Class 8 Maths Proportional Reasoning – 1 Notes

Class 8 Maths Proportional Reasoning – 1 Notes

Chapter 7: Proportional Reasoning – 1

Overview:
This chapter introduces the concept of proportion, ratios, and their applications. Students learn how quantities relate to each other, how to solve problems using proportional reasoning, and how ratios simplify real-life calculations. The chapter lays the foundation for understanding scaling, comparisons, and direct/indirect relationships in numbers.

Key Concepts

  1. Ratio:
    • A ratio compares two quantities of the same kind.
    • Example: If 8 pencils cost ₹24, the ratio of pencils to cost = 8:24 = 1:3
  2. Proportion:
    • Two ratios are in proportion if they are equal: a:b = c:d
    • Example: 2:3 = 4:6
  3. Unitary Method:
    • Solve problems by finding the value of one unit and then multiplying.
    • Example: If 5 kg sugar costs ₹60, 1 kg costs ₹60 ÷ 5 = ₹12
  4. Direct Proportion:
    • When one quantity increases, the other also increases at the same rate.
    • Example: 2 kg apples cost ₹40 → 4 kg apples cost ₹80
  5. Inverse Proportion:
    • When one quantity increases, the other decreases in such a way that their product is constant.
    • Example: 4 workers finish a job in 10 days → 8 workers finish it in 5 days
  6. Applications:
    • Problems related to speed, time, work, cost, recipe scaling, mixture problems.

Important Points to Remember:

  • Always simplify ratios before comparing.
  • Use the cross-multiplication method to check proportions.
  • Identify whether it’s direct or inverse proportion.
  • The unitary method is powerful for scaling and real-life calculations.

Examples:

  1. Ratio Problem:
    • A box has 12 red and 8 blue balls. Ratio of red to blue = 12:8 = 3:2
  2. Proportion Problem:
    • Check if 3:4 = 9:12 → Cross multiply: 3×12 = 36, 4×9 = 36 → Yes
  3. Unitary Method:
    • 5 pens cost ₹50 → 1 pen costs ₹50 ÷ 5 = ₹10 → 8 pens cost ₹80
  4. Direct Proportion:
    • 3 kg sugar → ₹90 → 5 kg → 5×(90 ÷ 3) = ₹150
  5. Inverse Proportion:
    • 6 workers → 12 days → 4 workers → 6×12 ÷ 4 = 18 days

Questions

A. Very Short Answer Questions (1–10)

  1. Define ratio.
  2. Simplify the ratio 18:24.
  3. Define proportion.
  4. Check if 3:5 = 9:15.
  5. What is the unitary method?
  6. Find the unit cost if 7 pens cost ₹56.
  7. Write an example of direct proportion.
  8. Write an example of inverse proportion.
  9. A recipe uses 3 cups of flour for 12 servings. How much flour for 1 serving?
  10. Express the ratio 10:15 in lowest terms.

B. Short Answer Questions (11–25)

  1. Find the fourth term in proportion: 5:8 = ? : 32
  2. If 6 workers can do a job in 10 days, find how many days 3 workers will take (inverse proportion).
  3. A car travels 150 km in 3 hours. Find distance in 1 hour.
  4. If 7 kg sugar costs ₹210, find the cost of 1 kg.
  5. 12 liters of juice are made using 3 liters concentrate. Find the ratio of concentrate to juice.
  6. Check if 7:14 = 3:6.
  7. Simplify the ratio 56:72.
  8. A man earns ₹450 in 5 days. How much will he earn in 1 day?
  9. 4 pens cost ₹28. Find the cost of 10 pens.
  10. A factory produces 120 toys in 8 hours. How many toys in 5 hours?
  11. Direct proportion: 5 notebooks cost ₹150 → find cost of 8 notebooks.
  12. Inverse proportion: 12 men can build a wall in 15 days → 20 men?
  13. If 2 kg apples cost ₹80, find cost of 7 kg.
  14. Find missing term: 3:4 = ? : 28
  15. The ratio of boys to girls in a class is 3:2. If there are 18 boys, how many girls?

C. Word Problems / Application Questions (26–40)

  1. A car travels 240 km in 4 hours. Find speed per hour.
  2. 8 workers finish a job in 12 days. How long will 6 workers take?
  3. 5 kg sugar costs ₹150. Find cost of 12 kg.
  4. A train travels 180 km in 3 hours. How far in 7 hours?
  5. 6 pencils cost ₹18. How many pencils for ₹54?
  6. Ratio of red balls to blue balls in a bag is 5:3. Total balls = 64. Find number of red and blue balls.
  7. Direct proportion: 7 books cost ₹210 → 10 books cost?
  8. A tank is filled by 3 pipes in 6 hours. How long with 2 pipes?
  9. Ratio of boys to girls = 7:5. Total students = 60 → find number of boys and girls.
  10. 120 km covered in 3 hours. Time for 200 km?
  11. Cost of 9 pens = ₹72. Find cost of 15 pens.
  12. 4 men can build a wall in 20 days → 5 men?
  13. A recipe uses 2 cups sugar for 5 servings. How much sugar for 12 servings?
  14. A car uses 12 liters petrol for 180 km → for 300 km?
  15. A shopkeeper sells 15 kg rice for ₹750. Price per kg?

D. Higher Order / Thinking Questions (41–50)

  1. A worker earns ₹480 in 8 days. How much in 12 days?
  2. Direct proportion: 9 pencils cost ₹45 → 20 pencils cost?
  3. Inverse proportion: 8 workers complete a job in 10 days → 4 workers?
  4. A tank is filled by 5 pipes in 8 hours. How long with 10 pipes?
  5. Ratio of two numbers = 7:9. Sum = 160. Find the numbers.
  6. The ratio of ages of A and B = 4:5. Sum = 36 → find ages.
  7. A car travels 360 km in 6 hours. Find time for 540 km.
  8. A recipe uses 3 cups sugar for 12 servings → for 20 servings?
  9. The cost of 12 notebooks = ₹480. Find cost of 20 notebooks.
  10. 10 men build a wall in 15 days → 6 men?

Answers

A. Very Short Answer Questions (1–10)

  1. Ratio: A comparison of two quantities of the same kind, expressed as a:b.
  2. 18:24 = 3:4 (divide both by 6)
  3. Proportion: Two ratios that are equal, e.g., a:b = c:d
  4. 3:5 = 9:15 → Yes, 3×3 = 9, 5×3 = 15 ✅
  5. Unitary method: Solve by finding the value of 1 unit and then multiplying to find the required quantity.
  6. 7 pens → ₹56 → 1 pen = 56 ÷ 7 = ₹8
  7. Example: More work → more wages. 2 kg apples → ₹40, 4 kg → ₹80 (direct proportion)
  8. Example: More workers → less days to complete work (inverse proportion).
  9. 3 cups flour → 12 servings → 1 serving → 3 ÷ 12 = 0.25 cups
  10. 10:15 = 2:3

B. Short Answer Questions (11–25)

  1. 5:8 = x:32 → 5×32 = 8×x → 160 = 8x → x = 20
  2. 6 workers → 10 days → 3 workers → x days (inverse)
  • 6×10 = 3×x → 60 = 3x → x = 20 days
  1. 150 km / 3 h → 1 h = 150 ÷ 3 = 50 km
  2. 7 kg → ₹210 → 1 kg → 210 ÷ 7 = ₹30
  3. 3:12 → concentrate:juice = 3:12 = 1:4
  4. 7:14 = 3:6 → 7×6 = 42, 14×3 = 42 → Yes ✅
  5. 56:72 → divide by 8 → 7:9
  6. 5 days → ₹450 → 1 day → 450 ÷ 5 = ₹90
  7. 4 pens → ₹28 → 1 pen = 28 ÷ 4 = ₹7 → 10 pens → 7×10 = ₹70
  8. 120 toys → 8 h → 1 h → 120 ÷ 8 = 15 → 5 h → 15×5 = 75 toys
  9. 5 notebooks → ₹150 → 1 notebook = 150 ÷ 5 = 30 → 8 notebooks → 30×8 = ₹240
  10. 12 men → 15 days → 20 men → x days (inverse)
  • 12×15 = 20×x → 180 = 20x → x = 9 days
  1. 2 kg → ₹80 → 1 kg → 80 ÷ 2 = ₹40 → 7 kg → 40×7 = ₹280
  2. 3:4 = x:28 → 3×28 = 4×x → 84 = 4x → x = 21
  3. Boys:girls = 3:2 → 18 boys → girls = (2/3)×18 = 12

C. Word Problems / Application Questions (26–40)

  1. 240 km / 4 h → speed = 240 ÷ 4 = 60 km/h
  2. 8 workers → 12 days → 6 workers → x (inverse)
  • 8×12 = 6×x → 96 = 6x → x = 16 days
  1. 5 kg → ₹150 → 1 kg → 150 ÷ 5 = 30 → 12 kg → 30×12 = ₹360
  2. 180 km / 3 h → 60 km/h → 7 h → 60×7 = 420 km
  3. 6 pencils → ₹18 → 1 pencil = 18 ÷ 6 = 3 → ₹54 → 54 ÷ 3 = 18 pencils
  4. Ratio 5:3, total balls = 64 → total parts = 5+3=8 → 1 part = 64 ÷ 8 = 8
  • Red balls = 5×8 = 40, Blue = 3×8 = 24
  1. 7 books → ₹210 → 1 book = 210 ÷ 7 = 30 → 10 books → 30×10 = ₹300
  2. 3 pipes → 6 h → total work = 3×6 = 18 → 2 pipes → x h → 2×x = 18 → x = 9 h
  3. Boys:girls = 7:5, total = 60 → total parts = 12 → 1 part = 60 ÷ 12 = 5
  • Boys = 7×5 = 35, Girls = 5×5 = 25
  1. 120 km / 3 h = 40 km/h → 200 km → 200 ÷ 40 = 5 h
  2. 9 pens → ₹72 → 1 pen = 8 → 15 pens → 8×15 = ₹120
  3. 4 men → 20 days → total work = 4×20 = 80 → 5 men → x → 5x = 80 → x = 16 days
  4. 2 cups → 5 servings → 1 serving = 2 ÷ 5 = 0.4 cups → 12 servings = 12×0.4 = 4.8 cups
  5. 12 liters → 180 km → 1 km = 12 ÷ 180 = 0.0667 liters → 300 km = 0.0667×300 ≈ 20 liters
  6. 15 kg → ₹750 → 1 kg = 750 ÷ 15 = ₹50

D. Higher Order / Thinking Questions (41–50)

  1. ₹480 / 8 days = 60/day → 12 days → 60×12 = ₹720
  2. 9 pencils → ₹45 → 1 pencil = 5 → 20 pencils → 5×20 = ₹100
  3. 8 workers → 10 days → 4 workers → x
  • 8×10 = 4×x → 80 = 4x → x = 20 days
  1. 5 pipes → 8 h → total work = 5×8 = 40 → 10 pipes → x → 10x = 40 → x = 4 h
  2. Ratio = 7:9, sum = 160 → total parts = 7+9=16 → 1 part = 160 ÷ 16 = 10
  • Numbers = 7×10=70, 9×10=90
  1. Ratio = 4:5, sum = 36 → total parts = 9 → 1 part = 36 ÷ 9 = 4
  • Ages = 4×4=16, 5×4=20
  1. 360 km / 6 h = 60 km/h → 540 km → 540 ÷ 60 = 9 h
  2. 3 cups → 12 servings → 1 serving = 0.25 → 20 servings = 0.25×20 = 5 cups
  3. 12 notebooks → ₹480 → 1 notebook = 480 ÷ 12 = 40 → 20 notebooks → 40×20 = ₹800
  4. 10 men → 15 days → total work = 10×15 = 150 → 6 men → x → 6x = 150 → x = 25 days