Chapter 11 — Direct and Inverse Proportions

Chapter 11 of Class 8 Mathematics, "Direct and Inverse Proportions", teaches how two quantities can vary together (direct proportion) or in opposite directions (inverse proportion), with applications to real-world problems like costs, speeds, distances, and resource distribution.

This chapter explores two fundamental concepts of variation: direct proportion (where quantities increase or decrease together at a constant ratio, like cost and weight) and inverse proportion (where one quantity increases as the other decreases at a constant product, like speed and travel time). Students learn to identify both types using ratios and equations, solve word problems using proportionality methods, and apply these concepts to practical scenarios including map scales, pricing, and work rates.

1. 01Direct proportion occurs when two quantities x and y maintain a constant ratio (x/y = k), meaning they increase or decrease together while their ratio stays the same
2. 02In direct proportion, if x and y correspond to values x₁, y₁ and x₂, y₂, then x₁/y₁ = x₂/y₂
3. 03Inverse proportion occurs when two quantities vary such that their product remains constant (xy = k), so when one increases, the other decreases proportionally
4. 04In inverse proportion, x₁y₁ = x₂y₂, or equivalently x₁/x₂ = y₂/y₁, showing the inverse relationship
5. 05Map scales demonstrate direct proportion: a scale of 1 cm : 8 km means distances on the map and actual distances are directly proportional
6. 06Practical applications include: cost and quantity purchased (direct), number of workers and time to complete a job (inverse), speed and travel time for a fixed distance (inverse)