Class 8 Maths Chapter 7: Proportional Reasoning Ratios, Rule of Three & Applications

 Class 8 Maths – Chapter 7: Proportional Reasoning – 1
(Ganita Prakash Part 1)

Introduction

This chapter introduces proportional reasoning through ratios, proportions, and the Rule of Three. Students learn how to compare quantities, solve real‑life problems, and apply proportionality in contexts like recipes, construction, unit conversions, and sharing profits.

Basic Knowledge Required

• Multiplication and division basics

• Concept of ratios and fractions

• Simplification using HCF

• Cross multiplication method

• Unit conversions

Important Definitions

Ratio

Comparison of two quantities by division, written as a : b.

Proportion

Two ratios are proportional if they are equal in simplest form.

Notation: a : b :: c : d.

Rule of Three

If a : b :: c : d, then ad = bc. Used to find the fourth term when three are known.

Simplest Form of Ratio

Divide both terms by their HCF.

Formulas / Concepts Used

• a : b :: c : d → ad = bc

• Simplest form of ratio = divide terms by HCF

• Sharing in ratio m : n → parts = (m/(m+n)) × total and (n/(m+n)) × total

• Unit conversions: 1 m = 3.281 ft, 1 L = 1000 mL, etc.

Solved Examples (Step‑by‑Step)

Example 1: Are 3 : 4 and 72 : 96 proportional?

HCF of 72 and 96 = 24 → 72 : 96 = 3 : 4.

Yes, proportional.

Example 2: Lemonade problem

6 glasses : 10 spoons sugar

18 glasses : ?

Factor = 18 ÷ 6 = 3 → 10 × 3 = 30 spoons sugar.

Example 3: Wall construction

Nitin: 60 : 3 = 20 : 1

Hari: 40 : 2 = 20 : 1

Ratios equal → walls equally strong.

Example 4: Teacher‑student ratio

5 : 170 = 1 : 34.

Compare with your school ratio.

Example 5: Neelima’s age

At 3 years: 3 : 30 = 1 : 10

At 12 years: 12 : 39 = 4 : 13

Ratio changes when same number added.

Example 6: Sharing 42 counters in ratio 4 : 3

Total groups = 7 → each group = 42 ÷ 7 = 6

Partner = 4 × 6 = 24, You = 3 × 6 = 18.

Example 7: Profit sharing

Investment ratio = 75000 : 25000 = 3 : 1

Profit = 4000 ÷ 4 = 1000 per part

Prashanti = 3000, Bhuvan = 1000.

Example 8: Sand‑cement mixture

40 kg mixture, ratio 3 : 1 → sand = 30, cement = 10

New ratio 5 : 2 → cement = 12

Add 2 kg cement.

Example 9: Car travel

150 min : 90 km :: 240 min : x

150x = 240 × 90 → x = 144 km.

Example 10: Tea price comparison

Himachal: 200 g : ₹200 → 1 kg = ₹1000

Meghalaya: 1 kg = ₹800

Tea from Himachal more expensive.

 FIGURE IT OUT — COMPLETE SOLUTIONS

1. True proportions

(i) 4 : 7 :: 12 : 21 → True

(iv) 21 : 6 :: 35 : 10 → True

(vi) 24 : 8 :: 9 : 3 → True

2. Ratios proportional to 4 : 9

8 : 18, 12 : 27, 20 : 45

3. Ratios proportional to 18 : 24

3 : 4, 12 : 16, 20 : 26.67, 27 : 36

4. Similar rectangles

Measure width : height → those with equal ratios are similar.

5. Drawing proportional rectangles

All students’ drawings differ in size but look similar if ratios equal.

6. Brick wall ratio

Simplify grey : coloured bricks → ratio in simplest form.

7. Human figure ratios

Head : torso, torso : arms, torso : legs → realistic if proportional.

8. Mid‑day meal

120 : 15 :: 80 : x → x = 10 kg rice.

9. Unit conversions

1 m = 3.281 ft, 1 L = 1000 mL, etc. Apply to problems.

 Conclusion

Proportional reasoning is a powerful tool for solving real‑life problems. Ratios, proportions, and the Rule of Three help in scaling, sharing, and comparing quantities effectively.

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FAQs – Chapter 7: Proportional Reasoning

1. What is a ratio?

A ratio compares two quantities by division, written as a : b.

2. What is proportion?

Two ratios are proportional if they are equal in simplest form.

3. What is the Rule of Three?

If a : b :: c : d, then ad = bc. Used to find the fourth term.

4. How do you simplify a ratio?

Divide both terms by their HCF.

5. Are 3 : 4 and 72 : 96 proportional?

Yes, both reduce to 3 : 4.

6. How to share 42 counters in ratio 4 : 3?

Partner = 24, You = 18.

7. How to share profit ₹4000 in ratio 3 : 1?

Prashanti = ₹3000, Bhuvan = ₹1000.

8. How to adjust sand‑cement mixture?

Add 2 kg cement to make ratio 5 : 2.

9. What is cross multiplication?

If a : b :: c : d, then ad = bc.

10. What is unit conversion for length?

1 metre = 3.281 feet.

11. What is unit conversion for area?

1 sq. m = 10.764 sq. ft.

12. What is unit conversion for volume?

1 litre = 1000 mL.

13. What is temperature conversion?

°F = (9/5 × °C) + 32.

14. Why does ratio change when same number added?

Because proportionality depends on multiplication, not addition.

15. Which tea is more expensive?

Himachal tea (₹1000 per kg) vs Meghalaya tea (₹800 per kg).