Inverse Variation Formula - Definition and Examples

Key Points

• The inverse variation formula is $y = \frac{k}{x}$, where $k$ is a non-zero constant called the constant of variation.

• Rearranged as $xy = k$, it says the product of the two quantities never changes, which is the quickest way to test a data table.

• Doubling $x$ halves $y$, so the two quantities move in opposite directions by matching factors.

• Direct variation is $y = kx$ instead, where the ratio $\frac{y}{x}$ stays constant rather than the product.

• The graph is a hyperbola that approaches both axes without touching them, since neither $x$ nor $y$ can be zero.

What Is the Inverse Variation Formula?

Two quantities vary inversely when one falls in proportion as the other rises:

$$y = \frac{k}{x}, \qquad \text{equivalently} \qquad xy = k$$

The constant $k$ is found from any single matching pair of values, after which the formula predicts every other pair.

Why the Product Stays Constant

In direct variation the ratio is fixed: double $x$ and $y$ doubles with it, so $\frac{y}{x}$ never moves. Inverse variation fixes the product instead.

Double $x$ and $y$ must halve, because their product has to stay at $k$. Treble $x$ and $y$ falls to a third. The factors are always reciprocals of each other.

That is what makes $xy = k$ the practical test. Given a table of paired values, multiply each pair: a constant product means inverse variation, a constant ratio means direct variation, and neither means some other relationship.

Variable Key

Examples of the Inverse Variation Formula

Example 1

$y$ varies inversely with $x$, and $y = 12$ when $x = 3$. Find $k$, then $y$ when $x = 4$.

$$k = xy = 3 \times 12 = 36, \qquad y = \frac{36}{4} = 9$$

Final answer: $k = 36$ and $y = 9$

Example 2

For the same relationship, find $y$ when $x = 4$ by treating it as direct variation.

Direct variation would give $y = kx$ with $k = \frac{12}{3} = 4$, so $y = 4 \times 4 = 16$.

But $x$ rose from 3 to 4 and the answer rose from 12 to 16. In an inverse relationship $y$ must fall when $x$ rises, so the direction alone shows the model is wrong.

Using the inverse form:

$$y = \frac{36}{4} = 9$$

Final answer: 9. Checking the direction of change is enough to rule out the direct model before any arithmetic.

Example 3

$y$ varies inversely with $x$. If $y = 5$ when $x = 8$, find $x$ when $y = 10$.

$$k = 8 \times 5 = 40, \qquad x = \frac{40}{10} = 4$$

Final answer: $x = 4$

The value of $y$ doubled and $x$ halved, which is the reciprocal pairing the formula requires.

Example 4

Six workers finish a job in 12 days. Assuming inverse variation, how long would nine workers take?

$$k = 6 \times 12 = 72, \qquad \text{days} = \frac{72}{9} = 8$$

Final answer: 8 days

Example 5

A table reads $x = 2, 4, 8$ with $y = 24, 12, 6$. Decide whether this is inverse variation.

Test the products rather than guessing:

$$2 \times 24 = 48, \qquad 4 \times 12 = 48, \qquad 8 \times 6 = 48$$

Final answer: Yes, it is inverse variation with $k = 48$, since every product is identical.

Had the ratios been constant instead, the relationship would have been direct.

Don't Confuse Inverse Variation With These Three

The last row catches people out. A straight line through the origin is direct variation, but add a constant and it no longer is, because doubling $x$ then fails to double $y$.

Inverse Variation, Common Confusions Cleared Up

Mistake 1: Using the direct variation formula

Where it slips in: Both relationships are set up the same way, by finding $k$ from one pair of values.

Don't do this: $y = kx$ for an inverse relationship.

The correct way: $y = \frac{k}{x}$. Check the direction: in inverse variation, raising $x$ must lower $y$.

Students who write down whether $y$ should rise or fall before computing anything catch the wrong model immediately. The direction is a property of the relationship, and it is decided before any number is substituted.

Mistake 2: Subtracting instead of dividing

Where it slips in: The idea that one quantity falls as the other rises suggests a subtraction.

Don't do this: $y = k - x$

The correct way: $y = \frac{k}{x}$. In a subtraction relationship the sum stays constant, not the product, and doubling $x$ would not halve $y$.

Mistake 3: Forgetting to find $k$ first

Where it slips in: The formula looks complete as written, so values get substituted straight into it.

Don't do this: Put $x = 4$ into $y = \frac{k}{x}$ while $k$ is still unknown.

The correct way: Use the given pair to compute $k$, then substitute. One pair of values is always enough to pin $k$ down.

The Short Version

• The inverse variation formula is $y = \frac{k}{x}$, equivalently $xy = k$.

• A constant product is the defining test, just as a constant ratio defines direct variation.

• Doubling one quantity halves the other, so the two change by reciprocal factors.

• Find $k$ from a given pair of values before predicting anything else.

• To work through proportional reasoning with a teacher, explore an algebra tutor or math tuition.

A Practical Next Step

If $y$ varies inversely with $x$ and $y = 7$ when $x = 6$, find $y$ when $x = 14$. Then test whether $x = 1, 2, 5$ with $y = 20, 10, 4$ is inverse variation.

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Read More

• Inverse Relation

• Inverse Functions

• What Is Proportion

• What Is Ratio

• Ratio Formula