Chapter 10 — Proportional Reasoning-2

Chapter 10 of the Class 8 Maths NCERT textbook (Ganita Prakash), "Proportional Reasoning-2", extends proportional reasoning concepts by teaching ratios with multiple terms, how to divide quantities in given ratios, pie chart construction using proportional angles, and inverse proportions where quantities change in opposite directions.

• Ratios with many terms — The chapter widens ratios beyond two terms, teaching a systematic way to divide a whole quantity across several parts so that every part scales by the same factor of the total.
• Proportion made visual — Pie charts turn proportional reasoning into pictures: each slice's angle is a share of 360° proportional to its data value, and map scales (RF) show ratios linking drawn and real distances.
• Direct versus inverse — Students contrast direct proportion (constant quotient) with inverse proportion (constant product, xy = k), where one quantity grows exactly as the other shrinks — two distinct signatures of related change.

1. 01Two ratios a : b and c : d are proportional if a × d = b × c (cross-multiplication test)
2. 02Representative Fraction (RF) on maps shows the ratio between map distance and actual geographical distance (e.g., 1 : 60,00,000 means 1 cm on map = 60 km on ground)
3. 03Ratios with multiple terms (a : b : c : d) are proportional if all terms scale by the same factor
4. 04To divide a quantity x in ratio p : q : r : s, each part is x × (term / sum of all terms)
5. 05Pie chart angles are proportional to data: angle = (value / total) × 360°
6. 06Inverse proportions: when one quantity increases by factor n, the other decreases by factor 1/n, with xy = k (constant product)
7. 07Direct proportion: quotient remains constant (x₁/y₁ = x₂/y₂). Inverse proportion: product remains constant (x₁y₁ = x₂y₂)