Class 9 Mathematics Chapter 10: Ratios, Proportions and Percentages – Important Questions with Solutions

Chapter 10 carries approximately 12–15 marks in CBSE Class 9 final exams, split across 1-mark MCQs, 2-mark short answers, 3-mark application problems, and 5-mark case studies. The 2024-25 rationalized syllabus emphasizes practical application: calculating discounts in retail, comparing investment returns, and simplifying real-world ratios. Board examiners focus on conceptual clarity over rote memorization. Common question types include: (1) simplifying ratios to lowest terms, (2) solving proportion problems using cross-multiplication, (3) converting between percentage, fraction, and decimal forms, (4) calculating profit/loss percentage given cost and selling prices, (5) deriving simple interest formulas and applying them to loan/deposit scenarios. Students who master ratio comparison and proportion handling typically score 10+ marks; those weak in percentage conversion and profit-loss logic average 6–7 marks. Expected question count: 3–4 objective questions (3–4 marks), 2 short-answer questions (4 marks), 1 long-answer question (5 marks), and 1 case-study or scenario-based question (3–5 marks).

These objective questions test definition recall, formula application, and quick calculation skills. Allocate 45 seconds per question in exam conditions. **Q1: If the ratio of two numbers is 3:5 and their sum is 48, what is the larger number?** A) 30 B) 25 C) 18 D) 20 **Answer: A) 30** Explanation: Let numbers be 3x and 5x. Then 3x + 5x = 48 → 8x = 48 → x = 6. Larger number = 5(6) = 30. **Q2: 25% of 80 is equal to:** A) 15 B) 18 C) 20 D) 25 **Answer: C) 20** Explanation: 25% = 1/4. One-fourth of 80 = 80 ÷ 4 = 20. **Q3: If cost price = ₹200 and selling price = ₹250, then profit percentage is:** A) 20% B) 25% C) 30% D) 35% **Answer: B) 25%** Explanation: Profit = SP − CP = 250 − 200 = ₹50. Profit% = (50/200) × 100 = 25%. **Q4: In a proportion, if 4:x = 8:10, then x equals:** A) 5 B) 3 C) 2 D) 8 **Answer: A) 5** Explanation: Cross-multiply: 4 × 10 = 8 × x → 40 = 8x → x = 5. **Q5: Simple interest on ₹1000 at 5% per annum for 2 years is:** A) ₹100 B) ₹125 C) ₹150 D) ₹200 **Answer: A) ₹100** Explanation: SI = (P × R × T) / 100 = (1000 × 5 × 2) / 100 = 10,000 / 100 = ₹100.

These require brief working and explanation. Aim for 2–3 minutes per question. **Q1: Simplify the ratio 36:48 to its lowest terms.** Solution: Find GCD of 36 and 48. 36 = 2² × 3² 48 = 2⁴ × 3 GCD = 2² × 3 = 12 36:48 = (36÷12):(48÷12) = 3:4 **Q2: A shopkeeper bought a pen for ₹15 and sold it for ₹18. Find the profit percentage.** Solution: Profit = SP − CP = 18 − 15 = ₹3 Profit% = (Profit / CP) × 100 = (3/15) × 100 = 20% **Q3: If 15% of a number is 45, find the number.** Solution: Let the number = x 15% of x = 45 (15/100) × x = 45 x = (45 × 100) / 15 = 300 **Q4: In a school, the ratio of boys to girls is 4:5. If there are 80 boys, how many girls are there?** Solution: Boys:Girls = 4:5 If boys = 80, then 4 units = 80 1 unit = 20 Girls = 5 units = 5 × 20 = 100 **Q5: Calculate simple interest on ₹5000 at 8% per annum for 1.5 years.** Solution: SI = (P × R × T) / 100 SI = (5000 × 8 × 1.5) / 100 SI = 60,000 / 100 = ₹600

These blend multiple concepts and require clear reasoning. Expect 3–4 minutes per solution. **Q1: A merchant bought two items for ₹200 each. He sold the first at 20% profit and the second at 15% loss. Find the overall profit or loss percentage.** Solution: First item: CP = ₹200, Profit% = 20% SP₁ = CP + (20/100) × CP = 200 + 40 = ₹240 Second item: CP = ₹200, Loss% = 15% SP₂ = CP − (15/100) × CP = 200 − 30 = ₹170 Total CP = 200 + 200 = ₹400 Total SP = 240 + 170 = ₹410 Overall Profit = 410 − 400 = ₹10 Profit% = (10/400) × 100 = 2.5% **Q2: If A:B = 3:4 and B:C = 5:6, find A:B:C.** Solution: A:B = 3:4 (multiply by 5) → A:B = 15:20 B:C = 5:6 (multiply by 4) → B:C = 20:24 Therefore, A:B:C = 15:20:24 **Q3: A student scored 35 marks out of 50 in one exam and 42 out of 60 in another. In which exam did the student perform better?** Solution: First exam percentage = (35/50) × 100 = 70% Second exam percentage = (42/60) × 100 = 70% The student performed equally well in both exams (both 70%). **Q4: A discount of 20% is offered on an item marked at ₹500. What is the selling price? If the cost price was ₹350, find the profit percentage.** Solution: Marked Price = ₹500 Discount = 20% of 500 = (20/100) × 500 = ₹100 Selling Price = 500 − 100 = ₹400 Cost Price = ₹350 Profit = 400 − 350 = ₹50 Profit% = (50/350) × 100 ≈ 14.29%

These are comprehensive, multi-step problems requiring detailed reasoning and formula application. Budget 5–6 minutes per question. **Q1: A person invested ₹10,000 at 9% per annum simple interest for 3 years. After 3 years, he withdrew the amount and invested it at 10% per annum compound interest for 2 more years. Find the total amount at the end of 5 years.** Solution: Step 1: Calculate Simple Interest for first 3 years. SI = (P × R × T) / 100 = (10,000 × 9 × 3) / 100 = 270,000 / 100 = ₹2,700 Amount after 3 years = Principal + SI = 10,000 + 2,700 = ₹12,700 Step 2: Use this amount (₹12,700) as principal for compound interest for 2 years at 10% p.a. For compound interest: A = P(1 + r/100)ᵗ A = 12,700 × (1 + 10/100)² A = 12,700 × (1.1)² A = 12,700 × 1.21 = ₹15,367 Total amount at the end of 5 years = ₹15,367 **Q2: In a mixture of 150 litres, the ratio of milk to water is 4:1. How much water must be added so that the ratio becomes 3:2?** Solution: Step 1: Find initial quantities of milk and water. Milk:Water = 4:1 Total parts = 4 + 1 = 5 Milk = (4/5) × 150 = 120 litres Water = (1/5) × 150 = 30 litres Step 2: Let x litres of water be added. New water quantity = 30 + x Milk remains = 120 litres Step 3: New ratio should be 3:2 120:(30 + x) = 3:2 Cross-multiply: 120 × 2 = 3 × (30 + x) 240 = 90 + 3x 3x = 150 x = 50 Water to be added = 50 litres Verification: New ratio = 120:80 = 3:2 ✓ **Q3: Two shopkeepers, A and B, bought goods for ₹5,000 each. Shopkeeper A sold the goods at a profit of 30%, while Shopkeeper B sold at a loss of 20%. How much more money did A earn than B?** Solution: Shopkeeper A: CP = ₹5,000 Profit% = 30% Profit = (30/100) × 5,000 = ₹1,500 SP = CP + Profit = 5,000 + 1,500 = ₹6,500 Shopkeeper B: CP = ₹5,000 Loss% = 20% Loss = (20/100) × 5,000 = ₹1,000 SP = CP − Loss = 5,000 − 1,000 = ₹4,000 Difference in earnings = Profit of A − Loss of B = 1,500 − (−1,000) = 1,500 + 1,000 = ₹2,500 Alternatively: SP of A − SP of B = 6,500 − 4,000 = ₹2,500 Shopkeeper A earned ₹2,500 more than B.

**Case Study: Raj's Online Shopping Decision** Raj saw a laptop with a marked price of ₹50,000 on an e-commerce platform. The platform offered a 20% discount. Additionally, his credit card provided a 5% cashback on the discounted price. Raj's friend suggested he could buy the same laptop from a physical store at ₹42,000 (no discounts). Raj has ₹45,000 in his account. **(a) Calculate the final price Raj pays online after discount and cashback.** Solution: Marked Price = ₹50,000 Discount = 20% of 50,000 = (20/100) × 50,000 = ₹10,000 Price after discount = 50,000 − 10,000 = ₹40,000 Cashback = 5% of 40,000 = (5/100) × 40,000 = ₹2,000 Final amount paid = 40,000 − 2,000 = ₹38,000 **(b) Compare the online and offline prices. Which is the better deal?** Online final price = ₹38,000 Physical store price = ₹42,000 Difference = 42,000 − 38,000 = ₹4,000 saving with online purchase Savings% = (4,000/42,000) × 100 ≈ 9.52% The online purchase is better. **(c) Can Raj afford the online purchase? What would be the effective percentage discount on the marked price?** Raj has ₹45,000 and needs to pay ₹38,000 online. Yes, he can afford it (38,000 < 45,000). Effective discount on marked price = (50,000 − 38,000) / 50,000 × 100 = 12,000 / 50,000 × 100 = 24% The effective discount combining both offers is 24% on the marked price.

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