Inverse proportion and constructing proportion equations
This topic covers writing proportionality statements as equations and using them to solve problems where one variable changes inversely with another (or with a power of another).
Proportionality notation
The symbol ∝ means "is proportional to". Translating to an equation:
- y ∝ x → y = kx.
- y ∝ x² → y = kx².
- y ∝ √x → y = k√x.
- y ∝ 1/x → y = k/x.
- y ∝ 1/x² → y = k/x² (also written y = k x⁻²).
- y ∝ 1/√x → y = k/√x.
The constant k is the constant of proportionality. It's fixed for a given relationship and is found from one (x, y) pair.
Three-step strategy
- Write the proportionality as an equation with k.
- Use a known (x, y) pair to find k.
- Use the equation with the new x to find y (or vice versa).
✦Worked example— Worked example — inverse
The time T to complete a task is inversely proportional to the number of workers W. With 5 workers, T = 18 hours.
- T = k/W.
- 18 = k/5 → k = 90.
- Equation: T = 90/W.
- With 9 workers: T = 90/9 = 10 hours.
✦Worked example— Worked example — inverse square
The gravitational force F between two objects is inversely proportional to the square of distance d. F = 200 N at d = 4 m.
- F = k/d².
- 200 = k/16 → k = 3200.
- Equation: F = 3200/d².
- At d = 8 m: F = 3200/64 = 50 N. (Note: doubling d halves F by factor 4.)
✦Worked example— Worked example — combined
y is directly proportional to x and inversely proportional to z. (Often written y ∝ x/z.)
- y = kx/z.
Sketching graphs
- y = k/x → hyperbola through 1st and 3rd quadrants; asymptotes: x = 0 and y = 0.
- y = k/x² → curve in 1st and 2nd quadrants (always positive y for k > 0); steeper near origin.
⚠Common mistakes
- Using x instead of x² when the relationship is "inversely proportional to the square".
- Forgetting the constant k — every proportionality must have one.
- Setting up y = x/k instead of y = k/x — k is always on the same side as the dependent variable when isolated.
- Confusing units — work in consistent units before substituting.
- Negative values — for inverse, x ≠ 0; for inverse square root, x > 0.
➜Try this— Quick check
y is inversely proportional to x. y = 12 when x = 5. Find y when x = 8.
- k = 12 × 5 = 60. y = 60/8 = 7.5.
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