Two quantities are directly proportional when their ratio never changes: double one and the other doubles too. The fixed ratio is the constant of proportionality k, also called the unit rate. Price per kilogram, kilometres per litre, pay per hour and the slope of a straight line through the origin are all constants of proportionality. On a graph, a direct proportion is a straight line through (0, 0) whose slope equals k.
In an inverse proportion the product stays constant instead: if one quantity doubles, the other halves. The time a journey takes at different speeds, or the number of workers and the days a job lasts, behave this way, and k = x·y.
Once you know k you can answer any “how much for…” question: multiply (direct) or divide (inverse) to predict y for a new x, or work backwards to find the x that produces a target y. The calculator does both and also gives the reciprocal 1 ÷ k, which is the rate expressed the other way round (for example kilograms per euro instead of euros per kilogram).
How to use the Constant of Proportionality Calculator
- Type of relationship
- Independent variable x (known value)
- Dependent variable y (known value)
- New x to predict y for
- Target y to find x for
Results update as you type, and everything is calculated in your browser.
How it works
Choose the type of relationship and enter one known pair (x, y). For direct proportion k = y ÷ x; for inverse proportion k = x × y. Then enter a new x to predict y, or a target y to find the matching x.
Worked example
4 kg of apples cost $14. For a direct proportion k = 14 ÷ 4 = 3.5 dollars per kilogram, so 10 kg cost 3.5 × 10 = $35 and $21 buys 21 ÷ 3.5 = 6 kg. The reciprocal 1 ÷ 3.5 = 0.2857 kg per dollar is the same rate turned around.
Inverse example: 4 painters need 14 days for a job, so k = 4 × 14 = 56 painter-days. 10 painters would need 56 ÷ 10 = 5.6 days, and finishing in 21 days needs 56 ÷ 21 ≈ 2.67 painters.
Checking a table for proportionality
Divide each y by its x. In the table x = 2, 5, 8 with y = 7, 17.5, 28 every ratio is 3.5, so the data are proportional with k = 3.5. If the ratios differ – for example 7 ÷ 2 = 3.5 but 18 ÷ 5 = 3.6 – the relationship is not exactly proportional, and a regression line is the better model.
Inputs explained
| Field | What to enter |
|---|---|
| Type of relationship | Choose one: Direct proportion (y = k·x), Inverse proportion (y = k ÷ x) |
| Independent variable x (known value) | A number |
| Dependent variable y (known value) | A number |
| New x to predict y for | A number |
| Target y to find x for | A number |
Example values
How the main result changes when Independent variable x changes and everything else stays at the example values above:
| Independent variable x (known value) | Constant of proportionality k |
|---|---|
| 2 | 7 |
| 3 | 4.6667 |
| 4 | 3.5 |
| 5 | 2.8 |
| 6 | 2.3333 |
| 8 | 1.75 |
Formula
Direct proportion: k = y ÷ x, so y = k·x · Inverse proportion: k = x·y, so y = k ÷ x
Worked example
Inputs — Type of relationship: Direct proportion (y = k·x); Independent variable x (known value): 4; Dependent variable y (known value): 14; New x to predict y for: 10; Target y to find x for: 21.
Results — Constant of proportionality k = 3.5; Reciprocal 1 ÷ k = 0.285714; Predicted y at the new x = 35; x needed for the target y = 6.
Key terms
- Direct proportion
- Two quantities whose ratio y ÷ x is always the same.
- Inverse proportion
- Two quantities whose product x·y is always the same.
- Unit rate
- The amount of y for one unit of x – the constant of proportionality.
Frequently asked questions
How do you find the constant of proportionality?
Divide y by x for a direct proportion. If 4 kg of apples cost $14, k = 14 ÷ 4 = $3.50 per kg. For an inverse proportion multiply instead: k = x × y.
Is the constant of proportionality the same as the slope?
Yes, for a direct proportion. The line y = kx has slope k and passes through the origin. A line y = mx + c with c ≠ 0 has a slope but is not a proportion.
How can I tell if a table shows a proportional relationship?
Divide every y by its x. If all the ratios are equal, the relationship is proportional and that common ratio is k.
Can the constant of proportionality be negative?
Yes, when one quantity rises as the other falls along a line through the origin, e.g. y = −2x. k is negative whenever exactly one of x and y is negative.
What if x and y are equal?
Then k = 1 for a direct proportion: y = x.
What is k in y = 7x?
k = 7 – the coefficient of x. In y = 12 ÷ x the constant of inverse proportion is 12.
What is a unit rate?
The constant of proportionality written as "per one unit", such as 60 km per hour or $3.50 per kilogram.