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

1. Find the percentage increase if the price of a shirt rises from ₹400 to ₹460.
2. A fan is bought at ₹1200 and sold at ₹1440. Find the profit %.
3. What is the SI on ₹5000 at 5% for 3 years?
4. Calculate the CI on ₹8000 for 2 years at 6% per annum.
5. A cycle marked ₹1500 is sold at 10% discount. Find the selling price.
6. 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.