Two quantities are inversely proportional when their product remains constant. The inverse proportion formula is y = k/x, or xy = k. When one quantity increases, the other decreases by a corresponding factor.
Enter three values and leave one blank to solve a proportion, or enter all four values to test whether two ratios are equal.
When x × y remains equal to 24, doubling x causes y to become half as large.
What Is Inverse Proportion?
Inverse proportion describes two quantities whose product remains constant.
When one quantity is multiplied by a factor, the other is divided by the same factor.
The values therefore move in opposite directions.
Inverse Proportion Formula
The standard formula is y = k/x.
It may also be written as xy = k.
The letter k represents the constant product.
How to Find the Constant of Proportionality
Multiply x by y.
For x = 3 and y = 8, multiply three by eight.
The constant is k = 24.
Example: Find Y in an Inverse Proportion
Suppose xy = 24 and x equals six.
Substitute into y = 24/6.
The result is y = 4.
Example: Find X in an Inverse Proportion
Suppose xy = 40 and y equals five.
Rearrange as x = 40/5.
The result is x = 8.
Inverse Proportion Table
Each row below has the same product of 24.
| x | y | x × y |
|---|---|---|
| 1 | 24 | 24 |
| 2 | 12 | 24 |
| 3 | 8 | 24 |
| 4 | 6 | 24 |
| 6 | 4 | 24 |
| 8 | 3 | 24 |
How to Recognise Inverse Proportion
Multiply each pair of corresponding values.
A constant product indicates inverse proportion.
The ratio y/x does not remain constant, so the direct-proportion test should not be used.
Write an Inverse Relationship as a Proportion
When x1y1 equals x2y2, the values can be rearranged into x1/x2 = y2/y1.
Notice that the order of the y-values is reversed.
This reversal reflects the fact that one quantity increases while the other decreases.
Inverse Proportion Graph
An inverse-proportion graph forms a curved hyperbola rather than a straight line.
As x becomes larger, y approaches zero without reaching it when k is nonzero.
The graph does not pass through the origin because division by zero is undefined.
Inverse Proportion in Work Problems
When workers perform at the same rate, the number of workers and completion time may be inversely proportional.
Doubling the workers can halve the time under an ideal simplified model.
Real work may include coordination limits, so the mathematical model should be applied only when the assumptions fit.
Inverse Proportion in Speed and Time
For a fixed distance, speed and travel time are inversely proportional.
Doubling the speed halves the time when the distance remains unchanged.
The constant product is speed multiplied by time, which equals distance.
Inverse Proportion in Sharing
When a fixed total is shared equally, the number of recipients and amount per recipient are inversely proportional.
If twice as many people share the same total, each person receives half as much.
The product remains equal to the fixed total.
Direct Versus Inverse Proportion
Direct proportion preserves a constant ratio.
Inverse proportion preserves a constant product.
Direct quantities move together, while inverse quantities move in opposite directions.
The quotient y/x stays constant.
The product x × y stays constant.
Can the Proportion Calculator Solve Inverse Problems?
The calculator solves equal-ratio equations in the form a/b = c/d.
An inverse problem can often be rearranged into an equivalent proportion before entering the values.
For example, x1/x2 = y2/y1 preserves the reversed relationship.
In an inverse relationship, the second quantity must be written in the opposite order when forming an equal-ratio equation.
Common Mistakes
Do not look for a constant quotient in an inverse relationship.
Do not write corresponding inverse values in the same order when creating ratios.
Do not use x = 0 in y = k/x.
Confirm that the product xy remains constant for multiple value pairs.
Conclusion
Use y = k/x or xy = k for inverse proportional relationships.
Find k by multiplying the corresponding values.
Rearrange an inverse relationship carefully before checking it with the Proportion Calculator.
FAQs
What is the inverse proportion formula?
The formula is y = k/x, or equivalently xy = k.
How do I find k?
Multiply x by y.
What stays constant in inverse proportion?
The product x × y remains constant.
What happens when x doubles?
The corresponding y-value becomes half as large.
Can x equal zero?
No. Division by zero is undefined in y = k/x.