.
Example 3 – Profit in Percentage
Question: A shopkeeper sells an item for Rs. 1 200. If the cost price was Rs. 960, what is the profit percentage?
Solution:
Profit (= 1,200 - 960 = 240)
Cost price (= 960)
[ \text{Profit}% = \frac{240}{960} \times 100 = 25% ]
Answer: 25 % profit.
Example 4 – New Salary After Decrease
Question: A person’s salary was reduced by 8 %. If the new salary is Rs. 58 000, find the original salary.
Solution:
Let original salary (= x).
After an 8 % decrease, new salary (= x \times (1 - 0.08) = 0.92x).
So, [ 0.92x = 58,000 ;\Rightarrow; x = \frac{58,000}{0.92} \approx 63,043.48 ]
Answer: Original salary ≈ Rs. 63 043 (rounded to the nearest rupee).
4. Quick Revision Summary
| Topic | Formula | Quick Tip |
|---|---|---|
| Part from Whole | (\text{Part} = \frac{% \times \text{Whole}}{100}) | “% of Whole” → multiply first, then divide by 100. |
| Whole from Part | (\text{Whole} = \frac{\text{Part} \times 100}{%}) | Multiply part by 100, then divide by % value. |
| Percentage of a Number | (%) of (N = \frac{N \times %}{100}) | Same as Part from Whole. |
| Percentage Increase | (\text{New} = \text{Old} \times (1 + \frac{\Delta%}{100})) | Add to 1, then multiply. |
| Percentage Decrease | (\text{New} = \text{Old} \times (1 - \frac{\Delta\ |