Inverse Variation Formula

The Formula

y = \frac{k}{x} \quad \text{equivalently } xy = k

When to use: More workers means less time: if 4 workers take 6 hours, 8 workers take 3 hours.

Quick Example

xy = 24 If x = 4, y = 6. If x = 8, y = 3. Product stays constant.

Notation

'y varies inversely as x' or 'y is inversely proportional to x'

What This Formula Means

A relationship where y = \frac{k}{x}: as one quantity doubles, the other halves—their product stays constant.

More workers means less time: if 4 workers take 6 hours, 8 workers take 3 hours.

Formal View

y \propto \frac{1}{x} \iff \exists\, k \neq 0: y = \frac{k}{x}, \; xy = k, \; x \neq 0

Worked Examples

Example 1

medium
\(y\) varies inversely with \(x\), and \(y = 8\) when \(x = 3\). Find \(k\) and the equation. Then find \(y\) when \(x = 6\).

Solution

  1. 1
    Inverse variation: \(y = k/x\), so \(k = xy\).
  2. 2
    \(k = 3 \times 8 = 24\).
  3. 3
    Equation: \(y = 24/x\).
  4. 4
    When \(x=6\): \(y = 24/6 = 4\).

Answer

\(k = 24\); \(y = 4\) when \(x = 6\)
In \(y = k/x\), as \(x\) doubles from 3 to 6, \(y\) halves from 8 to 4. The product \(xy = 24\) stays constant.

Example 2

hard
Speed and travel time are inversely proportional for a fixed distance. At 60 km/h the trip takes 4 hours. How long at 80 km/h? Identify \(k\).

Common Mistakes

  • Confusing inverse variation (y = \frac{k}{x}) with negative direct variation (y = -kx)
  • Forgetting that in inverse variation the product xy is constant, not the ratio \frac{y}{x}
  • Plugging in x = 0 — inverse variation is undefined at x = 0 since you cannot divide by zero

Why This Formula Matters

Models many physical relationships (pressure/volume, speed/time).

Frequently Asked Questions

What is the Inverse Variation formula?

A relationship where y = \frac{k}{x}: as one quantity doubles, the other halves—their product stays constant.

How do you use the Inverse Variation formula?

More workers means less time: if 4 workers take 6 hours, 8 workers take 3 hours.

What do the symbols mean in the Inverse Variation formula?

'y varies inversely as x' or 'y is inversely proportional to x'

Why is the Inverse Variation formula important in Math?

Models many physical relationships (pressure/volume, speed/time).

What do students get wrong about Inverse Variation?

Inverse variation is NOT y = -kx (that's direct with negative k).

What should I learn before the Inverse Variation formula?

Before studying the Inverse Variation formula, you should understand: proportionality, division.

← Back to Inverse Variation See All Examples Practice Problems