Two quantities are proportional when one is always a fixed multiple of the other — or, for an inverse proportion, when their product stays fixed. Recognizing which kind of relationship you have is half the work; the other half is finding the constant that links them. This proportion calculator does both, for direct, inverse, square and inverse-square relationships, and can test a table of data to see which pattern it follows.
How to use the proportion calculator
- Choose Find a missing value or Check whether a table of values is proportional.
- For a missing value, pick the type of proportion: direct, inverse, square or inverse square.
- Enter the known pair (x1, y1) and either the new x2 or the new y2.
- Read the answer, the constant k and the equation on the tape; the graph shows both points on the curve.
- To test a table, paste one “x, y” pair per line and read the verdict and the ratio and product columns.
Proportion formulas
| Relationship | Equation | Constant | Doubling x… |
|---|---|---|---|
| Direct | y = kx | k = y/x | doubles y |
| Inverse | y = k/x | k = xy | halves y |
| Square | y = kx2 | k = y/x2 | multiplies y by 4 |
| Inverse square | y = k/x2 | k = yx2 | divides y by 4 |
Once k is known from one pair, any other pair follows. For a direct proportion this is equivalent to the familiar cross-multiplied statement y1/x1 = y2/x2; for an inverse proportion it is x1y1 = x2y2.
Worked examples
Direct. You earn $10 for 4 hours of work at a fixed rate. How much for 7 hours? k = 10 ÷ 4 = 2.5, so y2 = 2.5 × 7 = $17.50. Hours rose by a factor of 1.75 and so did pay.
Inverse. Six workers finish a job in 12 days. How long would 8 workers take, working at the same pace? k = 6 × 12 = 72 worker-days, so y2 = 72 ÷ 8 = 9 days.
Inverse square. A lamp gives 100 lux at 2 m. At what distance is it 25 lux? k = 100 × 22 = 400, so x2 = 400 ÷ 25 = 16 and x = 4 m.
How to tell which proportion you have
Ask what happens when one quantity doubles:
- Cost and number of identical items — the cost doubles: direct.
- Speed and travel time for a fixed distance — the time halves: inverse.
- Area of a square and its side — the area quadruples: square law.
- Brightness and distance from a point light — the brightness drops to a quarter: inverse square.
With data, divide each y by its x. If the results match, the relationship is direct. If they don’t, multiply instead; matching products mean inverse proportion. The table checker does exactly this, and it also recognizes the common trap of a straight line that does not pass through the origin — linear, but not proportional. A taxi fare with a fixed starting charge is the classic example.
Related calculators
For a single statement A:B = C:D, the ratio calculator is quickest. The cross multiplication calculator solves proportions whose terms contain x, the unit rate calculator finds k as a price per unit, and linear interpolation handles straight-line data that does not pass through zero.
Frequently asked questions
What is the difference between direct and inverse proportion?
In a direct proportion, y = kx: when x doubles, y doubles. In an inverse proportion, y = k/x: when x doubles, y halves. The ratio y/x is constant in the first case and the product xy is constant in the second.
How do I find the constant of proportionality?
Use one known pair. For direct proportion k = y ÷ x; for inverse proportion k = x × y. With 4 hours of work earning $10, k = 10 ÷ 4 = 2.5 dollars per hour.
Is every straight line a direct proportion?
No. A direct proportion is a straight line through the origin. A line such as y = 2x + 5 has a constant slope but y/x is not constant, so doubling x does not double y.
What is an inverse square law?
A relationship y = k/x². Light intensity and gravitational pull follow it: doubling the distance cuts the effect to one quarter. Choose 'Inverse square' to work with it.
How is this different from the ratio calculator?
The ratio calculator solves a single A:B = C:D statement. This tool works with the underlying relationship: it finds k, applies it to new values, supports inverse and square laws, and tests whole tables.