Percentages – Quick‑Study Guide
For the Madhya Pradesh Patwari/Lekhpal Quantitative Aptitude Paper
1. Simple Explanation
A percentage tells how many parts of 100 are present in a given quantity.
- 20 % means 20 out of every 100 (i.e. 0.20 in decimal).
- Percentages are used to express parts of a whole, compare quantities, calculate discounts, tax, interest, growth, etc.
2. Key Points & Formulas
| Concept | Symbol | Formula | How to use it |
|---|---|---|---|
| Percentage of a number | (P) | (\displaystyle P = \frac{\text{part}}{\text{whole}}\times 100) | Find the part of a whole. |
| Part from a percentage | (x) | (\displaystyle x = \frac{P}{100}\times \text{whole}) | Find how much of the whole equals the percentage. |
| Whole from a percentage | (W) | (\displaystyle W = \frac{x\times 100}{P}) | Find the whole when part and percent are known. |
| Percentage change | (\Delta P) | (\displaystyle \Delta P = \frac{\text{new} - \text{old}}{\text{old}}\times 100) | Positive = increase, negative = decrease. |
| Percentage difference | (\Delta P_{\text{diff}}) | (\displaystyle \Delta P_{\text{diff}} = \frac{ | \text{A} - \text{B} |
| Converting between forms | (\displaystyle \text{percent} \to \text{fraction} = \frac{P}{100}) (\text{fraction} \to \text{percent} = \frac{\text{fraction}\times 100}{1}) |
Tip: Always cancel the 100 first in any fraction–percent conversion; it reduces mental calculation.
3. Worked Example (Multiple Steps)
Problem (typical Patwari question)
A landowner sold 30 % of his land. The sold portion was 18 ha.
- Find the total area of the land.
- If the market price is ₹ 12,000 per ha, what is the total revenue?
Step‑by‑step Solution
| Step | What to Find | Calculation | Result |
|---|---|---|---|
| 1 | Total area (Whole) | ( \text{Whole} = \frac{\text{Part}\times 100}{\text{Percent}}) (= \frac{18 \times 100}{30}) |
(= 60) ha |
| 2 | Revenue | ( \text{Revenue} = \text{Whole}\times \text{Price/ha}) (= 60 \times 12,000) |
(= ₹,720,000) |
Answer:
- Total land = 60 ha
- Revenue = ₹ 720,000
4. Quick Revision Summary
| Question | Formula | Quick Check |
|---|---|---|
| Find percent of a number | (P = \frac{x}{W}\times 100) | x = part, W = whole |
| Find part from percent | (x = \frac{P}{100}\times W) | P = percent |
| Find whole from percent | (W = \frac{x\times 100}{P}) | x = part |
| Percent change (increase) | (\Delta P = \frac{\text{New}-\text{Old}}{\text{Old}}\times 100) | >0 → increase |
| Percent change (decrease) | Same formula, result <0 | negative → decrease |
| Convert percent to decimal | Divide by 100 | 25 % → 0.25 |
| Convert decimal to percent | Multiply by 100 | 0.15 → 15 % |
Quick Test (1 Question)
A shopkeeper offers a 15 % discount on a ₹ 2,500 item.
What is the selling price?
Solution
Discount = (0.15 \times 2500 = ₹,375).
Selling price = (2500 - 375 = ₹,2125).
Final Tips for Patwari Exam
- Always keep the 100 handy – it appears in every conversion.
- Use the “part × 100 ÷ percent” rule for quick whole‑area or whole‑value calculations.
- Double‑check your sign for increases vs. decreases.
- Practice word problems where percentages describe profits, losses, tax, or growth.
Good luck and happy calculating!