Download Proportional Reasoning-2 – NCERT Class 8 Maths Chapter 10 PDF from the new NCERT Class 8 Maths book Ganita Prakash (2026-27). Read the chapter online or download the free PDF. Two ratios a : b and c : d are proportional when a × d = b × c. A ratio may have many terms: 8 : 4 : 2 : 1 says that whatever multiplies one term multiplies all of them.
Proportional Reasoning-2 PDF – Chapter 10 of Ganita Prakash (Part II)
Read the chapter online or download the PDF.
| Chapter | Download | |
|---|---|---|
| Chapter 10 PDF | Proportional Reasoning-2 | View PDF |
| Full book | NCERT Class 8 Maths – Ganita Prakash | View Book |
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What Is in Proportional Reasoning-2? (Class 8 Maths Chapter 10)
- Two ratios a : b and c : d are proportional when a × d = b × c. A ratio may have many terms: 8 : 4 : 2 : 1 says that whatever multiplies one term multiplies all of them.
- The RF (Representative Fraction) on a map, such as 1 : 60,00,000, is a ratio between two lengths, so it carries no unit — 1 cm on the map stands for 60,00,000 cm, that is 60 km, of straight-line ground distance.
- To divide a quantity x in the ratio a : b : c : …, add the terms and give each part its share: x × a/(a + b + c + …), x × b/(a + b + c + …), and so on.
- A pie chart divides 360° in the ratio of the data. Since 360° stands for the whole, each category gets an angle proportional to its count — one degree per person when 360 people are surveyed.
- Directly proportional quantities keep their quotient fixed: x₁/y₁ = x₂/y₂ = k. Inversely proportional quantities keep their product fixed: x₁y₁ = x₂y₂ = k.
- Whether a pair is direct or inverse is decided by the situation, not by a rule to memorise — ask what stays fixed. Fixed rate per item → direct. Fixed total job or fixed distance → inverse.
Quick revision – Proportional Reasoning-2
| Idea | What it says | How to use it | Quick example |
|---|---|---|---|
| Proportional ratios | A : b :: c : d when a × d = b × c | Cross-multiply and compare the two products | 6 : 3 and 4 : 2 → 6 × 2 = 12 = 3 × 4 |
| Ratio with many terms | A : b : c : d :: p : q : r : s when a/p = b/q = c/r = d/s | Find the factor from the one term you know, apply it to all | 8 : 4 : 2 : 1 halved is 4 : 2 : 1 : 0.5 |
| Representative Fraction | 1 : n means 1 unit on the map = n of the same units on the ground | Measure with a ruler, then multiply by the scale | 1 : 60,00,000 → 1 cm = 60 km |
| Dividing a whole | X in the ratio a : b : c gives x × a/(a+b+c), and so on | Add the terms, divide x by the sum, multiply back | 150 min in 3 : 4 : 3 : 5 → 30, 40, 30, 50 min |
| Pie chart angle | Angle = (part ÷ whole) × 360° | Convert each count to an angle, then draw with a protractor | 12 of 40 students → 12/40 × 360° = 108° |
| Direct proportion | Both change by the same factor; x/y stays constant | D = bc/a — the rule of three | 5 workers move 4500 bricks → 20 move 18000 |
| Inverse proportion | One changes by n, the other by 1/n; xy stays constant | X₁y₁ = x₂y₂ | 20 workers × 4 days = 10 workers × 8 days |
| The constant k | In xy = k, k is the fixed thing in the story | Name k before you calculate | Speed × time = distance (90 km) |
| Work in 1 hour | If a job takes t hours, 1/t of it is done in an hour | Add the one-hour shares, then invert | 1/3 + 1/2 = 5/6 → tank fills in 6/5 = 1.2 h |
| Triangle from a ratio | Angles: divide 180°. Sides: check a + b > c | Ratios fix the shape, not the size | Sides 3 : 4 : 5 work; 1 : 3 : 5 do not |
Old NCERT vs New – Class 8 Maths
This chapter is from Ganita Prakash, the new NCERT Class 8 Maths book (NCF-SE 2023, used from 2025-26). It replaced the old book “Mathematics”, so chapter numbers and names differ from older PDFs.
Solutions and All Chapters
Answers to the exercises and the full chapter list of Ganita Prakash:
Proportional Reasoning-2 – Overview
| Class | Class 8 |
| Subject | Maths |
| Book | Ganita Prakash – new NCERT book |
| Part | Part II |
| Chapter | 10 of 14 – Proportional Reasoning-2 |
| Book pages | Part II, 55 – 69 |
| Edition | NCERT Reprint 2026-27 |
| Session | 2026-27 |
| Free – read online or download |
Class 8 Maths Chapter 10 – FAQs
Where can I download Proportional Reasoning-2 Class 8 PDF?
From the download table on this page – the chapter PDF is free, and the full book is linked too.
What is Class 8 Maths Chapter 10 about?
Two ratios a : b and c : d are proportional when a × d = b × c. A ratio may have many terms: 8 : 4 : 2 : 1 says that whatever multiplies one term multiplies all of them.
Is Proportional Reasoning-2 from the new NCERT Class 8 Maths book?
Yes. It is chapter 10 of ganita prakash (part ii), the new NCERT Class 8 Maths book for 2026-27.
Is this the latest NCERT edition for 2026-27?
Yes – the PDF is NCERT’s latest edition for the 2026-27 session.
Where are the NCERT Solutions for Class 8 Maths Chapter 10?
Open the NCERT Solutions link above for answers to the exercise questions.