Inverse Variation Calculator
Calculate inverse variation relationships where xy = k (constant)
Cannot be zero
Cannot be zero
The constant of variation
Quick Examples
Using formula:
Step-by-Step Solution
Verification
Check: x × y = k
× = ✓ × = ✓ × = ✓
Real-World Inverse Variation Examples
| Scenario | Variable 1 | Variable 2 | Constant |
|---|---|---|---|
| Speed & Time | 60 mph | 2 hours | 120 miles |
| Workers & Days | 4 workers | 6 days | 24 worker-days |
| Pressure & Volume | 2 atm | 10 L | 20 atm·L |
| Gear Ratio | 20 teeth | 30 RPM | 600 |
| Price & Quantity | $5 each | 20 items | $100 budget |
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About Inverse Variation Calculator
What is Inverse Variation?
Inverse variation (also called inverse proportion) describes a relationship between two variables where their product is always constant. As one variable increases, the other decreases proportionally, and vice versa.
The Inverse Variation Formula
Primary Formula: xy = k
Alternate Forms:
- y = k/x (solve for y)
- x = k/y (solve for x)
- k = xy (find the constant)
Where:
- x and y are the two variables
- k is the constant of variation (always non-zero)
How to Use This Calculator
- Select a calculation mode:
- Find Constant (k): Enter x and y values
- Find Y: Enter x and k values
- Find X: Enter y and k values
- Enter your known values
- View the result with step-by-step solution
Real-World Examples of Inverse Variation
| Scenario | Variables | Relationship |
|---|---|---|
| Speed & Time | Faster speed = less travel time | Speed × Time = Distance |
| Workers & Days | More workers = fewer days to complete | Workers × Days = Total Work |
| Pressure & Volume | Higher pressure = smaller volume (Boyle's Law) | P × V = k |
| Gear Ratios | Larger gear = slower rotation | Teeth × RPM = k |
Understanding Inverse vs Direct Variation
| Type | Formula | Relationship | Graph |
|---|---|---|---|
| Inverse | xy = k | One increases, other decreases | Hyperbola |
| Direct | y = kx | Both increase or decrease together | Straight line |
Frequently Asked Questions
How do I know if a relationship is inverse variation?
Check if the product of the two variables is constant. If xy always equals the same value k, it's inverse variation.
Can k be negative in inverse variation?
Yes, k can be negative, but it cannot be zero. A negative k means the variables have opposite signs.
What does the graph of inverse variation look like?
The graph is a hyperbola with two branches in opposite quadrants. It never crosses the x or y axis (asymptotic).
How is inverse variation used in physics?
Boyle's Law (P₁V₁ = P₂V₂) for gases is a classic example. As pressure increases, volume decreases proportionally at constant temperature.
Note: This calculator handles positive and negative values. The constant k must be non-zero for inverse variation to exist.