Direct and Inverse Proportion: Class 8 Study Guide
In direct proportion both quantities rise together. In inverse proportion one rises as the other falls.
Two quantities are in direct proportion if their ratio stays the same, so when one doubles the other doubles. If 3 notebooks cost ₹60, then 5 notebooks cost ₹100, because the price of one is always ₹20. The rule is x ÷ y = constant.
Two quantities are in inverse proportion if their product stays the same, so when one doubles the other halves. If 6 workers finish a job in 10 days, then 12 workers need 5 days, because 6 × 10 = 12 × 5 = 60 worker-days. The rule is x × y = constant.
Comparing
| Situation | Type | Rule |
|---|---|---|
| Cost and number of items | Direct | Cost ÷ items is fixed |
| Distance and time at fixed speed | Direct | Distance ÷ time is fixed |
| Workers and days for a job | Inverse | Workers × days is fixed |
| Speed and time for a fixed distance | Inverse | Speed × time is fixed |
Solving
For direct proportion, use the unitary method or x₁ ÷ y₁ = x₂ ÷ y₂. For inverse proportion, use x₁ × y₁ = x₂ × y₂. A car going 60 km/h takes 4 hours for a trip. At 80 km/h it takes 60 × 4 ÷ 80 = 3 hours.
Do both rise together?
Ratio or product.
Find the unknown.
Direct: both rise. Inverse: one falls.
Graphs
A graph of direct proportion is a straight line through the origin. A graph of inverse proportion is a curve. Reading a distance–time graph with constant speed gives a straight line, and its steepness shows the speed.
Worked examples and common mistakes
Example: 8 pens cost ₹96. 5 pens cost 96 ÷ 8 × 5 = ₹60. Example: 4 pipes fill a tank in 9 hours. 6 pipes fill it in 4 × 9 ÷ 6 = 6 hours. Common mistakes: using direct proportion for a job with workers and days, and forgetting that more workers mean fewer days.
Check your understanding: model answers
1. ₹210. 2. 6 days. 3. Inverse.
✨ Key Takeaways
- Direct proportion: the ratio is constant.
- Inverse proportion: the product is constant.
- More workers mean fewer days.
- A direct proportion graph is a straight line through the origin.
💬 Questions for Discussion & Reflection
- 12 books cost ₹360. Find the cost of 7 books.
- 5 men build a wall in 12 days. How long for 10 men?
- Is speed and time for a fixed distance direct or inverse?