Direct & Inverse Proportion — Formulas & Examples

Both kinds of proportion lock two quantities together, but they keep something different fixed: direct proportion holds the ratio $\frac{y}{x}$ constant and uses $y = kx$, while inverse proportion holds the product $xy$ constant and uses $y = \frac{k}{x}$.

Direct vs inverse proportion side by side

The fastest way to tell them apart is to ask what stays the same as the numbers change.

Feature	Direct proportion	Inverse proportion
Formula	$y = kx$	$y = \frac{k}{x}$
What stays constant	the ratio $\frac{y}{x}$	the product $xy$
Double one quantity	the other doubles	the other halves
Typical example	cost vs number of items	time vs number of workers

Here $k$ is the constant of proportionality and $x \ne 0$.

If two quantities are in direct proportion, doubling one doubles the other. If they are in inverse proportion, doubling one halves the other.

When to read it as direct, and when as inverse

In direct proportion, one quantity is always a fixed multiple of the other. If pens cost $2$ dollars each, then total cost $C$ is directly proportional to the number of pens $n$:

$$C = 2n$$

Here the constant of proportionality is $k = 2$. The ratio $\frac{C}{n}$ stays equal to $2$ as long as the unit price stays the same. That condition matters. If there is a fixed delivery fee or a bulk discount, the relationship is no longer a direct proportion.

In inverse proportion, the product stays fixed instead of the ratio. A common example is time and number of workers for the same job, if every worker works at the same rate and coordination losses are ignored. If $w$ is the number of workers and $t$ is the time, then

$$wt = k$$

So doubling the number of workers cuts the time in half. This is only an inverse proportion model when total work stays fixed and all workers are equally effective. In real projects, adding workers does not always reduce time perfectly.

Worked example: choosing the right model

The difference is easiest to see when the same kind of question is answered both ways.

Direct case

Suppose $4$ notebooks cost $12$ dollars at a fixed price. The cost per notebook is

$$k = \frac{12}{4} = 3$$

so the direct proportion formula is $C = 3n$. If you buy $7$ notebooks, then

$$C = 3(7) = 21$$

so $7$ notebooks cost $21$ dollars.

Inverse case

Now suppose $4$ workers can finish the same job in $6$ hours, with equal work rates and the same total amount of work. The constant product is

$$k = wt = 4 \cdot 6 = 24$$

so the inverse proportion formula is $t = \frac{24}{w}$. If $8$ workers do the job, then

$$t = \frac{24}{8} = 3$$

so the job takes $3$ hours.

The contrast is the whole point. In the direct case the ratio stayed constant: $\frac{12}{4} = \frac{21}{7} = 3$. In the inverse case the product stayed constant: $4 \cdot 6 = 8 \cdot 3 = 24$.

A two-line test for which model fits

If you are unsure which model fits, test one known pair of values first.

1. Compute $\frac{y}{x}$. If it stays the same across valid data points, think direct proportion.
2. Compute $xy$. If that stays the same instead, think inverse proportion.
3. If neither stays constant, the relationship is probably neither one.

Confusion points to watch

Not every increasing relationship is direct proportion. The ratio must stay constant. For example, $y = x + 5$ increases as $x$ increases, but $\frac{y}{x}$ is not constant, so it is not a direct proportion.

Likewise, not every decreasing relationship is inverse proportion. The product must stay constant. For example, $y = 10 - x$ decreases, but $xy$ does not stay constant, so it is not inverse proportion.

Both formulas also depend on the situation staying simple. Fixed unit price supports direct proportion. Fixed total work with equal worker rate supports inverse proportion. If that condition changes, the model can fail.

Where each one shows up

Direct proportion appears in constant-price shopping, map scales, unit conversions, and distance traveled at a fixed speed. Inverse proportion appears in work-rate problems, speed and travel time for a fixed distance, and simple physics relationships where one quantity must decrease to keep another quantity fixed. In both cases, the key skill is noticing what stays constant.

To practice the distinction, take either example above, change one number while keeping its condition, and confirm that the ratio (or the product) still lands where you expect.