Direct Variation
Free Direct Variation calculator with clear step-by-step results.
About Direct Variation
Two quantities vary directly when their ratio never changes: y/x is a fixed constant k. This calculator recovers k from one known (x, y) pair by dividing y by x, then uses y = kx to predict the output at any other x. Because the relationship is purely multiplicative, multiplying x by any factor multiplies y by exactly the same factor — which is why the x scale factor and the ratio of the two y values always agree. A zero known x is rejected because k would be undefined.
Formulas
Constant of variation
k = y / x (from the known pair)
- k
- Constant of variation
- x
- Known input
- y
- Known output
Prediction
y_target = k x x_target
- x_target
- The x you want a prediction for
Worked example
With the default values below, Direct Variation returns Predicted y of 30.0000. Change any input above and every figure here is recalculated in your browser.
Inputs used
- Known x
- 4
- Known y
- 12
- Predict y at x =
- 10
Results
- Predicted y
- 30.0000
- Constant of variation k
- 3.0000
- x Scale FactorHow many times larger the target x is.
- 2.5000
- Variation Equation
- y = 3x
Step by step
- 1Values used
- 2Constant of variation
- 3Prediction
- 4Predicted y= 30.0000
- 5Constant of variation k= 3.0000
- 6x Scale Factor= 2.5000
- 7Variation Equation= y = 3x
Assumptions and limits
- The result is derived only from the 3 inputs listed above. Anything not among them is not part of the calculation.
- In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
- Figures are rounded for display to 4 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.
How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.
Frequently Asked Questions
How do I know if data varies directly?
Divide y by x for every pair you have. If the quotient is the same each time, the data varies directly and that quotient is k. If the quotient drifts, the relationship is not direct variation.
What does the constant k actually mean?
k is the amount y changes for each one-unit increase in x, and it is the slope of the line y = kx through the origin. In a unit-price problem k is the price per item; in a distance problem at constant speed k is the speed.
Why must the known x be non-zero?
k is defined as y divided by x, so a known x of zero leaves k undefined. Direct variation always passes through the origin, which means the point (0, 0) carries no information about k.