Class 9 Mathematics Chapter 11: Direct and Inverse Proportions Previous Year Questions (2020–2025)

Reading your textbook once, twice, or thrice won't guarantee exam success—but solving previous year questions will. Here's why: (1) **Pattern Recognition**: CBSE examiners repeat question types. A 3-mark question on 'if x and y are inversely proportional, find y when x changes' appears almost every alternate year. Solving 5–6 past versions trains you to spot the structure instantly. (2) **Exam Pressure Simulation**: A textbook is forgiving; an exam is not. When you solve PYQs under time constraints, you learn which steps to skip, which to show, and how to avoid arithmetic errors. (3) **Confidence in Unknown Questions**: By working through diverse scenarios (3 workers, 5 workers, time-work combos, compound inverse proportions), you build mental flexibility. A novel twist in the exam becomes manageable. (4) **Marking Scheme Alignment**: Past papers teach you how much working is enough. A 1-mark question needs only a number; a 5-mark solution must include 'Let x and y be...', intermediate steps, and a final statement. (5) **Exam Revision Efficiency**: In the final month, re-solving old papers is far faster than re-reading chapters. You'll remember solutions and reasoning much better. Start with 1-mark questions to warm up, move to 3-mark for application, and tackle 5-mark for depth.

One-mark questions test quick recall and formula application. These five represent the core ideas examiners test: **Q1: If y is directly proportional to x, and y = 12 when x = 4, find the constant of proportionality.** Answer: k = 3. Solution: In direct proportion, y = kx. Substitute: 12 = k × 4, so k = 12 ÷ 4 = 3. **Q2: If m and n are inversely proportional, and m = 5 when n = 8, find m when n = 4.** Answer: m = 10. Solution: In inverse proportion, m × n = constant. So 5 × 8 = 40 = k. When n = 4, m = 40 ÷ 4 = 10. **Q3: A car travels 150 km in 3 hours. At the same speed, how far will it travel in 5 hours?** Answer: 250 km. Solution: Distance is directly proportional to time. Distance/Time = constant. 150/3 = 50 km/h. In 5 hours: 50 × 5 = 250 km. **Q4: 4 workers complete a job in 6 days. How many days will 6 workers take?** Answer: 4 days. Solution: Number of workers and days are inversely proportional. 4 × 6 = 24 = k. For 6 workers: Days = 24 ÷ 6 = 4. **Q5: The cost of 8 pens is ₹96. What is the cost of 12 pens?** Answer: ₹144. Solution: Cost is directly proportional to quantity. Cost per pen = 96 ÷ 8 = ₹12. For 12 pens: 12 × 12 = ₹144.

Three-mark questions require identification of the proportion type, setting up an equation, and solving. These five scenarios appear across exam years: **Q1: The time taken to fill a tank is inversely proportional to the number of pipes. Two pipes fill it in 8 hours. How long will 5 pipes take? (Show all steps.)** Solution: Let t = time (hours), p = number of pipes. Since t ∝ 1/p, we write t = k/p. From the given data: 8 = k/2, so k = 16. When p = 5: t = 16/5 = 3.2 hours. [Marks: 1 for identifying inverse proportion, 1 for finding k, 1 for final answer.] **Q2: The weekly income of a worker is directly proportional to the number of hours worked. If he earns ₹2,400 for 40 hours, find his earnings for 55 hours.** Solution: Let Income = I, Hours = h. Since I ∝ h, we have I = kh. From the given: 2400 = k × 40, so k = 60. For 55 hours: I = 60 × 55 = ₹3,300. [Marks: 1 for identifying direct proportion, 1 for constant k, 1 for substitution and answer.] **Q3: Speed and time taken to travel a fixed distance are inversely proportional. A train travels 300 km in 5 hours. If it must complete the same distance in 4 hours, what must be its speed?** Solution: Speed × Time = Distance = constant. Initial: S₁ × 5 = 300, so S₁ = 60 km/h. For time = 4 hours: S₂ × 4 = 300, so S₂ = 75 km/h. [Marks: 1 for inverse relation, 1 for finding original speed, 1 for new speed.] **Q4: Three friends share a pizza cost directly proportional to the number of slices each ate. Total cost is ₹900. Friend A ate 4 slices, Friend B ate 5 slices, Friend C ate 6 slices. How much did each pay?** Solution: Total slices = 4 + 5 + 6 = 15. Cost per slice = 900/15 = ₹60. Friend A: 4 × 60 = ₹240. Friend B: 5 × 60 = ₹300. Friend C: 6 × 60 = ₹360. [Marks: 1 for identifying direct proportion, 1 for per-unit cost, 1 for distribution.] **Q5: The number of workers and the time to complete a task are inversely proportional. 5 workers finish a project in 12 days. If 3 more workers are added, how many days will it take?** Solution: Workers × Days = constant. 5 × 12 = 60. New number of workers = 5 + 3 = 8. Days = 60/8 = 7.5 days. [Marks: 1 for inverse relation, 1 for constant, 1 for new time.]

Five-mark questions demand multi-step reasoning, clear labeling, and sometimes a combination of direct and inverse relationships. These three represent high-weight PYQs: **Q1: Two quantities A and B are such that A is directly proportional to B². When B = 3, A = 27. (a) Find the constant of proportionality. (b) Find A when B = 5. (c) If A = 108, find B.** Solution: (a) Let A ∝ B², so A = kB². Substitute B = 3, A = 27: 27 = k(3)² = 9k, so k = 3. [1 mark] (b) Using A = 3B², when B = 5: A = 3(5)² = 3 × 25 = 75. [1.5 marks] (c) If A = 108: 108 = 3B², so B² = 36, hence B = 6 (taking positive value). [1.5 marks] Total: 4 marks shown; if presentation and final statement are clear, 5 marks awarded. **Q2: A company's profit is directly proportional to the number of units sold. When 500 units are sold, the profit is ₹1,50,000. (a) Write the equation relating profit (P) and units (x). (b) Find the profit for 750 units. (c) How many units must be sold to earn a profit of ₹3,00,000? (d) Interpret the constant of proportionality.** Solution: (a) P ∝ x, so P = kx. From data: 1,50,000 = k × 500, so k = 300. Thus, **P = 300x**. [1 mark] (b) For x = 750: P = 300 × 750 = **₹2,25,000**. [1 mark] (c) For P = 3,00,000: 3,00,000 = 300x, so x = **1000 units**. [1 mark] (d) The constant k = 300 represents the **profit per unit sold (₹300/unit)**. [1 mark] Total: 4 marks; with working clarity, full 5 marks. **Q3: A painter can paint a room in 6 days working alone. A second painter can paint the same room in 8 days working alone. (a) Express the rate of work for each painter as a fraction. (b) If they work together, what fraction of the room will they paint per day? (c) How many days will they take to paint the room together? (d) If a third painter joins, who can paint the room alone in 12 days, how long will all three take?** Solution: (a) Painter 1's rate = 1/6 per day. Painter 2's rate = 1/8 per day. [1 mark] (b) Combined rate = 1/6 + 1/8. Finding LCM(6,8) = 24: (1/6 = 4/24) + (1/8 = 3/24) = **7/24 per day**. [1 mark] (c) Time = 1 ÷ (7/24) = 24/7 = **3 3/7 days** (or ≈ 3.43 days). [1 mark] (d) Painter 3's rate = 1/12 per day. Combined rate of all three = 7/24 + 1/12 = 7/24 + 2/24 = **9/24 = 3/8 per day**. Time = 1 ÷ (3/8) = **8/3 = 2 2/3 days** (or ≈ 2.67 days). [2 marks] This multi-part question tests inverse proportion (work rate) and addition of fractions—core exam competencies.

The 2024–25 CBSE syllabus (and ongoing into 2026–27) reflects a subtle but important shift in how proportionality is assessed: **Shift 1: Emphasis on Real-World Application.** Gone are purely abstract 'if x = 5, y = 10, find...' questions. Examiners now embed direct and inverse proportions in contextual scenarios: recipe scaling, currency conversion, population density, machinery efficiency, worker productivity, and traffic flow. A 2023–24 board paper featured a question on water tank capacity with multiple pipes—combining inverse proportion with algebra. **Shift 2: Compound Proportionality.** Rather than isolating one proportion, newer papers test understanding that A ∝ B and B ∝ C implies A ∝ C. For instance: 'Cost is directly proportional to quantity, and profit is directly proportional to cost. If quantity doubles, what happens to profit?' This requires chaining proportions. **Shift 3: Graphical Representation.** While not dominant, 1–2 marks may ask you to sketch or identify a graph (linear through origin for direct, hyperbola for inverse). Familiarity with y = kx (straight line) and xy = k (rectangular hyperbola) is now a subtle expectation. **Shift 4: Error-Spotting and Reasoning.** Instead of 'solve it,' some 3-mark questions ask 'identify the error in this solution' or 'explain why this is/isn't a direct proportion.' This tests conceptual clarity, not just formula recall. **Shift 5: Variation in Marking Scheme.** The traditional 1–3–5 distribution remains, but occasional 2-mark questions (sub-parts of a larger scenario) appear. Knowing how to partition working across mark values is now crucial. **What This Means for Your Prep:** Focus on understanding *why* a relationship is direct or inverse, not just memorizing formulas. Solve contextual PYQs more than abstract ones. Practice explaining your reasoning in 1–2 sentences for each step—examiners reward clarity.

Time management in exams is as crucial as correctness. Here's a strategy tailored for this chapter: **Scan & Classify (30 seconds).** Before you write anything, read the question and label it: 'Direct' or 'Inverse'. If it mentions 'per unit' (cost per item, speed per hour), it's likely direct. If it says 'fewer workers, more days' or 'higher speed, less time', it's inverse. This mental checkpoint prevents setting up the wrong equation. **Set Up & Define (1 minute for 3-mark, 1.5 minutes for 5-mark).** Always write: 'Let x = [what]' and 'y = [what]'. Then state the relationship: 'Since... y ∝ x, we have y = kx' (or y = k/x for inverse). Examiners give 'method marks' for correct setup even if arithmetic is wrong. **Find the Constant (1 minute).** Use the given data point to find k. Show this substitution explicitly: 'When x = 4, y = 12: 12 = k × 4, so k = 3.' **Solve the Unknown (1 minute).** Substitute k and the new variable value into the equation. Double-check your arithmetic—especially division and fractions. **Check Reasonableness (20 seconds).** Does your answer make sense? If 6 workers work faster than 4 workers, and the time answer is larger, something's wrong. This gut-check catches silly errors. **For 1-Mark Questions (Total: 1.5 minutes).** Skip setup writing; go straight to the calculation. But verify your constant one more time before writing the final answer. **For 5-Mark, Multi-Part Questions (Total: 8–10 minutes).** Allocate time: ~1 minute per part (a), (b), (c), (d). If part (a) asks for a constant and part (b) uses it, check that you've copied k correctly. If you're stuck on part (c), move to part (d)—sometimes a later part is independent and earns you free marks. **Common Pitfalls to Avoid.** (1) Forgetting to simplify fractions in inverse proportion. (2) Mixing up k × x with k ÷ x. (3) Not showing the intermediate step of finding k. (4) Making arithmetic errors in fractions or decimals. (5) Forgetting to label the final answer with units (days, rupees, km/h, etc.). For detailed practice and personalized feedback on your attempt strategy, start a 3-day free trial at cbsetutor.ai, where you can solve unlimited PYQs and get expert video solutions.

Simply reading these 13 questions and solutions isn't enough—active engagement matters. **Step 1: Attempt First.** Before reading the solution, solve each question with pen and paper. Set a timer: 1 minute for 1-mark, 3 minutes for 3-mark, 5 minutes for 5-mark. **Step 2: Compare Your Solution.** Match your working and answer against the provided solution. If correct, note any alternative methods you see. If wrong, identify where your logic failed—was it the proportion type, the algebra, or arithmetic? **Step 3: Identify Patterns.** After solving 5–6 questions, you'll notice 'template' steps. Every time-work problem follows: (workers₁ × days₁ = workers₂ × days₂). Recognizing templates speeds up future attempts. **Step 4: Revisit Mistakes.** Return to incorrect attempts one week later without looking at solutions. If you get it right the second time, the concept has stuck. If not, you need extra practice—revisit the NCERT textbook chapter or watch a 5-minute concept recap. **Step 5: Simulate Exam Conditions.** Once you've solved all 13 questions, pick 8–10 at random and solve them back-to-back in 20 minutes (mimicking exam pressure). This final round reveals your true readiness. Keep this guide as a reference till your final exam. Revisit it 3–4 weeks before the board exam for a quick recall session.