NCERT Solutions for Class 8th Maths Chapter 3 Proportional Reasoning-2
Updated on 2026-09-19
About this chapter
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
- Proportionality
- Ratios in Maps
- Ratios in Maps · 3.3 Ratios with More than 2 Terms
- Ratios with More than 2 Terms
- Dividing a Whole in a Given Ratio
- A Slice of the Pie
- Inverse Proportions
Quick revision
| 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 |
Exercises
- In-text Questions — Proportionality Page 55
- In-text Questions — Ratios in Maps Page 56
- In-text Questions — Ratios in Maps · 3.3 Ratios with More than 2 Terms Page 57
- In-text Questions — Ratios with More than 2 Terms Page 58
- Figure it Out — Dividing a Whole in a Given Ratio Page 60
- In-text Questions — A Slice of the Pie Page 61
- Figure it Out — A Slice of the Pie Page 62
- In-text Questions — Inverse Proportions Page 63
- In-text Questions — Inverse Proportions Page 64
- Figure it Out — Inverse Proportions Page 65
- In-text Questions — Inverse Proportions Page 66
- Figure it Out — Inverse Proportions Page 67–68