Variation Calculator
Direct, inverse, joint and partial variation: find k (and a) from conditions, then compute the unknown.
Input Data
Results
At a glance:Variation describes a proportional relation. Direct y=kx; inverse y=k/x; joint y=kxz; partial y=a+kx (fixed part + proportional part). Find k (and a for partial) from known conditions, then substitute.
Formula
Direct: y = k x ⇒ k = y/x.
Inverse: y = k/x ⇒ k = x y.
Joint: y = k x z ⇒ k = y/(x z).
Partial: k = (y₂−y₁)/(x₂−x₁), a = y₁ − k x₁.
$$y=kx$$$$y=\frac{k}{x}$$$$y=kxz$$$$y=a+kx$$How to Use
- Pick a variation type.
- Enter the known condition x₁, y₁ (and z₁ for joint).
- For direct/inverse/joint, leave one of x₂, y₂ blank as the unknown.
- For partial, enter a second point (x₂,y₂) and a target x₃.
Variation types
| Type | Relation | Constant | Example |
|---|---|---|---|
| Direct | y = k x | k = y/x | distance ∝ time |
| Inverse | y = k/x | k = x y | time ∝ 1/speed |
| Joint | y = k x z | k = y/(xz) | volume ∝ l×w |
| Partial | y = a + k x | a,k from two points | cost = fixed + variable |
k is usually positive for strict direct/inverse; partial allows any sign.
Case Studies
Inverse: x=2→y=6, find y at x=4
k = 2×6 = 12; y₂ = 12/4 = 3.
Partial: (1,5) & (3,9), find y at x=5
k = (9−5)/(3−1)=2; a = 5−2=3; y = 3+2×5 = 13.
FAQ
Direct vs inverse?
Direct: y grows with x (y=kx). Inverse: y shrinks as x grows (y=k/x). The position of k is the key.
What is partial variation?
Partial variation is fixed part + proportional part: y = a + kx (e.g. cost = fixed + per-unit×quantity). Two conditions are needed to find both a and k.
What is joint variation?
Joint variation means y is proportional to several variables at once: y = k x z. Find k with y/(x z).
Can k be negative?
Strict direct/inverse take k>0, but algebra generally allows any sign; this tool computes from your input.
Related Tools
References
Content review: Calculatorism Science Team. The four variation constants and substitution steps have been verified. Results are for reference only.