Constant of Proportionality Calculator
Enter any two of the three values (x, y, and k) and this calculator solves for the missing one. Choose direct proportion (y = kx) or inverse proportion (y = k/x), then read the worked steps and a live table of corresponding values. The result updates as you type.
Formula
Worked example
A car travels at a constant speed. After 3 hours (x = 3) it has covered 210 km (y = 210). The constant of proportionality is k = 210 / 3 = 70. This means the car travels at 70 km/h. To find how far it travels in 5 hours: y = 70 * 5 = 350 km.
What is the constant of proportionality?
The constant of proportionality (k) is the fixed ratio that links two proportional quantities. In a direct proportion, dividing y by x always gives the same number, k. In an inverse proportion, multiplying x by y always gives the same number, k. You may already know k by other names: it is the slope of a proportional line through the origin, the unit rate in rate problems, and the spring constant in Hooke's Law. Whenever two quantities have a constant ratio or a constant product, k is the bridge between them.
Direct vs inverse proportion
In a direct proportion (y = kx), both variables increase or decrease together. Doubling x doubles y. The graph is a straight line through the origin with slope k. In an inverse proportion (y = k/x), one variable increases as the other decreases. Doubling x halves y, so the graph is a hyperbola. The sign of k matters too: a negative k means y and x move in opposite directions (direct) or that the product is negative (inverse). Real-world examples of direct proportion include speed-distance-time at constant speed, Ohm's Law at fixed resistance, and unit pricing. Real-world examples of inverse proportion include Boyle's Law (pressure and volume at fixed temperature) and work-rate problems where workers and time trade off against each other.
How to find k from a table of values
If you have a table of x and y values and want to test whether they are proportional, compute y/x for each row. If every row gives the same number, the data is in direct proportion and that number is k. If every row gives a different y/x but every row gives the same x*y, the data is in inverse proportion and that product is k. A single outlier means the relationship is not perfectly proportional (check for measurement error or a non-proportional model). This calculator's chart panel generates a reference table of y values for x from 1 to 10 so you can compare your data against the theoretical proportional relationship.
Solving for x, y, or k
The three-variable relationship y = kx (or y = k/x) lets you solve for any one variable when the other two are known. To find k: divide y by x (direct) or multiply x and y (inverse). To find y: multiply k by x (direct) or divide k by x (inverse). To find x: divide y by k (direct) or divide k by y (inverse). This calculator handles all three solves in both proportion types. Use the "Solve for" selector to pick your unknown, enter the two known values, and read the result instantly.
Common proportionality relationships in science and everyday life
| Context | Relationship | Type | k represents |
|---|---|---|---|
| Speed, distance, time (fixed speed) | d = s * t | Direct | Speed (m/s or km/h) |
| Ohm's Law (fixed resistance) | V = I * R | Direct | Resistance (ohms) |
| Unit price | Total = price * quantity | Direct | Price per unit |
| Hooke's Law (spring) | F = k * x | Direct | Spring constant (N/m) |
| Boyle's Law (fixed temperature) | P * V = k | Inverse | Pressure-volume product |
| Work rate (fixed work) | workers * time = k | Inverse | Total work units |
| Wavelength and frequency | f * lambda = c | Inverse | Speed of light |
Examples of direct and inverse proportion from real-world contexts.
Frequently asked questions
What is the constant of proportionality?
The constant of proportionality (k) is the unchanging ratio between two proportional quantities. For a direct proportion y = kx, dividing any y by its paired x always equals k. For an inverse proportion y = k/x, multiplying any x by its paired y always equals k. It tells you exactly how one quantity scales relative to the other.
How do you find the constant of proportionality from a graph?
For a direct proportion, the graph is a straight line through the origin. The constant k equals the slope of that line: pick any point (x, y) on the line (other than the origin) and divide y by x. For an inverse proportion, the graph is a hyperbola, and k equals the product x * y for any point on the curve.
What is the difference between direct and inverse proportion?
In a direct proportion (y = kx), y and x increase or decrease together. If x doubles, y doubles. In an inverse proportion (y = k/x), y and x move in opposite directions. If x doubles, y halves. The constant k measures the slope in direct proportion and the product in inverse proportion.
Can the constant of proportionality be negative?
Yes. A negative k means that as x increases, y decreases (direct proportion) or that the product x * y is negative (inverse proportion). For example, if a bank charges a fee proportional to a withdrawal, the fee might be represented as a negative y value relative to the account balance x.
Is the constant of proportionality the same as the slope?
For a direct proportion y = kx, yes. The line passes through the origin and k is the slope. For a general linear equation y = mx + b where b is not zero, m is the slope but the relationship is not a direct proportion because y/x is not constant across all values of x.
How do you check if a table of values shows direct proportion?
Compute y/x for every row in the table. If every row gives the same value, the data is in direct proportion and that value is k. If the ratios differ, the relationship is either inverse proportionality (check x*y for a constant product) or a more complex function.