In this chapter, you will learn
- —Understand the concept of percentage as a fraction with denominator 100
- —Convert percentages to fractions and decimals, and vice versa
- —Calculate percentage of a quantity
- —Apply percentages to real-life scenarios like profit, loss, and discount
- —Calculate simple interest and tax
- —Solve word problems involving percentages
What is Percentage?
Percentage means 'per hundred'. It expresses a number as a fraction of 100.
Definition: Percentage (%) = (Part / Whole) × 100
Example: 25% means 25 out of 100 = 25/100 = 1/4
Why Percentages?
- Easier to compare quantities (50% vs 0.5 vs 1/2 - all same)
- Universal standard for expressing ratios
- Used in discounts, interest, taxes, grades, surveys
Real-Life Applications:
- Discount: "50% off" means pay only 50% of original price
- Interest: Bank gives "8% per annum" on savings
- Grades: Scored 85% means 85 marks out of 100
- Tax: GST is 18% on goods
Exam Tip
Percentage is fundamental to real-world math. Always convert % to fraction or decimal for calculations.
Common Mistake
Confusing the part and the whole. Example: 20% of 50 is 10, not 20. (20/100 × 50 = 10)
Percentage to Fraction and Decimal Conversions
Percentage to Fraction: Remove % and put value over 100, then simplify.
Examples:
- 25% = 25/100 = 1/4
- 50% = 50/100 = 1/2
- 75% = 75/100 = 3/4
- 20% = 20/100 = 1/5
Percentage to Decimal: Divide percentage by 100 or move decimal 2 places left.
Examples:
- 25% = 25 ÷ 100 = 0.25
- 50% = 50 ÷ 100 = 0.5
- 8% = 8 ÷ 100 = 0.08
- 150% = 150 ÷ 100 = 1.5
Fraction to Percentage: Multiply by 100.
- 1/4 = 0.25 = 25%
- 3/5 = 0.6 = 60%
Exam Tip
Remember key percentages: 25%=1/4, 50%=1/2, 75%=3/4, 20%=1/5, 10%=1/10
Common Mistake
Moving decimal in wrong direction. 25% → 0.25 (left), NOT 2.5. Move left to convert % to decimal.
Percentage of a Quantity
Finding Percentage of a Number:
Percentage of = (Percentage / 100) × Quantity
Examples:
- 20% of 50 = (20/100) × 50 = 0.2 × 50 = 10
- 15% of 200 = (15/100) × 200 = 0.15 × 200 = 30
- 5% of 1000 = (5/100) × 1000 = 50
Quick Method: 10% = Divide by 10. Then multiply to find other percentages.
- 10% of 80 = 8
- 20% of 80 = 2 × 8 = 16
- 5% of 80 = 8 ÷ 2 = 4
Exam Tip
Find 10% first by dividing by 10. Then use it to calculate other percentages quickly.
Common Mistake
Writing 20% of 50 = 20 × 50. Wrong! It's (20/100) × 50 = 10, not 1000.
Profit, Loss, and Discount
Profit and Loss:
Profit = Selling Price - Cost Price (if positive)
Loss = Cost Price - Selling Price (if positive)
Profit % = (Profit / Cost Price) × 100
Loss % = (Loss / Cost Price) × 100
Example: Cost Rs. 100, Sell Rs. 120. Profit = 20. Profit% = (20/100) × 100 = 20%
Discount: Reduction in marked price.
Discount = Marked Price - Selling Price
Discount % = (Discount / Marked Price) × 100
Selling Price = Marked Price - Discount
Example: Marked Rs. 200, Discount 20%. Discount amount = (20/100) × 200 = 40. Selling Price = 200 - 40 = 160
Exam Tip
Always calculate percentages on the cost price for profit/loss. On marked price for discount.
Common Mistake
Calculating loss % on selling price instead of cost price. Loss % = (Loss / Cost Price) × 100, NOT denominator = SP.
Simple Interest and Tax
Simple Interest (SI): Interest earned on principal without compounding.
SI = (Principal × Rate × Time) / 100
Amount = Principal + SI
Example: Principal Rs. 1000, Rate 10% per annum, Time 2 years.
- SI = (1000 × 10 × 2) / 100 = 200
- Amount = 1000 + 200 = 1200
Tax: Percentage-based charge on goods/services.
Tax Amount = (Tax % / 100) × Original Price
Final Price = Original Price + Tax
Example: Item costs 100, GST is 18%. Tax = 18. Final = 118
Exam Tip
SI formula: Principal × Rate × Time / 100. Amount = P + SI. Read 'per annum' carefully (usually yearly).
Common Mistake
Confusing simple interest with amount. SI is only interest earned, not total amount. Amount = P + SI.
Chapter Summary
Percentage and its Applications connects math to real world. Key topics:
- Percentage: Out of 100. Percent = Part/Whole × 100
- Conversions: % ↔ Fraction ↔ Decimal
- Percentage of: (P/100) × Quantity
- Profit/Loss: Profit% = (Profit/CP) × 100
- Discount: Discount% = (Discount/MP) × 100
- Simple Interest: SI = (P × R × T) / 100
- Tax: Final = Original + (Tax% × Original)
Exam Focus: Real-life problems on profit/loss, discount, SI, and tax. Always identify CP, MP, and what percentage to calculate on.