Introduction
In our daily lives, we often compare prices, discounts, profits, weights, and more. This chapter introduces the tools and concepts to compare such quantities: percentages, ratios, profit & loss, simple and compound interest, taxes, and discounts.
1. Ratio and Percentage
1.1. Ratio
A ratio compares two quantities of the same kind by division. Example: Ratio of 3 boys to 2 girls is 3:2.
1.2. Percentage
A percentage is a ratio expressed as a fraction of 100. Example: 25 out of 50 = (25/50) × 100 = 50%.
2. Converting Quantities
- 25% = 25/100 = 1/4
- 3/4 = (3/4) × 100 = 75%
- 0.2 = 20%
- 12.5% = 0.125
3. Increase and Decrease as Percentage
Percentage Increase = [(Increase) / Original] × 100
Percentage Decrease = [(Decrease) / Original] × 100
4. Profit and Loss
- Profit = SP − CP
- Loss = CP − SP
- Profit % = (Profit / CP) × 100
- Loss % = (Loss / CP) × 100
5. Discounts
Discount = Marked Price − Selling Price
Discount % = (Discount / MP) × 100
6. Sales Tax and GST
GST = (Tax Rate × CP) / 100
Total Price = CP + GST
7. Value Added Tax (VAT)
Similar to GST, VAT was used earlier to tax goods before GST replaced it.
8. Simple Interest (SI)
SI = (P × R × T) / 100
Amount = Principal + Interest
9. Compound Interest (CI)
Amount = P(1 + R/100)T
CI = Amount − Principal
10. CI Compounded Half-Yearly
- R → R/2
- T → 2T
11. Increase or Decrease in Population
Increase: Final = Initial × (1 + R/100)T
Decrease: Final = Initial × (1 − R/100)T
12. Depreciation
Depreciated Value = Initial × (1 − R/100)T
13. Successive Discounts
Net Discount = d1 + d2 − (d1 × d2)/100
14. Application-Based Word Problems
Includes marked price and discount, profit/loss with GST, investment growth, price change over years, and more.
15. Tips and Tricks
- Profit/loss % is always on CP
- Use correct formula for SI or CI
- In CI, adjust time and rate if compounded semi-annually
16. Summary Table
| Concept | Formula |
|---|---|
| Percentage | (Part / Whole) × 100 |
| Profit | SP − CP |
| Profit % | (Profit / CP) × 100 |
| Loss | CP − SP |
| Loss % | (Loss / CP) × 100 |
| Discount % | (Discount / MP) × 100 |
| Simple Interest | (P × R × T) / 100 |
| Compound Interest | P(1 + R/100)T |
| Depreciation | Initial × (1 − R/100)T |
| Population Growth | Initial × (1 + R/100)T |
| Population Decline | Initial × (1 − R/100)T |
17. Practice Questions
- Find the percentage increase if the price of a shirt rises from ₹400 to ₹460.
- A fan is bought at ₹1200 and sold at ₹1440. Find the profit %.
- What is the SI on ₹5000 at 5% for 3 years?
- Calculate the CI on ₹8000 for 2 years at 6% per annum.
- A cycle marked ₹1500 is sold at 10% discount. Find the selling price.
- What amount is obtained on ₹10000 compounded yearly at 5% for 3 years?
Conclusion
The chapter “Comparing Quantities” enhances real-world math skills and financial literacy. Mastering it allows students to handle percentages, interests, taxes, discounts, and much more confidently.