📚 Gujarat Patwari / Lekhpal – Quantitative Aptitude
📌 Topic: Percentages
Percentages are a convenient way to express parts of a whole and to compare ratios. Mastery of this topic is essential for any quantitative aptitude section, especially in competitive exams like the Patwari/Lekhpal. Below are concise notes, key points, a worked example, and a quick revision summary.
1. Simple Explanation
- Percent = “per hundred.”
- 20 % of 50 = 0.20 × 50 = 10.
- Basic Idea: Convert a fraction to a percent by multiplying by 100, or convert a percent to a decimal by dividing by 100.
2. Key Points & Formulas
| Concept | Formula | Example |
|---|---|---|
| Percent of a number | ( \text{Percentage} = \frac{\text{Part}}{\text{Whole}}\times100 ) | 25 % of 200 = ( \frac{25}{100}\times200 = 50 ) |
| Percentage increase/decrease | ( \text{New} = \text{Old}\times\left(1\pm\frac{\text{Percent}}{100}\right) ) | 50 % increase → (1+0.5=1.5) |
| Finding the original when percent increase/decrease known | ( \text{Old} = \frac{\text{New}}{1\pm\frac{\text{Percent}}{100}} ) | 30 % decrease to 84 → ( \frac{84}{0.7}=120 ) |
| Percentage change | ( \text{Change} = \frac{\text{New}-\text{Old}}{\text{Old}}\times100 ) | From 80 to 110 → ( \frac{30}{80}\times100=37.5% ) |
| Percent of a percent | Multiply the two percentages (as decimals) | 20 % of 30 % of 200 = (0.2\times0.3\times200=12) |
| Finding percent when whole and part known | ( \text{Percent} = \frac{\text{Part}}{\text{Whole}}\times100 ) | 45 out of 150 → 30 % |
| Percent relation to ratio | If (a:b = m:n) then percent of (a) in total (= \frac{m}{m+n}\times100) | 3:5 → ( \frac{3}{8}\times100=37.5% ) |
Common Tricks
- Work backwards: For percent increase/decrease, factor out (1 \pm \frac{p}{100}).
- Cross–multiplication: When comparing two percent problems, set up a ratio and cross–multiply.
- Unit Rate: “Per 100” is a unit; treat 100 as a constant.
3. Worked Example
Problem
A shopkeeper sells 120 kg of rice at ₹45 per kg. A customer buys 30 % of this stock at a 10 % discount. What is the total amount the customer pays?
Step 1 – Quantity bought
30 % of 120 kg
( = 0.30 \times 120 = 36 \text{ kg} )
Step 2 – Original price for 36 kg
( 36 \text{ kg} \times ₹45 = ₹1,620 )
Step 3 – Discount amount
10 % of ₹1,620
( = 0.10 \times ₹1,620 = ₹162 )
Step 4 – Final amount payable
( ₹1,620 - ₹162 = ₹1,458 )
Answer: ₹1,458
4. Quick Revision Summary
| Topic | Quick Recall |
|---|---|
| Percent of a number | (\frac{p}{100}\times N) |
| Increase/Decrease | Multiply/divide by (1 \pm \frac{p}{100}) |
| Percent change | (\frac{\text{New-Old}}{\text{Old}}\times100) |
| Finding original | (\frac{\text{New}}{1 \pm \frac{p}{100}}) |
| Percent of a percent | Multiply percentages (as decimals) |
| Tip | Always convert percentages to decimals before multiplying. |
| Common Pitfall | Forgetting to divide by 100 when converting percent to decimal. |
Practice Tip: Convert every problem into a basic “part‑whole” relation. Once the numbers are in the form part ÷ whole, the rest follows from the formulas above.
Happy studying! 🚀