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CBSE Class 8 Mathematics Chapter 11 Direct and Inverse Proportions Worksheet with Answers

Direct and Inverse Proportions form a critical foundation in CBSE Class 8 Mathematics, bridging arithmetic with algebraic thinking. This chapter from NCERT Class 8 Mathematics equips students to solve real-world problems involving speed, time, distance, cost, work completion, and resource allocation. This worksheet provides structured practice across all question types appearing in CBSE examinations.

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Key takeaways

  • ✓Direct proportion: when one quantity increases, the other increases proportionally (x₁/y₁ = x₂/y₂)
  • ✓Inverse proportion: when one quantity increases, the other decreases proportionally (x₁y₁ = x₂y₂)
  • ✓Time-work problems follow inverse proportion — more workers complete the same job in less time
  • ✓Speed-distance-time problems often demonstrate direct proportion between distance and time at constant speed
  • ✓Unitary method is the foundational technique for solving proportion problems in CBSE Class 8
  • ✓Case-study questions test real-world application skills, carrying 4 marks in recent CBSE patterns
  • ✓HOTS questions require students to identify whether a relationship is direct, inverse, or neither

Quick Chapter Recap: Direct and Inverse Proportions

Chapter 11 of NCERT Class 8 Mathematics introduces two fundamental relationships between quantities. In direct proportion, two quantities x and y are related such that their ratio remains constant: x/y = k (constant). When x increases, y increases proportionally. Common examples include cost and quantity of items, distance and time at constant speed, and wages and number of days worked. In inverse proportion, the product of two quantities remains constant: xy = k. When one quantity increases, the other decreases proportionally. Typical examples include speed and time for a fixed distance, number of workers and days to complete work, and number of pipes and time to fill a tank. The unitary method serves as the primary problem-solving tool. Students must first identify the type of proportion, then set up equations accordingly. Direct proportion uses x₁/y₁ = x₂/y₂ while inverse proportion uses x₁y₁ = x₂y₂. Time-work problems almost always demonstrate inverse relationships, while cost-quantity problems show direct relationships. Mastering these distinctions is essential for scoring well in CBSE Class 8 Mathematics examinations.
  • Direct proportion: ratio constant, both quantities change in same direction
  • Inverse proportion: product constant, quantities change in opposite directions
  • Use unitary method to find value of one unit before scaling up
  • Always verify if the relationship is direct, inverse, or neither before solving

Worksheet Instructions and Difficulty Level

This CBSE Class 8 Mathematics Chapter 11 worksheet is designed for medium difficulty, suitable for students who have completed the NCERT textbook exercises. Suggested time allocation is 90 minutes under exam conditions. The worksheet contains 25+ questions across multiple formats mirroring the latest CBSE examination pattern. Section A tests conceptual understanding through multiple-choice questions. Section B evaluates recall and application via fill-in-the-blanks. Section C uses true/false statements to identify common misconceptions. Section D features short-answer questions worth 2-3 marks each, requiring method display. Section E includes higher-order thinking questions and one case-study question that simulates real CBSE board exam patterns. Students should attempt all questions sequentially, showing complete working for calculation-based problems. Use a pencil for graphs or diagrams if required. A complete answer key with step-by-step explanations follows the question sections, enabling effective self-assessment and identification of weak areas for targeted revision.
  • Total Questions: 25+ across six sections
  • Suggested Time: 90 minutes
  • Difficulty Level: Medium (aligned with CBSE Class 8 standards)
  • Marking Scheme: MCQs (1 mark), Short (2-3 marks), Long (4-5 marks)
  • Materials needed: ruled paper, calculator (optional), pencil, eraser

Section A: Multiple Choice Questions (1 mark each)

This section contains six multiple-choice questions testing fundamental concepts from NCERT Class 8 Mathematics Chapter 11. Each question has four options with exactly one correct answer. Students should read each question carefully, identify whether the situation describes direct proportion, inverse proportion, or neither, then apply the appropriate formula. MCQs in CBSE Class 8 Mathematics often test conceptual clarity rather than lengthy calculations, so focus on understanding the relationship between quantities before selecting your answer. Common traps include confusing direct and inverse relationships or applying incorrect formulas. Mark your answers clearly and avoid spending more than one minute per question. These questions mirror the objective-type questions increasingly appearing in CBSE assessments and help build speed and accuracy essential for competitive examinations later. Review the answer key explanations to understand why incorrect options are wrong, strengthening your conceptual foundation for Class 8 Mathematics solutions.
  • Q1. If 15 articles cost ₹45, what will 25 articles cost? (a) ₹65 (b) ₹75 (c) ₹85 (d) ₹95
  • Q2. Which of the following is an inverse proportion? (a) Number of books and their cost (b) Speed and time for fixed distance (c) Area of circle and radius (d) Distance and time at constant speed
  • Q3. If 8 workers can complete work in 12 days, how many days will 6 workers take? (a) 9 days (b) 16 days (c) 14 days (d) 10 days
  • Q4. In direct proportion, if x increases, y will: (a) decrease (b) remain constant (c) increase (d) become zero
  • Q5. The relationship between circumference and diameter of a circle is: (a) direct proportion (b) inverse proportion (c) neither (d) cannot be determined
  • Q6. If x and y are in inverse proportion, then: (a) x + y = constant (b) x - y = constant (c) xy = constant (d) x/y = constant

Section B: Fill in the Blanks (1 mark each)

Fill-in-the-blank questions test recall and application of direct and inverse proportion formulas from Class 8 Mathematics Chapter 11. Students must complete each statement with the precise mathematical term, number, or expression. These questions assess understanding of proportion terminology, formulae, and numerical relationships without providing answer choices. Read each sentence carefully and ensure your answer fits grammatically and mathematically. Common topics include identifying the type of proportion, completing proportion equations, and stating key properties. Write answers clearly in the space provided. Avoid spelling errors in mathematical terms. These questions develop precision in mathematical language, an important skill for CBSE board examinations where examiners award marks only for exactly correct answers. NCERT Class 8 Mathematics emphasizes accurate use of terminology throughout Chapter 11, so review definitions before attempting this section. Time management is crucial — spend no more than 30 seconds per blank. If unsure, move on and return later rather than wasting time guessing randomly.
  • Q7. Two quantities are in _______ proportion if their ratio remains constant.
  • Q8. If 12 men can dig a pond in 8 days, then 18 men will dig it in _______ days.
  • Q9. In inverse proportion, the product of two quantities is _______.
  • Q10. The cost of 5 kg of rice is ₹200. The cost of 1 kg is _______.
  • Q11. Speed and time for a fixed distance are in _______ proportion.
  • Q12. If x/y = 3/4 and they are in direct proportion, then 2x/2y = _______.

Section C: True or False (1 mark each)

True/false questions identify common misconceptions about direct and inverse proportions that appear frequently in CBSE Class 8 Mathematics assessments. Read each statement carefully and determine whether it is mathematically correct or incorrect. Write 'True' or 'False' clearly beside each statement number. These questions often contain subtle errors in wording or mathematical relationships designed to test deep understanding rather than superficial memorization. Pay special attention to absolute terms like 'always', 'never', and 'only' which often signal false statements. For example, stating that all time-based problems are inverse proportions is false because distance and time at constant speed show direct proportion. NCERT Class 8 Mathematics Chapter 11 emphasizes careful identification of proportion types before solving. If marking a statement false, be prepared to mentally justify why — this helps reinforce correct understanding. The answer key provides detailed explanations for each statement, turning this assessment into a powerful learning tool for Class 8 Mathematics notes revision.
  • Q13. More the number of workers, less the time taken to finish a job (assuming same efficiency).
  • Q14. The cost of articles and the number of articles purchased are always in direct proportion.
  • Q15. If x and y are in inverse proportion, doubling x will halve y.
  • Q16. Direct proportion can be represented as y = kx where k is a constant.
  • Q17. The relationship between side of a square and its area is direct proportion.
  • Q18. If speed is doubled, time taken to cover the same distance is halved.

Section D: Short Answer Questions (2-3 marks each)

Short-answer questions require complete working and method display for full marks in CBSE Class 8 Mathematics examinations. Each question is worth 2 to 3 marks depending on complexity. Students must show all calculation steps clearly, write relevant formulas, identify the type of proportion, and present the final answer with appropriate units. Simply writing the final answer without working fetches zero marks even if correct. Use the unitary method systematically: find the value for one unit, then multiply for the required quantity. Clearly state whether you are using direct proportion formula (x₁/y₁ = x₂/y₂) or inverse proportion formula (x₁y₁ = x₂y₂). Underline or box your final answer for clarity. These questions form the backbone of CBSE scoring, typically contributing 15-20 marks in a 40-mark chapter test. Practice writing neat, logical solutions that an examiner can follow easily. Time management is critical — allocate 3-4 minutes per question. The Class 8 Mathematics solutions provided in the answer key demonstrate the exact presentation expected by CBSE examiners.
  • Q19. A car travels 240 km in 4 hours. How far will it travel in 7 hours at the same speed? (3 marks)
  • Q20. If 15 workers can build a wall in 48 hours, how many workers are needed to build it in 30 hours? (3 marks)
  • Q21. The cost of 12 pens is ₹156. Find the cost of 8 such pens. (2 marks)
  • Q22. A hostel has food for 150 students for 40 days. If 30 more students join, for how many days will the food last? (3 marks)
  • Q23. If 35 meters of cloth costs ₹1,470, what length can be purchased for ₹882? (2 marks)

Section E: Long Answer and HOTS Questions (4-5 marks each)

Long-answer questions and higher-order thinking skill (HOTS) problems test deeper understanding of proportion concepts from NCERT Class 8 Mathematics Chapter 11. Each question carries 4 to 5 marks and requires multi-step reasoning, integration of concepts, or application to unfamiliar contexts. HOTS questions may ask students to determine whether a given relationship is direct proportion, inverse proportion, or neither — testing analytical skills beyond rote application. Show all working methodically: state given information, identify the type of proportion with justification, write the applicable formula, substitute values, perform calculations step-by-step, and state the conclusion clearly. These questions often combine proportion with other mathematical concepts like percentages, fractions, or algebra. Allocate 6-8 minutes per question. CBSE marking schemes award partial credit for correct method even if the final answer is wrong, so never leave a question blank. The detailed Class 8 Mathematics solutions in the answer key demonstrate how to structure extended responses that maximize marks. Practice these regularly to build confidence for board examinations.
  • Q24. A factory requires 42 machines to produce a given number of articles in 56 days. How many machines would be required to produce the same number of articles in 48 days? Explain the type of proportion involved. (4 marks)
  • Q25. Two quantities x and y are related as: when x = 6, y = 18; when x = 10, y = 30. Determine if x and y are in direct proportion, inverse proportion, or neither. Justify your answer mathematically. (5 marks)
  • Q26. Rashmi can walk 8.2 km in 2 hours. How long will she take to walk 14.35 km at the same speed? Also find the distance she covers in 3.5 hours. (5 marks)

Section F: Case Study Question (4 marks)

Case-study questions have become a staple in CBSE Class 8 Mathematics examinations since 2020, testing application skills in real-world contexts. Students read a short scenario describing a practical situation involving direct or inverse proportions, then answer 3-4 sub-questions based on the passage. Each case study typically carries 4 marks total. Read the passage carefully, underline key numerical data, and identify the relationships between quantities before attempting sub-questions. These questions assess comprehension, data interpretation, and mathematical application simultaneously. Sub-questions may be MCQ format or very short answer type. Common contexts include shopping scenarios, construction projects, travel planning, or resource allocation — all areas where proportion concepts apply daily. The case study below mirrors actual CBSE board exam patterns. Spend about 8-10 minutes on the complete case study. Show working for calculation-based sub-questions even if the format is MCQ. This section helps students see why NCERT Class 8 Mathematics concepts matter beyond textbooks, building mathematical literacy for real life and competitive examinations.

Complete Answer Key with Explanations

This comprehensive answer key provides correct answers with step-by-step explanations for every question in the worksheet. Students should attempt all questions independently before consulting this section. Use the answer key for self-assessment: mark your answers, calculate your score, and identify topics requiring revision. The explanations demonstrate the exact working and presentation expected in CBSE Class 8 Mathematics examinations. For MCQs, explanations clarify why the correct option is right and why others are wrong. For numerical problems, every calculation step is shown using proper mathematical notation. Pay special attention to how proportion type is identified and which formula is applied. If you made errors, revisit the relevant NCERT Class 8 Mathematics Chapter 11 sections and practice similar problems. Common mistakes include confusing direct and inverse proportions, calculation errors, and omitting units in final answers. Regular worksheet practice with answer key review significantly improves examination performance. CBSETUTOR.ai offers unlimited personalized worksheets with instant photo-upload doubt solving at just ₹999/month for Classes 6-12, with a 3-day free trial to experience AI-powered learning.
  • Section A Answers: 1(b), 2(b), 3(b), 4(c), 5(a), 6(c)
  • Section B Answers: 7-direct, 8-5.33 or 16/3, 9-constant, 10-₹40, 11-inverse, 12-3/4
  • Section C Answers: 13-True, 14-True, 15-True, 16-True, 17-False, 18-True
  • Section D worked solutions provided below
  • Section E detailed solutions provided below
  • Case study answers: 27-24 students, 28-8 buses, 29-24 students, 30-direct proportion

Section A Detailed Solutions

A2. Answer: (b) Speed and time for fixed distance. Explanation: When distance is constant, if speed increases, time decreases proportionally. This is inverse proportion: speed × time = constant distance. Option (a) is direct proportion, option (c) shows quadratic relationship (area = πr²), option (d) is direct proportion (distance = speed × time). A3. Answer: (b) 16 days. Explanation: Workers and days are inversely proportional for the same work. Using formula: 8 × 12 = 6 × d, therefore d = 96/6 = 16 days. More workers complete work faster; fewer workers take longer. A4. Answer: (c) increase. Explanation: In direct proportion, both quantities change in the same direction. When x increases, y also increases proportionally. The ratio x/y remains constant. A5. Answer: (a) direct proportion. Explanation: Circumference = πd where d is diameter. Since π is constant, C and d are directly proportional. When diameter doubles, circumference doubles. A6. Answer: (c) xy = constant. Explanation: This is the defining property of inverse proportion. The product of inversely proportional quantities remains constant even as individual values change.

Section B, C, and D Detailed Solutions

B8. Answer: 5.33 days or 5⅓ days or 16/3 days. Explanation: Men and days are inversely proportional. 12 × 8 = 18 × d, so d = 96/18 = 16/3 ≈ 5.33 days. More men finish work faster. B12. Answer: 3/4. Explanation: In direct proportion, the ratio remains constant regardless of scalar multiplication. Multiplying both numerator and denominator by the same number (2) does not change the ratio. C17. Answer: False. Explanation: Side and area of square show quadratic relationship (Area = side²), not direct proportion. When side doubles, area becomes four times, not double. D20. Solution: Workers and hours are inversely proportional for same work. Let required workers = w. Then 15 × 48 = w × 30. Therefore w = (15 × 48)/30 = 720/30 = 24 workers needed. Answer: 24 workers. D21. Solution: Cost and number of pens are directly proportional. 12 pens cost ₹156, so 1 pen costs ₹156/12 = ₹13. Therefore 8 pens cost ₹13 × 8 = ₹104. Answer: ₹104. D22. Solution: More students consume food faster — inverse proportion. Food lasts for 150 students for 40 days. Total students now = 150 + 30 = 180. Let food last for d days. 150 × 40 = 180 × d, so d = 6000/180 = 33.33 days or 33⅓ days. Answer: 33⅓ days.

Section E and Case Study Detailed Solutions

E25. Solution: Check if x/y is constant (direct) or xy is constant (inverse). When x = 6, y = 18: ratio = 6/18 = 1/3, product = 6 × 18 = 108. When x = 10, y = 30: ratio = 10/30 = 1/3, product = 10 × 30 = 300. Since x/y remains constant (1/3) but xy does not, x and y are in direct proportion. We can also write y = 3x, confirming direct proportion. Answer: Direct proportion. E26. Solution: Distance and time are directly proportional at constant speed. (i) 8.2 km in 2 hours means 1 km in 2/8.2 hours. Therefore 14.35 km takes (2/8.2) × 14.35 = 28.7/8.2 = 3.5 hours. (ii) In 2 hours, distance = 8.2 km. In 1 hour, distance = 4.1 km. In 3.5 hours, distance = 4.1 × 3.5 = 14.35 km. Answers: (i) 3.5 hours (ii) 14.35 km. Case Study: Q27. 120 students in 5 buses means 120/5 = 24 students per bus. Q28. For 192 students: 192/24 = 8 buses needed. Q29. 7 buses accommodate 7 × 24 = 168 students. Students left = 192 - 168 = 24. Q30. More students require more buses in the same ratio — direct proportion.

Frequently asked questions

What is the difference between direct and inverse proportion in CBSE Class 8 Mathematics?+
In direct proportion, two quantities increase or decrease together, maintaining a constant ratio (x/y = k). In inverse proportion, when one quantity increases, the other decreases such that their product remains constant (xy = k). Cost-quantity problems typically show direct proportion, while worker-time problems show inverse proportion.
How do I identify if a word problem involves direct or inverse proportion?+
Ask yourself: if one quantity increases, does the other increase (direct) or decrease (inverse)? Common direct proportion indicators include cost and items, distance and time at constant speed. Inverse proportion signals include more workers finishing work in less time, higher speed covering distance in less time.
What is the unitary method and why is it important for Class 8 Mathematics Chapter 11?+
The unitary method finds the value of one unit first, then scales up to the required quantity. It is the foundational approach taught in NCERT Class 8 Mathematics for solving proportion problems. For example, if 5 books cost ₹100, find cost of 1 book (₹20), then multiply for 8 books (₹160).
Are time and work problems always inverse proportion in CBSE Class 8?+
Yes, when the amount of work is fixed and worker efficiency is constant. More workers complete the job in less time (inverse). However, if work amount changes or efficiency varies, the relationship may differ. Always read the question carefully to identify what remains constant.
How many marks does Chapter 11 Direct and Inverse Proportions carry in CBSE Class 8 exams?+
Chapter 11 typically carries 8-12 marks in an 80-mark CBSE Class 8 Mathematics annual examination. Questions range from 1-mark MCQs and fill-in-the-blanks to 4-5 mark application problems and case studies. Proper practice with worksheets ensures strong performance on this scoring chapter.
What are common mistakes students make in proportion problems?+
Common errors include: confusing direct and inverse proportions, incorrect formula application, forgetting to use unitary method, calculation mistakes, omitting units in final answers, and not showing working steps. Always identify proportion type first, write the formula, then solve systematically.
How can CBSETUTOR.ai help with NCERT Class 8 Mathematics Chapter 11 practice?+
CBSETUTOR.ai provides unlimited AI-generated worksheets tailored to your learning level, instant doubt resolution through photo upload, and step-by-step solutions for proportion problems. At ₹999/month for all classes 6-12, it is affordable for every student. The 3-day free trial lets you experience personalized learning before committing.
Should I memorize formulas for direct and inverse proportion?+
Yes, memorize the core formulas: direct proportion x₁/y₁ = x₂/y₂ or y = kx, and inverse proportion x₁y₁ = x₂y₂ or y = k/x. However, understanding when to apply each formula is more important than rote memorization. Practice diverse problems to build intuition about proportion relationships.
How much time should I spend on this worksheet?+
The suggested time is 90 minutes under exam conditions. Allocate approximately 10 minutes for Section A, 8 minutes for Section B, 7 minutes for Section C, 20 minutes for Section D, 25 minutes for Section E, 10 minutes for the case study, and 10 minutes for review and checking.
Are all real-world relationships either direct or inverse proportion?+
No. Many relationships are neither. For example, the relationship between side of a square and its area is quadratic (Area = side²), not direct proportion. Population growth is exponential. Students must test whether ratio or product remains constant before concluding proportion type.

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