Math & Statistics

Inverse Proportion Calculator

Preserve a constant product from one known x-y pair, calculate the corresponding y at a target x and x at a target y, compare the calculated y with an entered reference, and display both reciprocal scale ratios.

Known xKnown yTarget xEntered comparison y at target xTarget y for inverse solutionEntered scale multiplier

Calculated y at target x-

Inverse-proportion constant-

Calculated x at target y-

Calculated y minus entered comparison-

Target x divided by known x-

Calculated y divided by known y-

Constant after entered multiplier-

Decision view

Constant-product reciprocal curve

Constant-product reciprocal curveThe known pair and both solved targets sit on the same inverse-proportion hyperbola, making reciprocal scale changes visible.

Exact scenario comparisonTarget x changes while all other entered assumptions remain constant.

Target x	Calculated y at target x	Inverse-proportion constant	Calculated x at target y	Calculated y minus entered comparison	Target x divided by known x	Calculated y divided by known y	Constant after entered multiplier

How to use Inverse Proportion Calculator

Enter one defensible known x-y pair.
Enter a target x to solve its reciprocal y.
Enter a target y to solve its reciprocal x.
Use the ratio check and curve to confirm the modeled inverse relationship.

Calculator guide

Understanding Inverse Proportion Calculator

In an inverse proportion, x and y move in opposite directions while their product remains constant. This calculator solves y from a target x, solves x from a target y, and checks the reciprocal ratios against the known pair.

Product stays fixed Every modeled pair multiplies to k.

Variables move oppositely Larger x produces smaller y.

Solve by division Divide k by the known target variable.

Ratios reconcile The two scale factors multiply to one.

Calculation method

How the calculation works

Preserve the constant product x times y, then solve separately for y at a target x and x at a target y while exposing reciprocal ratios. In the Inverse Proportion Calculator, the live scenario varies target x and tracks calculated y at target x while the remaining results preserve the reconciliation path. Multiply the known x and y to obtain k. Divide k by a target x to solve y, or divide k by a target y to solve x.

Detailed calculation process

Preserve one constant product in both solution directions

The default known pair is x = 8 and y = 15, so every point on the modeled curve has product 120.

General formula: xy = k; y_target = k/x_target; x_target = k/y_target; (x_target/x_known)(y_target/y_known) = 1 Multiplying one variable by a factor requires dividing the other by the same factor. The fixed product k defines the entire reciprocal curve.

What each symbol means

x, y Positive paired quantities in their respective units.

k Constant product in x-units times y-units.

x_known, y_known Entered pair used to establish k.

x_target Entered x value whose paired y is requested.

y_target Entered y value whose paired x is requested.

s Entered multiplier applied only to the displayed scaled constant.

Worked substitution with the default inputs

1. Calculate the invariant product: k = 8 x 15 = 120 The product carries combined x-y units.

2. Solve y at the target x: y = 120 / 12 = 10 Increasing x from 8 to 12 reduces y from 15 to 10.

3. Solve x at the target y: x = 120 / 6 = 20 A smaller target y requires a larger x to preserve the product.

4. Check reciprocal ratios: x ratio = 12/8 = 1.5; y ratio = 10/15 = 0.666667; 1.5 x 0.666667 = 1 The scale factors are reciprocals, confirming the inverse relationship.

5. Compare and scale: 10 - 10.5 = -0.5; scaled k = 120 x 1 = 120 The calculated y is 0.5 below the entered comparison; the multiplier does not alter the base solution.

The default pair fixes k = 120, gives y = 10 at x = 12 and x = 20 at y = 6, and reconciles reciprocal scale ratios of 1.5 and 0.666667.

Model check

Confirm the hyperbola represents the real system

A mathematically exact reciprocal curve is useful only when its constant-product assumption is justified.

Invariant Identify what product stays constant.

Domain Define valid positive x and y ranges.

Limits Check capacity, overhead, and efficiency changes.

Units Record the combined units carried by k.

Worked situations

Practical examples

The known pair 8 and 15 establishes product 120.
At x = 12, the matching inverse value is y = 10.
At y = 6, the matching x is 20.

Better inputs

Useful tips

Keep x and y strictly positive when using the displayed positive-domain curve.
State the physical meaning and units of the constant product.
Verify inverse proportionality from subject knowledge, not appearance alone.

Before relying on the result

Limitations and common mistakes

Many systems include fixed overhead, limits, or changing efficiency and are not pure inverse proportions.
Zero inputs make reciprocal division undefined.
Measurement error and uncertainty in the known pair propagate to every solved value.

Reference

Key terms

Inverse proportion: Relationship in which xy stays constant.
Constant product: Invariant k defining the reciprocal curve.
Reciprocal ratio: Scale factor whose product with the other variable's scale factor is one.

Important note

Calculated directly from the entered values using the displayed formula and rounding settings.

Frequently asked questions

Why does y decrease when x increases?

Because their product must remain k, so y = k/x.

Can either variable equal zero?

No. A positive constant product cannot be preserved with a zero variable, and reciprocal division would be undefined.

Is an inverse proportion the same as a negative linear slope?

No. The reciprocal curve is nonlinear and its slope changes with x.

What does the scale multiplier change?

It changes the displayed scaled constant only; the base known-pair solutions remain tied to k.