Class 9 Mathematics Chapter 11: Direct and Inverse Proportions MCQ with Answers

The rationalized CBSE 2024-25 curriculum shifted from lengthy subjective questions to data-driven, time-bound assessments. Multiple-choice questions now constitute 25-40% of Mathematics papers across Classes 9 and 10, particularly in units testing proportional reasoning. Chapter 11 (Direct and Inverse Proportions) is ideal for MCQ-based testing because: (1) it requires quick conceptual judgment, not lengthy derivations; (2) trap options mirror common student misconceptions—identifying the correct answer reveals depth of understanding; (3) real-world applications (work-rates, speeds, costs) lend themselves to scenario-based MCQs. The CBSE question papers increasingly blend assertion-reason formats, which test not just the answer but the reasoning pathway. By practicing these 30 MCQs, you'll internalize the distinction between direct proportion (when one quantity doubles, the other doubles: y ∝ x, or y = kx) and inverse proportion (when one quantity doubles, the other halves: y ∝ 1/x, or y = k/x). MCQ success here also trains pattern recognition—a skill essential for competitive exams later.

**Q1.** If x and y are in direct proportion and x = 5 when y = 15, find y when x = 8. (A) 20 (B) 24 (C) 30 (D) 40 **Answer:** (B) 24 **Reason:** Direct proportion: y/x = constant. Here, k = 15/5 = 3, so y = 3 × 8 = 24. **Q2.** Which pair represents direct proportion? (A) Distance and time at constant speed (B) Speed and time to cover fixed distance (C) Number of workers and days to finish a job (D) Both (B) and (C) **Answer:** (A) Distance and time at constant speed **Reason:** At constant speed, d = v × t; as t increases, d increases proportionally. Options (B) and (C) are inverse. **Q3.** If a and b are inversely proportional with ab = 24, find b when a = 6. (A) 4 (B) 6 (C) 8 (D) 12 **Answer:** (A) 4 **Reason:** Inverse proportion: ab = constant = 24, so b = 24/6 = 4. **Q4.** The cost of 8 kg sugar is ₹240. What is the cost of 12 kg? (A) ₹280 (B) ₹320 (C) ₹360 (D) ₹400 **Answer:** (C) ₹360 **Reason:** Direct proportion: cost/weight = 240/8 = 30 per kg, so cost of 12 kg = 30 × 12 = ₹360. **Q5.** If y = k/x (inverse proportion), and y = 10 when x = 2, find k. (A) 5 (B) 12 (C) 20 (D) 30 **Answer:** (C) 20 **Reason:** Substitute into y = k/x: 10 = k/2, so k = 20. **Q6.** Two variables are in inverse proportion. When one is 8, the other is 6. Find the other when the first is 3. (A) 12 (B) 16 (C) 18 (D) 20 **Answer:** (B) 16 **Reason:** Constant = 8 × 6 = 48. When first = 3, other = 48/3 = 16. **Q7.** Which statement is true for inverse proportion? (A) Both quantities increase together (B) Both quantities decrease together (C) As one increases, the other decreases (D) The quantities remain equal **Answer:** (C) As one increases, the other decreases **Reason:** Inverse means y ∝ 1/x; they move in opposite directions. **Q8.** If 5 workers can paint a wall in 12 days, how many workers are needed to paint it in 6 days? (A) 8 (B) 9 (C) 10 (D) 12 **Answer:** (C) 10 **Reason:** Time inversely proportional to workers; constant = 5 × 12 = 60, so workers = 60/6 = 10. **Q9.** The equation for direct proportion between p and q is: (A) p × q = k (B) p/q = k (C) p + q = k (D) p² = kq **Answer:** (B) p/q = k **Reason:** Direct proportion: p = kq, or p/q = k (constant ratio). **Q10.** If x ∝ y and x = 3 when y = 9, find x when y = 15. (A) 5 (B) 5.5 (C) 6 (D) 7 **Answer:** (A) 5 **Reason:** x/y = constant = 3/9 = 1/3, so x = 15/3 = 5.

**Q11.** A recipe uses 3 cups flour for 2 cups sugar. To make 15 cups flour, how much sugar is needed? (A) 8 cups (B) 9 cups (C) 10 cups (D) 12 cups **Answer:** (C) 10 cups **Reason:** Direct proportion: flour/sugar = 3/2. For 15 cups flour, sugar = 15 × (2/3) = 10 cups. **Q12.** The speed of a car and the time to travel 300 km are in inverse proportion. At 60 km/h, it takes 5 hours. At what speed will it take 3 hours? (A) 90 km/h (B) 100 km/h (C) 110 km/h (D) 120 km/h **Answer:** (B) 100 km/h **Reason:** Inverse: speed × time = constant = 60 × 5 = 300. At 3 hours, speed = 300/3 = 100 km/h. **Q13.** If m and n are in direct proportion, and (m₁, n₁) = (4, 12) and (m₂, n₂) = (7, ?), find n₂. (A) 18 (B) 19 (C) 20 (D) 21 **Answer:** (D) 21 **Reason:** Constant ratio = 12/4 = 3, so n₂ = 7 × 3 = 21. **Q14.** 6 men can complete a project in 20 days. How many days will 8 men take? (A) 12 days (B) 15 days (C) 16 days (D) 18 days **Answer:** (B) 15 days **Reason:** Days inversely proportional to men: constant = 6 × 20 = 120. Days = 120/8 = 15. **Q15.** The total cost of notebooks is directly proportional to the quantity. 8 notebooks cost ₹96. Find the cost of 25 notebooks. (A) ₹250 (B) ₹280 (C) ₹300 (D) ₹320 **Answer:** (C) ₹300 **Reason:** Cost per notebook = 96/8 = ₹12. Total cost = 25 × 12 = ₹300. **Q16.** If p ∝ q and q ∝ r, then what is the relationship between p and r? (A) p ∝ r (B) p ∝ 1/r (C) p = q + r (D) No relationship **Answer:** (A) p ∝ r **Reason:** If p = k₁q and q = k₂r, then p = k₁k₂r, so p ∝ r (transitivity of proportion). **Q17.** The pressure of a gas is inversely proportional to its volume at constant temperature. When V = 4 L, P = 30 atm. Find P when V = 6 L. (A) 15 atm (B) 18 atm (C) 20 atm (D) 25 atm **Answer:** (C) 20 atm **Reason:** Inverse: PV = constant = 30 × 4 = 120. When V = 6, P = 120/6 = 20 atm. **Q18.** A typist types 40 pages in 8 hours. At this rate, how many pages will be typed in 15 hours? (A) 70 (B) 72 (C) 75 (D) 80 **Answer:** (C) 75 **Reason:** Pages directly proportional to hours. Rate = 40/8 = 5 pages/hour. In 15 hours: 5 × 15 = 75 pages. **Q19.** If x and y satisfy 3x = 5y, what is the relationship? (A) x ∝ y (B) x ∝ 1/y (C) x = y (D) x + y = constant **Answer:** (A) x ∝ y **Reason:** 3x = 5y implies x/y = 5/3 = constant, so x ∝ y (direct proportion). **Q20.** Four painters can paint 8 rooms in 6 days. How many painters are needed to paint 12 rooms in 6 days? (A) 5 (B) 6 (C) 7 (D) 8 **Answer:** (B) 6 **Reason:** Painters ∝ rooms (time constant). If 4 painters → 8 rooms, then x painters → 12 rooms: x = 4 × (12/8) = 6.

**Q21.** **Assertion (A):** If two quantities are in inverse proportion, their product is always constant. **Reason (R):** Inverse proportion is defined as y = k/x, where k is the constant of proportionality. (A) Both A and R true; R explains A (B) Both A and R true; R doesn't explain A (C) A true, R false (D) Both A and R false **Answer:** (A) Both A and R true; R explains A **Reason:** Inverse proportion y = k/x directly implies xy = k (product constant). R is the definition and fully justifies A. **Q22.** **Assertion (A):** In a direct proportion, if x doubles, y doubles. **Reason (R):** Direct proportion means y/x = k (constant). (A) Both A and R true; R explains A (B) Both A and R true; R doesn't explain A (C) A true, R false (D) Both A and R false **Answer:** (A) Both A and R true; R explains A **Reason:** From y = kx, if x → 2x, then y → k(2x) = 2(kx) = 2y. R (definition) fully explains A. **Q23.** A company's profit p is directly proportional to the number of items sold n, and inversely proportional to the cost per item c. If profit doubles when items sold double (at constant cost), which equation fits? (A) p = kn/c (B) p = knc (C) p = k/(nc) (D) p = nc/k **Answer:** (A) p = kn/c **Reason:** Direct proportion to n and inverse to c means p ∝ n/c, so p = kn/c. Doubling n doubles p. **Q24.** **Assertion (A):** If 10 workers finish a job in 8 days, then 20 workers will finish it in 4 days. **Reason (R):** The number of workers and days are inversely proportional if the total work is constant. (A) Both A and R true; R explains A (B) Both A and R true; R doesn't explain A (C) A true, R false (D) Both A and R false **Answer:** (A) Both A and R true; R explains A **Reason:** Workers × Days = constant (total work). Here, 10 × 8 = 80 = 20 × 4. R (inverse relation) explains A. **Q25.** A train travels 240 km in 4 hours. Another train travels at the same speed. Their distance and time are in: (A) Direct proportion (B) Inverse proportion (C) No proportion (D) Partial proportion **Answer:** (A) Direct proportion **Reason:** At constant speed (60 km/h), distance = speed × time. As time increases, distance increases proportionally (d ∝ t). **Q26.** **Assertion (A):** The equation y = 3/x represents an inverse proportion. **Reason (R):** In inverse proportion, the product of the two variables is a non-zero constant. (A) Both A and R true; R explains A (B) Both A and R true; R doesn't explain A (C) A true, R false (D) Both A and R false **Answer:** (A) Both A and R true; R explains A **Reason:** y = 3/x means xy = 3 (constant). This is the definition of inverse proportion; R explains A perfectly. **Q27.** If x is directly proportional to y, and y is inversely proportional to z, then x is: (A) Directly proportional to z (B) Inversely proportional to z (C) Neither (A) nor (B) (D) Proportional to z² **Answer:** (B) Inversely proportional to z **Reason:** x = k₁y and y = k₂/z, so x = k₁(k₂/z) = K/z. Thus x ∝ 1/z (inverse). **Q28.** A man completes 1/4 of work in 6 days. How many days for the entire work, assuming constant work rate? (A) 20 days (B) 24 days (C) 26 days (D) 30 days **Answer:** (B) 24 days **Reason:** Work directly proportional to days at constant rate. If 1/4 work = 6 days, total work = 4 × 6 = 24 days. **Q29.** **Assertion (A):** If p ∝ q² (p is directly proportional to the square of q), and p = 8 when q = 2, then p = 32 when q = 4. **Reason (R):** In p ∝ q², the constant of proportionality k satisfies p = kq². (A) Both A and R true; R explains A (B) Both A and R true; R doesn't explain A (C) A true, R false (D) Both A and R false **Answer:** (A) Both A and R true; R explains A **Reason:** From 8 = k(2²), k = 2. So p = 2(4²) = 2(16) = 32. R (definition) justifies A. **Q30.** Three quantities a, b, and c are related such that a ∝ b and b ∝ c. If a = 12 when b = 3, and b = 3 when c = 6, find a when c = 12. (A) 16 (B) 20 (C) 24 (D) 28 **Answer:** (C) 24 **Reason:** From a ∝ b: a = 4b. From b ∝ c: b = c/2. Combining: a = 4(c/2) = 2c. When c = 12, a = 2(12) = 24.

CBSE MCQ design intentionally includes plausible-sounding wrong answers that exploit common misconceptions. Recognizing these traps will prevent last-minute careless errors. **Trap 1: Confusing Direct and Inverse Relationships** Many students reverse the proportion. Example: "5 workers finish in 20 days; 10 workers finish in 40 days." This reverses the inverse relationship. The correct logic: workers ∝ 1/days, so doubling workers halves the days to 10 days. Watch for options that double/halve in the wrong direction. **Trap 2: Forgetting to Find the Constant First** Students rush and use wrong constants. Example: "If y = 2 when x = 5, find y when x = 10." A student might guess y = 4 (thinking 10/5 = 2), but the constant k = 2/5 = 0.4, so y = 0.4 × 10 = 4. Here the answer coincidentally matches, but in other setups, it won't. Always isolate k explicitly. **Trap 3: Mixing Direct and Inverse in Combined Scenarios** Problems involving work, speed, or cost often combine proportions. Example: "Cost ∝ weight. Weight ∝ quantity of items." Students confuse which quantity goes where. Trace through: Cost = k₁ × Weight, Weight = k₂ × Items, so Cost = k₁k₂ × Items. Cost ∝ Items (direct)—not inverse. **Trap 4: Ignoring Units and Real-World Context** Trick options ignore units. Example: "A car travels 300 km in 5 hours. Find the time for 450 km." A student might compute 300/5 = 60 and think 450/60 = 7.5, confusing speed with time. The correct approach: speed = 60 km/h is constant, so time = 450/60 = 7.5 hours. Check that your answer makes physical sense. **Trap 5: Misapplying Proportions to Non-Linear Relationships** Options may include y ∝ x², y ∝ √x, or y ∝ 1/x² (power laws), which look like proportion but have different constants. Assertion-reason MCQs test this: an answer like "p = 3/q represents inverse proportion" is true, but "p ∝ 1/q and p ∝ q represent the same relationship" is false (one is inverse, one is direct). Read the assertion carefully. **Trap 6: Incorrect Constant Calculation from Paired Values** When given multiple (x, y) pairs, one option uses the wrong pair to find k. Example: Points (2, 8) and (3, 12) both follow y = 4x. An incorrect option might use y = 8 and x = 3 to get k = 8/3, leading to a wrong answer. Always verify k with at least two pairs. **Strategy:** Before selecting an answer, ask: (1) Is this direct or inverse? (2) Did I find k correctly? (3) Does the magnitude make sense? (4) Are units consistent? This metacognitive check catches 70% of trap options.

In CBSE Class 9 Mathematics exams, MCQs on Direct and Inverse Proportions typically appear in Section A (1-mark, 30 seconds per question) or Section B (2-marks, 90 seconds per question). Effective time allocation is crucial. **Pre-Exam Preparation (1 week before):** Work through all 30 questions in this guide with a timer: 20 seconds for easy MCQs, 45 seconds for medium, and 90 seconds for hard/assertion-reason. Identify which question types you stumble on. If assertion-reason MCQs cost you 2+ minutes, practice identifying the logical link between A and R separately before committing to an answer. **During the Exam (Real-Time Strategy):** **Minutes 1-5:** Scan all MCQ questions on Direct and Inverse Proportions. Mark 2-3 "quick-win" questions (usually setup problems: "Cost of 8 kg sugar is ₹240; find cost of 12 kg"). These don't require deep conceptual reasoning—just plug numbers into the constant-ratio formula. Solve them first. **Minutes 6-12:** Tackle medium-difficulty questions (real-world scenarios, work-rate problems). These require identifying whether the relationship is direct or inverse, then computing the constant. Budget 50-60 seconds per question. If stuck after 50 seconds, mark and return. **Minutes 13-18:** Approach hard/assertion-reason MCQs methodically. For assertion-reason: first evaluate A (true/false), then evaluate R (true/false), then check if R explains A. Don't rush the "explanation" step—this is where most students lose marks. Example: "A: 3x = 5y implies direct proportion. R: Direct proportion means y/x = constant." Both are true, but does R explain A? Yes, because 3x = 5y → x/y = 5/3 = constant, which is the definition in R. **When You're Stuck:** - For proportion problems: Rewrite the relationship algebraically (e.g., y = kx or y = k/x). This clarifies the type instantly. - For time-work problems: Use the mantra "constant = workers × days" or "constant = speed × time." Solve for the unknown. - For assertion-reason: If the answer isn't obvious, eliminate options where A and R are both true but R doesn't explain A—these are frequent distractors. **Last 2 Minutes:** Review flagged questions. If time is running out, guess intelligently: In a randomly designed MCQ, option (B) or (C) appears slightly more often as the correct answer. However, never rely on this for questions where you've worked partway through—mark your best guess based on calculation, not pattern. **Post-Exam Review:** After the exam, identify which questions you got wrong. Was it a conceptual gap (didn't know direct vs. inverse) or a computational error (found k but miscalculated the final value)? Conceptual gaps require re-reading NCERT Chapter 11; computational errors require slower, more deliberate practice. **Bonus Tip for cbsetutor.ai Users:** Start a 3-day free trial at cbsetutor.ai to access adaptive MCQ quizzes that simulate real exam conditions. The platform tracks your weak question types and adjusts difficulty dynamically, saving you hours of unfocused practice.

Mastering Chapter 11 requires internalizing five core concepts and their algebraic forms: **1. Direct Proportion (y ∝ x)** Definition: y = kx, where k is the constant of proportionality. Key property: y/x = k (ratio is constant); if x doubles, y doubles. Quick check: Graph passes through origin (0, 0). Example: Cost of apples: if 3 kg costs ₹90, then k = 90/3 = 30 per kg. For 7 kg, cost = 30 × 7 = ₹210. **2. Inverse Proportion (y ∝ 1/x)** Definition: y = k/x, where k is the constant. Key property: xy = k (product is constant); if x doubles, y halves. Quick check: Graph is a hyperbola, never touching axes. Example: Workers and days: if 6 workers finish in 20 days, then k = 6 × 20 = 120. For 10 workers, days = 120/10 = 12. **3. Time-Work Formula** No. of workers × No. of days = Total work (constant). Variation: If work increases, either workers increase or days increase (or both). Example: 8 workers, 15 days → total work unit = 8 × 15 = 120. To finish in 10 days, workers needed = 120/10 = 12. **4. Speed-Distance-Time Relationship** Speed = Distance / Time. At constant speed, distance ∝ time (direct). At constant distance, speed ∝ 1/time (inverse). Example: A car covers 300 km in 5 hours (speed = 60 km/h). To cover 450 km at the same speed, time = 450/60 = 7.5 hours. **5. Combined Proportions** If x ∝ y and y ∝ z, then x ∝ z (transitive). If x ∝ y and y ∝ 1/z, then x ∝ 1/z. Example: Profit ∝ revenue, revenue ∝ sales. Thus, profit ∝ sales (direct). **Checkpoint: Test Yourself** Before an exam, solve these three problems without referring back: 1. If 12 m of cloth costs ₹480, find the cost of 18 m. [Answer: ₹720. Method: k = 480/12 = 40/m.] 2. If 9 taps fill a tank in 8 hours, how many taps fill it in 6 hours? [Answer: 12 taps. Method: taps × hours = constant = 72.] 3. Write the equation: x is directly proportional to y and inversely proportional to z. [Answer: x = ky/z.] If all three are correct and fast, you're exam-ready. If not, revisit the corresponding section above and redo those 10 MCQs.