Inverse Variation Calculator

Solve for the constant of variation (k) or a missing variable in the equation y = k/x

This tool helps you understand and solve problems involving inverse proportionality, a fundamental concept in algebra and physics.

Initial Values (Point 1)

Solve For (Point 2)

Examples

Click on any example to load it into the calculator

Find y₂ given x₁, y₁, and x₂

solveForY2

If y is 10 when x is 2, find y when x is 4.

x₁: 2, y₁: 10

Find value for: 4

Find x₂ given x₁, y₁, and y₂

solveForX2

If y is 6 when x is 5, find x when y is 3.

x₁: 5, y₁: 6

Find value for: 3

Physics: Speed and Time

solveForY2

A car takes 3 hours at 60 km/h. How long would it take at 90 km/h?

x₁: 60, y₁: 3

Find value for: 90

Economics: Price and Demand

solveForY2

If 500 units are sold at $10 each, how many would be sold at $8?

x₁: 10, y₁: 500

Find value for: 8

Other Titles
Understanding Inverse Variation: A Comprehensive Guide
Explore the concept of inverse variation, how to find the constant of proportionality, and its applications in real-world scenarios.

What is Inverse Variation? Core Concepts

  • Defining the relationship where two variables move in opposite directions.
  • Understanding the constant of variation 'k'.
  • The fundamental formula: y = k/x.
Inverse variation, also known as inverse proportion, describes a relationship between two variables where as one variable increases, the other variable decreases, and vice-versa. The key characteristic is that their product remains constant.
This constant product is called the 'constant of variation' or 'constant of proportionality,' denoted by 'k'. The relationship is mathematically expressed by the formula y = k/x, which can also be written as xy = k.

Basic Examples

  • If y = 10 when x = 5, then k = 10 * 5 = 50. The equation is y = 50/x.
  • If y = 2 when x = 8, then k = 2 * 8 = 16. The equation is y = 16/x.

Step-by-Step Guide to Using the Inverse Variation Calculator

  • Entering your initial known values (x₁ and y₁).
  • Selecting the variable you wish to solve for (x₂ or y₂).
  • Interpreting the calculated constant, equation, and final result.
1. Input the Initial Values
In the 'Initial Values (Point 1)' section, enter the values for your known pair of variables, x₁ and y₁.
2. Choose What to Solve For
From the 'Solve For' dropdown menu, select whether you want to find a new y-value (y₂) or a new x-value (x₂).
3. Enter the Known Variable
An input field will appear for the corresponding variable (either x₂ or y₂). Enter its value.
4. Calculate and See the Result
Click 'Calculate'. The tool will display the constant of variation (k), the full inverse variation equation, and the final calculated value for your unknown variable.

Usage Examples

  • Problem: y is 15 when x is 3. Find y when x is 5.
  • Solution: Enter x₁=3, y₁=15. Select 'Find y₂ given x₂' and enter x₂=5. Result: k=45, y₂=9.

Real-World Applications of Inverse Variation

  • Physics: Understanding relationships like speed-time and pressure-volume.
  • Economics: Modeling concepts such as price and demand.
  • Project Management: Relating the number of workers to project completion time.
Speed and Travel Time
For a fixed distance, speed and travel time are inversely proportional. The faster you go, the less time it takes. Formula: Time = Distance / Speed.
Pressure and Volume (Boyle's Law)
In physics, Boyle's Law states that for a fixed amount of gas at a constant temperature, the pressure and volume are inversely proportional. When you increase the pressure, the volume decreases. Formula: P ∝ 1/V.
Work and Time
The number of people working on a project is often inversely proportional to the time it takes to complete it. More workers lead to less time.

Practical Scenarios

  • Traveling 240 miles: At 60 mph, it takes 4 hours. At 80 mph, it takes 3 hours. (60 * 4 = 80 * 3 = 240)
  • A gas in a piston: If the pressure is 1 atm at 2L, increasing the volume to 4L will decrease the pressure to 0.5 atm.

Common Misconceptions vs. Correct Methods

  • Distinguishing inverse variation from direct variation.
  • Setting up the proportion correctly.
  • Avoiding common algebraic errors.
Misconception 1: Confusing with Direct Variation
Incorrect: Thinking that as one variable increases, the other must also increase. This describes direct variation (y = kx).
Correct: In inverse variation, as one variable goes up, the other goes down. The relationship is y = k/x.
Misconception 2: Incorrectly Setting up the Equation
Incorrect: Using a ratio like x₁/y₁ = x₂/y₂. This is a common mistake for those new to the concept.
Correct: The correct relationship is derived from the constant product: x₁y₁ = x₂y₂. This is the foundation for solving for an unknown.

Clarification Examples

  • Direct Variation: The more hours you work, the more you get paid. (Pay = Rate × Hours)
  • Inverse Variation: The more people who share a pizza, the smaller each slice is. (Slice Size = Pizza Size / Number of People)

Mathematical Derivation and Formula

  • Deriving the formula for the constant 'k'.
  • Deriving the formula to solve for a new variable.
  • A step-by-step solved example.
The core principle of inverse variation is that the product of the two variables is constant. Let's use this to derive the formulas used in the calculator.
Derivation
1. Definition: y varies inversely as x.
2. Formula: y = k/x
3. Finding k: To find the constant, rearrange the formula: k = x * y. For any point (x₁, y₁) on the curve, k = x₁ * y₁.
4. Solving for y₂: We know that x₁y₁ = k and x₂y₂ = k. Therefore, x₁y₁ = x₂y₂. To find y₂, simply rearrange this equation: y₂ = (x₁y₁) / x₂.
5. Solving for x₂: Similarly, to find x₂, rearrange the equation: x₂ = (x₁y₁) / y₂.

Solved Example

  • Problem: If y = 8 when x = 3, find x when y = 6.
  • 1. Find k: k = x₁y₁ = 3 * 8 = 24.
  • 2. Use the formula for x₂: x₂ = (x₁y₁) / y₂ = 24 / 6.
  • 3. Solve: x₂ = 4.