Proportional Reasoning – 2

Overview

This chapter deepens proportional reasoning with direct and inverse proportion and their everyday applications such as speed, work and pricing. It builds on ratios from the earlier proportion chapter.

Key concepts

• In direct proportion, two quantities increase or decrease together; their ratio stays constant.
• In inverse proportion, one increases as the other decreases; their product stays constant.
• Speed, time and distance problems use proportional thinking.
• Graphs of direct proportion are straight lines through the origin.

Important formulae

• Direct: x ÷ y = constant, so x₁ ÷ y₁ = x₂ ÷ y₂.
• Inverse: x × y = constant, so x₁ × y₁ = x₂ × y₂.
• Speed = distance ÷ time.

Solved example

1. If 6 workers build a wall in 8 days, how long for 4 workers (same rate)?
2. Workers and days are in inverse proportion: 6 × 8 = 4 × d.
3. d = 48 ÷ 4 = 12 days.

Important questions

1. A car travels 180 km on 12 litres. How far on 20 litres (direct proportion)?
2. If 15 pipes fill a tank in 4 hours, how many pipes fill it in 3 hours?
3. A map scale is 1 cm to 50 km. What distance does 7 cm represent?
4. State whether the number of people and time taken to eat fixed food is direct or inverse.

Quick revision

Decide first whether a situation is direct (ratio constant) or inverse (product constant). Direct proportion graphs as a line through the origin. Speed-time-distance and work problems are classic applications.