Inverse Variation Calculator

Solve for the constant of variation (k) or a missing variable

Solve for

Value of x (x₁)

Value of y (y₁)

New value of x (x₂)

Examples

If y varies inversely as x, and y = 10 when x = 2, find y when x = 4. (k = 20, y₂ = 5)
If y varies inversely as x, and y = 6 when x = 3, find k. (k = 18)
y = 40 / x. Find y when x = 10. (y = 4)

Important Note

Inverse variation describes a relationship where as one variable increases, the other decreases, such that their product is always a constant (k). The relationship is represented by the equation y = k/x.

Understanding Inverse Variation: A Comprehensive Guide

Inverse variation (or inverse proportion) is a relationship between two variables in which the product is a constant. When one variable increases, the other decreases in proportion so that the product remains the same.

This relationship is mathematically expressed as y = k/x, where 'y' and 'x' are the variables and 'k' is the constant of variation. The formula 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

1. Choose What to Solve For

Select the desired calculation from the 'Solve for' dropdown menu. You can find the constant 'k' and a new y value, 'k' and a new x value, or solve for one of the initial variables.

2. Input the Known Values

Fill in the text fields with the numbers from your problem. For example, if y = 12 when x = 4, and you want to find y when x = 6, you would select 'Find k and y₂' and input x₁=4, y₁=12, and x₂=6.

3. Calculate and Interpret the Result

Click 'Calculate'. The calculator will show the value of the constant of variation (k), the full inverse variation equation, and the value of the variable you were solving for.

Usage Examples

To solve 'y=15 when x=3, find y when x=5': Select 'Find k and y₂', input x₁=3, y₁=15, x₂=5. Result: k=45, y₂=9.
To find the equation if y=20 when x=2: Input x₁=2, y₁=20. The calculator finds k=40 and the equation y=40/x.

Real-World Applications of Inverse Variation

Speed and Travel Time

The time it takes to travel a fixed distance varies inversely with speed. If you increase your speed, the travel time decreases. (Time = Distance / Speed).

Pressure and Volume (Boyle's Law)

In physics, Boyle's Law states that the pressure of a gas varies inversely with its volume at a constant temperature (P = k/V).

Supply and Demand

In economics, the price of a product can sometimes vary inversely with its supply. When supply is high, the price might be low, and vice-versa.

Practical Examples

Traveling 120 miles: At 60 mph, it takes 2 hours. At 30 mph, it takes 4 hours. (60*2 = 30*4 = 120).
A piston in a cylinder: If the volume is 1 liter and pressure is 2 atm, doubling the pressure to 4 atm halves the volume to 0.5 liters.

Common Misconceptions and Correct Methods in Inverse Variation

Misconception 1: Confusing with Direct Variation

Wrong: Assuming that as one variable increases, the other also increases. That is direct variation (y=kx).

Correct: In inverse variation, as one variable increases, the other decreases. The relationship is y = k/x.

Misconception 2: Incorrectly Setting up the Equation

Wrong: Setting up the proportion as x₁/y₁ = x₂/y₂.

Correct: The correct relationship is x₁y₁ = x₂y₂. From this, you can derive y₂ = (x₁y₁) / x₂.

Correction Examples

Direct Variation: More hours worked means more pay. (y=kx)
Inverse Variation: More workers on a job means less time to complete it. (y=k/x)

Mathematical Derivation and Examples

The core of inverse variation is the equation xy = k, where k is the constant of variation. This means that for any two points (x₁, y₁) and (x₂, y₂) that satisfy the variation, the product of their coordinates will be the same.

Derivation

1. Start with the definition: y varies inversely as x.

2. Write the formula: y = k/x.

3. To find k, rearrange the formula: k = xy.

4. Given a set of points (x₁, y₁), you can find k. k = x₁y₁.

5. To find a new value y₂ given x₂, use the constant k: y₂ = k/x₂, which is the same as y₂ = (x₁y₁)/x₂.

Derivation Example

Problem: If y = 8 when x = 3, find x when y = 6.
1. Find k: k = x₁y₁ = 3 * 8 = 24.
2. Use k to find x₂: y₂ = k/x₂ => 6 = 24/x₂.
3. Solve for x₂: x₂ = 24 / 6 = 4.