Last updated: July 31, 2026
Inverse Variation Calculator
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Formula
k = x1 × y1, then y2 = k / x2
Where:
- x_1=Known x-value
- y_1=Known y-value
- k=Constant of variation
- x_2=Target x-value
- y_2=Predicted y-value
Worked Examples
Solve a basic inverse variation
Start from a known pair and predict a new y-value.
- 1Compute the constant: k = 4 × 3 = 12.
- 2Use the inverse formula y₂ = 12 / 6.
- 3The predicted value is y₂ = 2.
Work with larger y-values
The product stays fixed even when x changes.
- 1Multiply the known pair: k = 2 × 10 = 20.
- 2Substitute x₂ = 5 into y = k/x.
- 3y₂ = 20 / 5 = 4.
Use decimal values
Inverse variation also works cleanly with decimals.
- 1Find the constant: k = 1.5 × 8 = 12.
- 2Divide by the new x-value: y₂ = 12 / 2.4.
- 3The result is y₂ = 5.
Invalid zero input
A zero x-value breaks the inverse variation model.
- 1The constant needs k = x₁ × y₁.
- 2The model also needs y₂ = k / x₂ with non-zero x-values.
- 3Because x₁ = 0, the calculator returns an error.
Introduction
The Inverse Variation Calculator helps you model relationships where one quantity increases as the other decreases. It first finds the constant product k = xy from a known pair, then uses that constant to predict new values. This is useful in algebra, physics, and rate-based situations where a fixed product describes the system.
What inverse variation means
Inverse variation links two variables so their product stays constant instead of their ratio.
Equation form is y = k/x.
The product x × y stays fixed.
When x increases, y decreases for positive k.
Graphs form hyperbola-like curves.
You need one known pair to determine k.
Both x-values used in the model must be non-zero.
How to find the constant
Multiply the known x-value and y-value to get the constant of variation.
Use the pair (x₁, y₁).
Compute k = x₁ × y₁.
Keep sign changes exactly as entered.
Decimals are allowed.
The same k must fit every point in the relationship.
How the calculator solves y₂
After determining k, the calculator substitutes your target x₂ into y = k/x.
Enter a non-zero x₂.
Divide the constant by x₂.
Read the new dependent value y₂.
Rounded output keeps the result readable.
You can reuse the same k for many target values.
Validation rules
The calculator checks the exact conditions needed for inverse variation to make sense.
x₁ must not be zero.
x₂ must not be zero.
All three inputs must be valid numbers.
Blank fields trigger an error.
Undefined division is blocked automatically.
How to interpret the results
Use k to understand the strength of the relationship and y₂ to answer the specific prediction question.
Larger |k| gives larger outputs for the same x.
Positive k keeps x and y with the same sign.
Negative k flips the sign of y.
Comparing k across models shows scale differences.
y₂ changes nonlinearly as x₂ changes.
Common applications
Inverse variation appears whenever two values trade off while maintaining a constant product.
Gas laws with fixed temperature assumptions.
Speed and travel time for fixed distance.
Workers and time for a fixed job estimate.
Electrical relationships in simplified models.
Resource sharing problems in algebra classes.
Common mistakes to avoid
Most errors come from using the wrong variation model or entering zero for an x-value.
Using k = y/x instead of k = xy.
Forgetting that x₂ cannot be zero.
Mixing direct and inverse variation ideas.
Ignoring the sign of negative values.
Rounding too early during manual checks.
FAQs
What is inverse variation?
It is a relationship where two variables multiply to the same constant, written as y = k/x.
How do I find k?
Multiply the known x-value and y-value: k = x₁ × y₁.
Why can’t x₁ or x₂ be zero?
The formula y = k/x requires division by x, so zero would make the expression undefined.
Can the constant of variation be negative?
Yes. If one of the known values is negative, k becomes negative and the predicted y-values follow that sign pattern.
Do decimals work in this calculator?
Yes. Decimal inputs are accepted and outputs are rounded to six decimal places.
How is inverse variation different from direct variation?
Direct variation keeps y/x constant, while inverse variation keeps x × y constant.
What does the output k tell me?
It shows the fixed product that defines the inverse relationship for every valid point in the model.