Proportion
2.8 Proportion – Cheat Sheet
1. Proportional Symbol and Meaning
- ∝ means “is proportional to” – two quantities change in a consistent ratio.
- To change from proportional form to equation form, introduce a constant k:
- If y ∝ x → y = kx
- If y ∝ 1/x → y = k/x
2. Types of Proportion
| Type | Proportional Form | Equation Form | Example |
|---|---|---|---|
| Direct (linear) | y ∝ x | y = kx | If y = 12 when x = 4 → k = 3 → y = 3x |
| Direct square | y ∝ x² | y = kx² | If y = 20 when x = 2 → k = 5 → y = 5x² |
| Direct square root | y ∝ √x | y = k√x | If y = 12 when x = 9 → k = 4 → y = 4√x |
| Direct cube | y ∝ x³ | y = kx³ | If y = 54 when x = 3 → k = 2 → y = 2x³ |
| Direct cube root | y ∝ ³√x | y = k³√x | If y = 10 when x = 8 → k = 5 → y = 5³√x |
| Inverse (linear) | y ∝ 1/x | y = k/x | If y = 8 when x = 4 → k = 32 → y = 32/x |
| Inverse square | y ∝ 1/x² | y = k/x² | If y = 5 when x = 2 → k = 20 → y = 20/x² |
| Inverse square root | y ∝ 1/√x | y = k/√x | If y = 6 when x = 9 → k = 18 → y = 18/√x |
| Inverse cube | y ∝ 1/x³ | y = k/x³ | If y = 4 when x = 2 → k = 32 → y = 32/x³ |
| Inverse cube root | y ∝ 1/³√x | y = k/³√x | If y = 15 when x = 27 → k = 45 → y = 45/³√x |
3. Direct Proportion Method
- Write y ∝ xⁿ (n depends on type: 1, 2, 1/2, 3, 1/3).
- Change to equation y = kxⁿ.
- Use given values to find k.
- Use k to find unknowns.
Example:
y ∝ x², y = 18 when x = 3
→ 18 = k(3²) → 18 = 9k → k = 2 → y = 2x²
If x = 5 → y = 2(25) = 50
4. Inverse Proportion Method
- Write y ∝ 1/xⁿ.
- Change to equation y = k/xⁿ.
- Use given values to find k.
- Use k to find unknowns.
Example:
y ∝ 1/x, y = 10 when x = 4
→ 10 = k/4 → k = 40 → y = 40/x
If x = 5 → y = 8
5. Examples Table for Quick Reference
| Problem | Type | Equation | Answer |
|---|---|---|---|
| y ∝ x, y = 12 when x = 4 | Direct linear | y = 3x | y = 21 when x = 7 |
| y ∝ x², y = 20 when x = 2 | Direct square | y = 5x² | y = 80 when x = 4 |
| y ∝ √x, y = 12 when x = 9 | Direct square root | y = 4√x | y = 8√x when k = 8 |
| y ∝ 1/x, y = 8 when x = 4 | Inverse linear | y = 32/x | y = 16 when x = 2 |
| y ∝ 1/x², y = 5 when x = 2 | Inverse square | y = 20/x² | y = 5/4 when x = 4 |