Proportional Reasoning-1 – Class 8 Maths Chapter 7 Notes & NCERT PDF

Class 8MathematicsChapter 7NCERT book: Ganita Prakash - Mathematics Class 8

Notes and a simple summary of Chapter 7 of the NCERT Class 8 Mathematics book, with the official chapter PDF, flashcards, an MCQ quiz and an AI tutor for your doubts.

🎧 Listen to the content

Chapter 7: Proportional Reasoning-1

Introduction

Proportional Reasoning-1 begins with digital images of a tiger. Images A, C and D look alike even though their sizes differ, while B looks stretched and E looks squashed. Why? Because in A, C and D the width and height changed by the same factor (multiplication), not by the same amount (subtraction).

Key Ideas

7.1–7.2 Observing Similarity and Ratios

Image A is 60 mm wide and 40 mm high; C is 30 × 20 (both halved); D is 90 × 60.

Image B is 20 mm less in width and height than A, yet it looks distorted — subtracting the same amount does not keep things similar.

A ratio a : b means for every a units of the first quantity there are b units of the second. a and b are the terms of the ratio.

Ratios are proportional when their terms change by the same factor.

7.3 Ratios in their Simplest Form

Divide both terms by their HCF: 60 : 40 → 3 : 2 (HCF 20); 90 : 60 → 3 : 2 (HCF 30). Image B is 40 : 20 = 2 : 1 and E is 60 : 60 = 1 : 1 — different, so they are not proportional to A.

a : b :: c : d means the ratios a : b and c : d are proportional. Example: 60 : 40 :: 30 : 20.

7.4 Problem Solving with Proportional Reasoning

Lemonade: 6 glasses need 10 spoons of sugar. 18 is 3 × 6, so 6 : 10 :: 18 : 30 — 30 spoons.

Compound wall: Nitin 60 ft with 3 bags, Hari 40 ft with 2 bags: both 20 : 1, so the walls are equally strong.

Ages: Neelima 3, mother 30 → 1 : 10. Nine years later 12 : 39 = 4 : 13. Adding (or subtracting) the same number to both terms changes the ratio.

Filter coffee: regular 15 mL decoction : 35 mL milk; 'strong' 20 : 30 has more decoction per cup; 'light' 10 : 40 has less.

Trairāśika — the Rule of Three. When a : b :: c : d, then c = fa and d = fb for the same factor f, which gives

Cross multiplication: a : b :: c : d ⟹ ad = bc, so d = bc ÷ a.

Āryabhaṭa named the terms pramāṇa (measure, a), phala (fruit, b), ichchhā (requisition, c) and ichchhāphala (yield, d):

"Multiply the phala by the ichchhā and divide the resulting product by the pramāṇa."

Mid-day meal: 120 students need 15 kg of rice; 80 students → 120 : 15 :: 80 : 10, so 10 kg.

Units must match: 90 km in 150 minutes; for 4 hours use 240 minutes: 150 × x = 240 × 90 ⟹ x = 144 km.

Tea prices: Himachal 200 g for ₹200 means ₹1000 per kg; Meghalaya 1 kg for ₹800. The weight-to-price ratios (1 : 1 and 5 : 4) are not proportional, and Himachal tea is costlier.

Not every problem is a Rule of Three! At higher speed a trip takes less time, so 50 : 2 :: 75 : __ is wrong.

Shampoo prices in the book's table (6 mL for ₹2, 180 mL for ₹154, 1000 mL for ₹540) are not proportional to volume — the price per mL differs from pack to pack.

7.5 Sharing, but Not Equally!

To divide a quantity x in the ratio m : n, make m + n equal groups:

First part = m × x ÷ (m + n); second part = n × x ÷ (m + n).

12 counters in 3 : 1 → 9 and 3.

42 counters in 4 : 3 → 7 groups of 6 → 24 and 18.

Profit ₹4,000 shared in the investment ratio 75,000 : 25,000 = 3 : 1 → ₹3,000 and ₹1,000.

7.6 Unit Conversions

1 metre = 3.281 feet

1 square metre = 10.764 square feet

1 acre = 43,560 square feet

1 hectare = 10,000 square metres = 2.471 acres

1 mL = 1 cc; 1 litre = 1,000 mL

Fahrenheit = 9/5 × Celsius + 32, and Celsius = 5/9 × (Fahrenheit – 32). So 25 °C = 77 °F.

Worked Examples

Are 3 : 4 and 72 : 96 proportional? HCF(72, 96) = 24, so 72 : 96 = 3 : 4. Yes.

Fill in 14 : 21 :: 6 : __. The factor is 6/14 = 3/7, and 21 × 3/7 = 9, so the ratio is 6 : 9.

Sand and cement. 40 kg mixture in 3 : 1 has 30 kg sand and 10 kg cement. For 5 : 2 with 30 kg sand, cement = 30 × 2/5 = 12 kg. Add 2 kg.

Divide ₹4,500 in 2 : 3. 5 parts of ₹900: ₹1,800 and ₹2,700.

Līlāvatī problem. If 2½ palas of saffron cost 3/7 niska, then 9 niskas buy 9 ÷ 3/7 × 2½ = 52½ palas.

Figure it Out — Hints

Always convert to the same units before writing a proportion (minutes with minutes, grams with grams).

Check that the two quantities increase together; if one decreases as the other increases, the Rule of Three does not apply.

Use cross multiplication (ad = bc) to test statements like 21 : 6 :: 35 : 10 (21 × 10 = 6 × 35).

For ages, set up (x + 1) : (x + 5) = 1 : 2 rather than scaling the old ratio.

Maths Words

Ratio (अनुपात)

Proportion / proportional (समानुपात / समानुपाती)

Terms of a ratio (अनुपात के पद)

Simplest form (सरलतम रूप)

HCF (महत्तम समापवर्तक)

Factor of change (परिवर्तन गुणक)

Cross multiplication (वज्र गुणन)

Rule of Three (त्रैराशिक)

Unit conversion (इकाई रूपांतरण)

Similar figures (समरूप आकृतियाँ)

Share / part (भाग)

Historical and Indian Connections

Trairāśika (Rule of Three) was used by Āryabhaṭa and other Indian mathematicians to solve complex proportion problems. The saffron problem comes from Bhaskaracharya's Līlāvatī.

Common Mistakes

Adding or subtracting the same number to both terms to 'keep' a ratio — you must multiply or divide.

Mixing units (km with m, hours with minutes).

Using the Rule of Three for inverse situations (speed and time for a fixed distance).

Forgetting to reduce ratios to their simplest form before comparing.

Maths in Daily Life

Recipes, resizing photos, mixing paints or concrete, comparing prices per kg, sharing profits and converting land measures (acre, hectare) all use proportional reasoning.

💡 Key Learning Points

✓Understand that things look similar (proportional) when quantities change by the same factor (multiplication), not by the same amount.
✓Write ratios a : b, reduce them to simplest form using the HCF, and test proportionality (a : b :: c : d).
✓Solve Rule of Three (Trairāśika) problems by cross multiplication (ad = bc), converting to the same units first.
✓Divide a quantity in a given ratio m : n using m × x ÷ (m + n) and n × x ÷ (m + n).
✓Use common unit conversions (acre, hectare, square feet, litres, °C/°F) in proportion problems, and recognise when a situation is not proportional.

👨‍🏫 Teaching Tips

→Show a photo resized correctly and incorrectly on a projector, then measure the widths and heights with students.
→Make lemonade or sherbet in class and scale the recipe for more glasses to feel proportionality.
→Share 12, then 42, counters in pairs in different ratios before introducing the m × x ÷ (m + n) rule.
→Tell the Trairāśika story and have students label pramāṇa, phala, ichchhā and ichchhāphala in each word problem.
→Run the shampoo or rice price survey as a group project and discuss why prices are not proportional to size.
→Give a 'trap' problem (speed and time) and let students discover why the Rule of Three fails.
→Use the Neelima age example to show that adding the same number changes a ratio.

📋 Assessment Questions

Can the student reduce a ratio to its simplest form and decide whether two ratios are proportional?
Can the student find a missing term using a factor of change or cross multiplication?
Can the student divide a quantity in a given ratio?
Does the student convert units correctly before comparing or scaling?
Can the student tell whether a situation is directly proportional or not?

Ask the AI tutor about Class 8 Mathematics Chapter 7

Get question answers, important questions, a quick MCQ quiz or a simple explanation of “Proportional Reasoning-1”. Free — sign in with Google.

How the AI tutor works →

👩‍🏫 Class 8 Mathematics classes near you — online or at home

Prefer a real teacher? TalentJR teachers take live online classes anywhere in India and visit homes in selected cities. Tell us what you need and our team will call you with timings and fees.

Live online classesSmall groups or 1:1 with a teacher, from home, any city.
Teacher at homeA TalentJR teacher visits your home (selected cities).
At a centreClasses at a TalentJR partner centre near you, where available.

Keep learning

← Previous: Chapter 6We Distribute, Yet Things Multiply