📚 Study Notes – Percentages
For the Sikkim Patwari / Lekhpal Quantitative Aptitude Paper
1️⃣ Simple Explanation
- Percent = “per hundred.”
- It tells us how many parts of 100 a particular quantity represents.
- Used to compare parts of a whole, express changes (increase/decrease), and solve many everyday math problems.
2️⃣ Key Points (Quick Reference)
| Concept | Formula | What It Means |
|---|---|---|
| Percent of a Number | ( \displaystyle P% \text{ of } N = \frac{P}{100} \times N ) | The part of N that equals P percent. |
| Part from Whole | ( \displaystyle \text{Part} = \frac{P}{100} \times \text{Whole} ) | Find the part when whole and percentage are known. |
| Whole from Part | ( \displaystyle \text{Whole} = \frac{\text{Part} \times 100}{P} ) | Determine the whole when part and percentage are known. |
| Percentage Increase | ( \displaystyle % \text{ Increase} = \frac{\text{New} - \text{Old}}{\text{Old}} \times 100 ) | How much New exceeds Old as a percent of Old. |
| Percentage Decrease | ( \displaystyle % \text{ Decrease} = \frac{\text{Old} - \text{New}}{\text{Old}} \times 100 ) | How much New falls short of Old as a percent of Old. |
| Percentage Change (overall) | ( \displaystyle % \text{ Change} = \frac{\text{New} - \text{Old}}{ | \text{Old} |
| Unit Rate from Percent | ( \displaystyle \text{Unit Rate} = \frac{\text{Amount}}{100} ) | Useful when you know a percent value and want a per‑unit amount. |
Quick Tips
- Multiplication by 10 ⇒ move decimal one place right.
- Division by 10 ⇒ move decimal one place left.
- To find a percent of a number → multiply by the percentage then divide by 100.
- To find what percent one number is of another → divide the first by the second and multiply by 100.
3️⃣ Worked Example
Problem:
A shopkeeper sold 1200 ₹ worth of goods at a 15 % profit.
- What was the cost price?
- What is the selling price?
- If he sold the same goods at a 10 % discount on the selling price, what would be the final amount received?
Step‑by‑Step
-
Let CP = cost price, SP = selling price.
Profit = 15 % of CP
[ \text{Profit} = 0.15 \times CP ] Therefore
[ SP = CP + 0.15,CP = 1.15,CP ] -
Given SP = 1200 ₹.
[ 1.15,CP = 1200 ;\Rightarrow; CP = \frac{1200}{1.15} ] [ CP = 1043.48,₹ \quad(\text{approx.}) ] -
Selling price at 10 % discount
Discount = 10 % of SP
[ \text{Discount} = 0.10 \times 1200 = 120,₹ ] Final amount received
[ \text{Final} = SP - \text{Discount} = 1200 - 120 = 1080,₹ ]
Summary of Answers
| Quantity | Value (₹) |
|---|---|
| Cost Price (CP) | ≈ 1043.48 |
| Selling Price (SP) | 1200 |
| Final amount after 10 % discount | 1080 |
4️⃣ Quick Revision Summary
- Percent of a Number →
P% of N = (P/100) × N. - Part from Whole →
Part = (P/100) × Whole. - Whole from Part →
Whole = (Part × 100) / P. - Percent Increase →
((New-Old)/Old) × 100. - Percent Decrease →
((Old-New)/Old) × 100. - Tip: To avoid mistakes, write the formula first, then plug in numbers.
- Common errors:
- Mixing up “percentage of” vs “percentage change”.
- Forgetting to divide by 100 when converting a percentage to a decimal.
- Not distinguishing between “increase” and “decrease” (sign matters).
Quick Practice Problems
- 40 % of what number is 80?
- A price is increased from 250 ₹ to 300 ₹. What is the percent increase?
- A salary was reduced by 12 % to 2640 ₹. What was the original salary?
Solve these before you check the answers to cement the concepts.
Good luck with your preparation!