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Complete Solutions and Summary of Direct and Inverse Proportions – NCERT Class 8 Mathematics Chapter 11

Comprehensive explanations, examples, and exercises on direct and inverse proportions, including real-life applications, tables, and problems from NCERT Class 8 Mathematics Chapter 11.

Updated: 10 months ago

Categories: NCERT, Class 8, Mathematics, Proportion, Direct Proportion, Inverse Proportion, Chapter 11

Tags: Direct Proportion, Inverse Proportion, Ratio, Proportionality, NCERT Class 8, Mathematics, Chapter 11, Real-life Applications, Exercises, Solutions

Direct and Inverse Proportions - Complete Study Guide

Chapter Overview

Direct Proportion Type

Inverse Proportion Type

x = ky Direct Formula

xy = k Inverse Formula

What You'll Learn

Direct Proportion

Understanding how quantities increase or decrease together, with formulas like \( \frac{x}{y} = k \).

Inverse Proportion

Exploring when one quantity increases as the other decreases, using \( xy = k \).

Real-Life Applications

Applying proportions to scenarios like cost, speed, and work.

Problem Solving

Solving examples involving scales, shadows, and mixtures.

Historical Context

This chapter introduces variations in quantities, starting with everyday examples like preparing tea or arranging chairs. It explains direct proportion where ratios remain constant, and inverse proportion where products are constant, building on basic math concepts.

Key Highlights

Key formulas include direct: \( x = ky \), inverse: \( xy = k \). Applications range from cost calculations to time-speed-distance problems, with activities to reinforce understanding.

Comprehensive Chapter Summary

1. Introduction to Proportions

The chapter begins with examples like Mohan preparing tea for varying numbers of people, illustrating how quantities change together. It lists situations where one quantity affects another, such as cost increasing with articles or time decreasing with speed. More content: Five additional situations include fuel consumption with distance, wages with hours worked, crop yield with land area, electricity bill with usage, and pages printed with ink used.

2. Direct Proportion

Definition and Examples

Direct proportion occurs when \( \frac{x}{y} = k \) (constant). Examples: Cost of sugar (\( \frac{36}{1} = 36 \)), petrol and distance (\( \frac{4}{60} = \frac{1}{15} \)). Formula: \( \frac{x_1}{y_1} = \frac{x_2}{y_2} \).

Activities and Observations

Clock hand activity shows angle proportional to time (\( \frac{T}{A} \) constant). Age table shows not always proportional. More formulas: Simple interest \( I = P \times \frac{r}{100} \times t \), compound interest \( A = P(1 + \frac{r}{100})^t \).

Solved Examples

Example 1: Cloth cost. Example 2: Pole shadow (\( x = 21 \) m). Example 3: Paper weight (\( x = 750 \)). Example 4: Train speed (\( x = 25 \) km, \( y = 200 \) min). Example 5: Map scale (\( y = 1200 \) km).

3. Inverse Proportion

Definition and Examples

Inverse proportion: \( xy = k \). Examples: Workers and time, speed and time. Zaheeda's travel: Time decreases as speed increases. Books and price: \( 40 \times 150 = 6000 \).

Activities

Counters arrangement shows rows and columns inverse. Tables check inverse pairs.

Solved Examples

Example 7: Pipes (\( x = 96 \) min). Example 8: Food provisions (\( y = 16 \) days). Example 9: Workers (\( y = 24 \)).

4. Additional Formulas and Content

Key Formulas Expanded

Direct: \( x \propto y \), \( x = ky \). Inverse: \( x \propto \frac{1}{y} \), \( xy = k \). Unitary method for solving. Scale in maps: \( \frac{\text{map distance}}{\text{actual distance}} = \text{scale} \).

5. Applications and Discussions

Real-World Use

Proportions in interest, maps, shadows, work rates. Discussion: Variables not always proportional (e.g., human growth).

Challenges

Identifying proportion type, solving with constants.

6. Summary of Variations

Direct: Ratios constant. Inverse: Products constant. More content: Examples like pressure-volume (inverse), force-acceleration (direct).

Questions and Answers from Chapter

Short Questions

Q1. Check if the parking charges are in direct proportion to the parking time. (Ex 11.1 Q1)

Answer: No.

Q2. In a mixture of paint, find parts of base for 20 parts red pigment. (Ex 11.1 Q2)

Answer: 160.

Q3. How much red pigment with 1800 mL base? (Ex 11.1 Q3)

Answer: 24.

Q4. Bottles filled in five hours? (Ex 11.1 Q4)

Answer: 700.

Q5. Actual length of bacteria? (Ex 11.1 Q5)

Answer: \( 10^{-4} \) cm.

Q6. Length of model ship? (Ex 11.1 Q6)

Answer: 21 cm.

Q7. Crystals in 5 kg sugar? (Ex 11.1 Q7 i)

Answer: \( 2.25 \times 10^7 \).

Q8. Distance covered in map? (Ex 11.1 Q8)

Answer: 4 cm.

Q9. Shadow length for 10 m 50 cm pole? (Ex 11.1 Q9 i)

Answer: 5.6 m.

Q10. Distance in 5 hours? (Ex 11.1 Q10)

Answer: 42 km.

Q11. Which are in inverse proportion? (Ex 11.2 Q1 i)

Answer: Yes.

Q12. Prize for 20 winners? (Ex 11.2 Q2)

Answer: 5000.

Q13. Angle for 12 spokes? (Ex 11.2 Q3)

Answer: 30°.

Q14. Sweets each if 20 children? (Ex 11.2 Q4)

Answer: 6.

Q15. Food last for 30 animals? (Ex 11.2 Q5)

Answer: 4 days.

Medium Questions

Q1. Find enlarged length if 20,000 times. (Ex 11.1 Q5)

Answer: 2 cm. Calculation: Actual length \( 10^{-4} \) cm, enlarged 20,000 times: \( 10^{-4} \times 20000 = 2 \) cm. (3 marks)

Q2. Crystals in 1.2 kg sugar? (Ex 11.1 Q7 ii)

Answer: \( 5.4 \times 10^6 \). Proportion: \( \frac{9 \times 10^6}{2} \times 1.2 = 5.4 \times 10^6 \). (3 marks)

Q3. Height of pole for 5m shadow. (Ex 11.1 Q9 ii)

Answer: 9.375 m. Ratio: \( \frac{5.6}{3.2} = \frac{x}{5} \), \( x = 9.375 \). (3 marks)

Q4. Which are in inverse proportion? (Ex 11.2 Q1 ii)

Answer: No, time and distance at uniform speed are direct. (3 marks)

Q5. Prize for 8 winners. (Ex 11.2 Q2)

Answer: 12500. Inverse: \( 100000 \div 8 = 12500 \). (3 marks)

Q6. Spokes for 40° angle. (Ex 11.2 Q3 iii)

Answer: 9. Inverse: \( 360 \div 40 = 9 \). (3 marks)

Q7. Time for 4 persons to rewire. (Ex 11.2 Q6)

Answer: 3 days. Inverse: \( 4 \div 3 \times 3 = 3 \). (3 marks)

Q8. Boxes with 20 bottles. (Ex 11.2 Q7)

Answer: 15. Inverse: \( 25 \times 12 \div 20 = 15 \). (3 marks)

Q9. Machines for 54 days. (Ex 11.2 Q8)

Answer: 36. Inverse: \( 42 \times 63 \div 54 = 36 \). (3 marks)

Q10. Time at 80 km/h. (Ex 11.2 Q9)

Answer: 1.5 hours. Inverse: \( 2 \times 60 \div 80 = 1.5 \). (3 marks)

Q11. Time for one person. (Ex 11.2 Q10 i)

Answer: 6 days. Inverse: \( 3 \times 2 = 6 \). (3 marks)

Q12. Persons for one day. (Ex 11.2 Q10 ii)

Answer: 6. Inverse: \( 3 \times 2 = 6 \). (3 marks)

Q13. Period length for 9 periods. (Ex 11.2 Q11)

Answer: 40 min. Inverse: \( 45 \times 8 \div 9 = 40 \). (3 marks)

Q14. Check inverse for area and crop. (Ex 11.2 Q1 iii)

Answer: No, direct. (3 marks)

Q15. Angle for 15 spokes. (Ex 11.2 Q3 ii)

Answer: 24°. Inverse: \( 360 \div 15 = 24 \). (3 marks)

Long Questions

Q1. Following are the car parking charges near a railway station up to 4 hours ₹60, 8 hours ₹100, 12 hours ₹140, 24 hours ₹180. Check if in direct proportion. (Ex 11.1 Q1)

Answer: Ratios: \( \frac{60}{4} = 15 \), \( \frac{100}{8} = 12.5 \), \( \frac{140}{12} \approx 11.67 \), \( \frac{180}{24} = 7.5 \). Not constant, so not direct proportion. Explanation: In direct proportion, ratio should be constant, but here it decreases, indicating not proportional.

Q2. Mixture of paint: 1 red with 8 base. Find base for 4,7,12,20 red. If 1 red needs 75 mL base, how much red for 1800 mL base? (Ex 11.1 Q2, Q3)

Answer: Base: 32,56,96,160. Red for 1800 mL: 24. Explanation: Ratio 1:8, so base = 8 × red. For Q3: \( \frac{1}{75} = \frac{x}{1800} \), x=24.

Q3. Machine fills 840 bottles in 6 hours. How many in 5 hours? (Ex 11.1 Q4)

Answer: 700. Explanation: Rate = 840/6 = 140 per hour. In 5 hours: 140 × 5 = 700. Direct proportion: \( \frac{840}{6} = \frac{x}{5} \), x=700.

Q4. Bacteria enlarged 50,000 times is 5 cm. Actual length? Enlarged 20,000 times length? (Ex 11.1 Q5)

Answer: Actual: \( 10^{-4} \) cm. Enlarged: 2 cm. Explanation: Actual = 5 / 50000 = 0.0001 cm. For 20000: 0.0001 × 20000 = 2 cm.

Q5. Model mast 9 cm, actual 12 m. Ship length 28 m, model length? (Ex 11.1 Q6)

Answer: 21 cm. Explanation: Ratio 9/1200 = x/2800, wait no: Mast ratio 9 cm : 12 m = 9:1200 = 3:400. Ship: 28 m = 2800 cm. Model = 2800 × (3/400) / (wait, correct: \( \frac{9}{12} = \frac{x}{28} \), but in m/cm mix. Actual: Ratio 9 cm / 12 m = 9/1200 = 0.0075. Model ship = 28 × 0.0075 × 100 = 21 cm.

Q6. 2 kg sugar has 9 × 10^6 crystals. In 5 kg? In 1.2 kg? (Ex 11.1 Q7)

Answer: 5 kg: 2.25 × 10^7. 1.2 kg: 5.4 × 10^6. Explanation: Per kg: 4.5 × 10^6. 5 × 4.5 × 10^6 = 2.25 × 10^7. 1.2 × 4.5 × 10^6 = 5.4 × 10^6.

Q7. Map scale 1 cm = 18 km. Distance on map for 72 km? (Ex 11.1 Q8)

Answer: 4 cm. Explanation: \( \frac{1}{18} = \frac{x}{72} \), x=4.

Q8. 5.6 m pole shadow 3.2 m. Shadow for 10.5 m pole? Height for 5 m shadow? (Ex 11.1 Q9)

Answer: i) 6 m. ii) 8.75 m. Explanation: Ratio 5.6/3.2. For i: (10.5 × 3.2)/5.6 = 6. For ii: (5 × 5.6)/3.2 = 8.75.

Q9. Truck 14 km in 25 min. Distance in 5 hours? (Ex 11.1 Q10)

Answer: 168 km. Explanation: Speed = 14/25 km/min = 33.6 km/h. In 5 h: 168 km.

Q10. Which in inverse? Workers and time. (Ex 11.2 Q1 i)

Answer: Yes. Explanation: More workers, less time; product constant.

Q11. Prize money table for winners. (Ex 11.2 Q2)

Answer: Inversely proportional. Table: 25000,20000,12500,10000,5000. Explanation: Product 100000 constant.

Q12. Spokes table, angle for 15, spokes for 40°. (Ex 11.2 Q3)

Answer: i) Yes inverse. ii) 24°. iii) 9. Explanation: Angle = 360/spokes.

Q13. Sweets if children reduced by 4. (Ex 11.2 Q4)

Answer: 6. Explanation: 24 × 5 / 20 = 6. Inverse.

Q14. Food for 20 animals last 6 days, for 30? (Ex 11.2 Q5)

Answer: 4 days. Explanation: 20 × 6 / 30 = 4. Inverse.

Q15. 3 persons rewire in 4 days, time for 4? (Ex 11.2 Q6)

Answer: 3 days. Explanation: 3 × 4 / 4 = 3. Inverse.

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Complete Solutions and Summary of Direct and Inverse Proportions – NCERT Class 8 Mathematics Chapter 11

Comprehensive explanations, examples, and exercises on direct and inverse proportions, including real-life applications, tables, and problems from NCERT Class 8 Mathematics Chapter 11.

Direct and Inverse Proportions

Chapter Overview

What You'll Learn

Direct Proportion

Inverse Proportion

Real-Life Applications

Problem Solving

Historical Context

Key Highlights

Comprehensive Chapter Summary

1. Introduction to Proportions

2. Direct Proportion

Definition and Examples

Activities and Observations

Solved Examples

3. Inverse Proportion

Definition and Examples

Activities

Solved Examples

4. Additional Formulas and Content

Key Formulas Expanded

5. Applications and Discussions

Real-World Use

Challenges

6. Summary of Variations

Key Concepts and Definitions

Direct Proportion

Inverse Proportion

Constant of Proportion

Unitary Method

Scale in Maps

Variation

Important Facts and Figures

Questions and Answers from Chapter

Short Questions

Q1. Check if the parking charges are in direct proportion to the parking time. (Ex 11.1 Q1)

Q2. In a mixture of paint, find parts of base for 20 parts red pigment. (Ex 11.1 Q2)

Q3. How much red pigment with 1800 mL base? (Ex 11.1 Q3)

Q4. Bottles filled in five hours? (Ex 11.1 Q4)

Q5. Actual length of bacteria? (Ex 11.1 Q5)

Q6. Length of model ship? (Ex 11.1 Q6)

Q7. Crystals in 5 kg sugar? (Ex 11.1 Q7 i)

Q8. Distance covered in map? (Ex 11.1 Q8)

Q9. Shadow length for 10 m 50 cm pole? (Ex 11.1 Q9 i)

Q10. Distance in 5 hours? (Ex 11.1 Q10)

Q11. Which are in inverse proportion? (Ex 11.2 Q1 i)

Q12. Prize for 20 winners? (Ex 11.2 Q2)

Q13. Angle for 12 spokes? (Ex 11.2 Q3)

Q14. Sweets each if 20 children? (Ex 11.2 Q4)

Q15. Food last for 30 animals? (Ex 11.2 Q5)

Medium Questions

Q1. Find enlarged length if 20,000 times. (Ex 11.1 Q5)

Q2. Crystals in 1.2 kg sugar? (Ex 11.1 Q7 ii)

Q3. Height of pole for 5m shadow. (Ex 11.1 Q9 ii)

Q4. Which are in inverse proportion? (Ex 11.2 Q1 ii)

Q5. Prize for 8 winners. (Ex 11.2 Q2)

Q6. Spokes for 40° angle. (Ex 11.2 Q3 iii)

Q7. Time for 4 persons to rewire. (Ex 11.2 Q6)

Q8. Boxes with 20 bottles. (Ex 11.2 Q7)

Q9. Machines for 54 days. (Ex 11.2 Q8)

Q10. Time at 80 km/h. (Ex 11.2 Q9)

Q11. Time for one person. (Ex 11.2 Q10 i)

Q12. Persons for one day. (Ex 11.2 Q10 ii)

Q13. Period length for 9 periods. (Ex 11.2 Q11)

Q14. Check inverse for area and crop. (Ex 11.2 Q1 iii)

Q15. Angle for 15 spokes. (Ex 11.2 Q3 ii)

Long Questions

Q1. Following are the car parking charges near a railway station up to 4 hours ₹60, 8 hours ₹100, 12 hours ₹140, 24 hours ₹180. Check if in direct proportion. (Ex 11.1 Q1)

Q2. Mixture of paint: 1 red with 8 base. Find base for 4,7,12,20 red. If 1 red needs 75 mL base, how much red for 1800 mL base? (Ex 11.1 Q2, Q3)

Q3. Machine fills 840 bottles in 6 hours. How many in 5 hours? (Ex 11.1 Q4)

Q4. Bacteria enlarged 50,000 times is 5 cm. Actual length? Enlarged 20,000 times length? (Ex 11.1 Q5)

Q5. Model mast 9 cm, actual 12 m. Ship length 28 m, model length? (Ex 11.1 Q6)

Q6. 2 kg sugar has 9 × 10^6 crystals. In 5 kg? In 1.2 kg? (Ex 11.1 Q7)

Q7. Map scale 1 cm = 18 km. Distance on map for 72 km? (Ex 11.1 Q8)

Q8. 5.6 m pole shadow 3.2 m. Shadow for 10.5 m pole? Height for 5 m shadow? (Ex 11.1 Q9)

Q9. Truck 14 km in 25 min. Distance in 5 hours? (Ex 11.1 Q10)

Q10. Which in inverse? Workers and time. (Ex 11.2 Q1 i)

Q11. Prize money table for winners. (Ex 11.2 Q2)