Inverse Proportionality — Обратная пропорциональность
What is Inverse Proportionality?
Two quantities are inversely proportional when one increases while the other decreases by the same factor. Their product always stays constant!
Real-Life Example: Workers and Time
Imagine you need to paint a fence:
- 4 workers can finish in 6 days
- 8 workers (double) finish in 3 days (half)
- 2 workers (half) need 12 days (double)
Notice: When workers double, time halves!
4 × 6 = 8 × 3 = 2 × 12 = 24 (constant!)
4 × 6 = 8 × 3 = 2 × 12 = 24 (constant!)
vs Direct Proportionality
| Direct | Inverse |
|---|---|
| Both increase together | One ↑, other ↓ |
Ratios are equal: x₁/x₂ = y₁/y₂ |
Products are equal: x₁ × y₁ = x₂ × y₂ |
| y = kx (straight line) | y = k/x (curved line) |
The Constant Product (k)
The Golden Rule:
x × y = k
or equivalently: y = k/x
The constant k represents:
- Total work (worker-days)
- Total distance (speed × time)
- Total budget (price × quantity)
- Any fixed total amount
Interactive Table: Workers and Days (k = 24)
Interactive visualization — available in the full lesson
The curve is called a hyperbola — it never touches the axes!
How to Recognize Inverse Proportionality
More of one → Less of the other
Less of one → More of the other
Doubling one → Halves the other
Product remains constant
Quick Test: Multiply any pair of corresponding values. If you always get the same number, it's inverse proportionality!
Examples of Inverse Proportionality
| Quantity 1 | Quantity 2 | Constant |
|---|---|---|
| Workers | Time | Total work |
| Speed | Time | Distance |
| Pipe width | Fill time | Tank volume |
| Price/item | Quantity | Budget |
| Gear teeth | Rotations | Total teeth engaged |