Inverse Variation Calculator
Calculate inverse proportionality relationships with the equation y = k/x
Inverse Variation Calculator
The constant of inverse proportionality
Independent variable (cannot be zero)
Example Problems
Example 1: Speed and Time
Problem: A car traveling at 60 mph takes 2 hours to cover a certain distance. How long will it take at 40 mph?
Solution:
- First, find k: k = speed × time = 60 × 2 = 120
- Then find time at 40 mph: time = k/speed = 120/40 = 3 hours
Answer: It will take 3 hours at 40 mph.
Example 2: Pressure and Volume
Problem: If y = 14/x, find y when x = 4.
Solution:
- Given: k = 14, x = 4
- Calculate: y = k/x = 14/4 = 3.5
Answer: y = 3.5 when x = 4.
Example 3: Workers and Time
Problem: 8 workers can complete a job in 15 days. How many days will 12 workers take?
Solution:
- Find k: k = workers × days = 8 × 15 = 120
- Find days for 12 workers: days = k/workers = 120/12 = 10
Answer: 12 workers will take 10 days.
Inverse Variation Formula
Basic Formula
Where y is inversely proportional to x
Variables:
- y: Dependent variable
- x: Independent variable (≠ 0)
- k: Constant of proportionality
Key Properties:
- • x and y cannot be zero
- • k = x × y (constant)
- • As x increases, y decreases
- • Graph is a hyperbola
Real-World Examples
Speed & Time
Distance = Speed × Time
Pressure & Volume
Boyle's Law: PV = constant
Workers & Time
More workers, less time needed
Gravitational Force
Force ∝ 1/distance²
Quick Tips
For inverse variation: xy = k (constant)
Neither x nor y can equal zero
The graph never touches the axes
When one variable doubles, the other halves
The product xy remains constant
Understanding Inverse Variation
What is Inverse Variation?
Inverse variation describes a relationship between two variables where one variable increases as the other decreases proportionally. The product of the two variables remains constant.
Mathematical Definition
Two variables x and y are inversely proportional if their relationship can be expressed as:
y = k/x or xy = k
where k is a non-zero constant
Key Properties
Constant Product
For any two points (x₁, y₁) and (x₂, y₂) on an inverse variation curve: x₁y₁ = x₂y₂ = k
Graph Shape
The graph of inverse variation is a hyperbola that approaches but never touches the x and y axes (asymptotes).
Domain and Range
Both domain and range are all real numbers except zero: (-∞, 0) ∪ (0, ∞)
How to Identify Inverse Variation
Method 1: Check the Product
- Calculate xy for each data point
- If all products are equal, it's inverse variation
- The constant product is your k value
Method 2: Check the Ratio
- Calculate y₁/y₂ for two points
- Calculate x₂/x₁ for the same points
- If y₁/y₂ = x₂/x₁, it's inverse variation