How can I tell whether a table of values shows direct or inverse proportion?

Compute the ratio $y/x$ for every row. If that ratio is constant, the proportion is direct. If instead the product $x \times y$ is constant, the proportion is inverse. If neither value stays the same across all rows, the relationship is neither direct nor inverse proportion.

Direct and Inverse Proportion — Definition, Formula & Examples

Direct and inverse proportion describe two ways that quantities can be linked. In a direct proportion, when one quantity increases the other increases at the same rate; in an inverse proportion, when one quantity increases the other decreases so that their product stays constant.

Two variables $x$ and $y$ are directly proportional if $y = kx$ for some nonzero constant $k$ . They are inversely proportional if $y = \dfrac{k}{x}$ for some nonzero constant $k$ . In both cases, $k$ is called the constant of proportionality.

Key Formula

\text{Direct: } y = kx \qquad \text{Inverse: } y = \frac{k}{x}

Where:

$y$ = The dependent variable
$x$ = The independent variable
$k$ = The constant of proportionality (nonzero)

How It Works

To decide which type of proportion applies, ask: does doubling one quantity double the other (direct), or does doubling one quantity halve the other (inverse)? For direct proportion, divide

y

x

— if the ratio

\dfrac{y}{x}

is always the same, the relationship is direct. For inverse proportion, multiply

x

and

y

— if the product

xy

is always the same, the relationship is inverse. Once you identify the constant

k

, you can use the appropriate formula to find any missing value.

Worked Example

Problem: A car travels at a constant speed. It covers 120 km in 2 hours. How far does it travel in 5 hours?

Identify the type: Distance increases as time increases at a constant speed, so distance is directly proportional to time.

d = kt

Find k: Use the known pair: 120 km in 2 hours.

k = \frac{d}{t} = \frac{120}{2} = 60

Solve for the unknown: Substitute

t = 5

into the formula.

d = 60 \times 5 = 300 \text{ km}

Answer: The car travels 300 km in 5 hours.

Another Example

Problem: Six workers can paint a building in 10 days. How many days would it take 15 workers to paint the same building?

Identify the type: More workers means fewer days to finish, so workers and days are inversely proportional.

w \times d = k

Find k: Use the known pair: 6 workers and 10 days.

k = 6 \times 10 = 60

Solve for the unknown: Substitute

w = 15

and solve for

d

d = \frac{60}{15} = 4 \text{ days}

Answer: 15 workers can paint the building in 4 days.

Visualization

Why It Matters

Direct and inverse proportion appear throughout algebra, science, and everyday life — from converting currencies and scaling recipes to calculating speed, pressure, and electrical resistance. In middle-school math and GCSE/pre-algebra courses, proportion problems are among the most commonly tested topics. Understanding these relationships also prepares you for more advanced work with linear functions, rational functions, and joint variation.

Common Mistakes

Mistake: Mixing up the two types — using

y = kx

when the relationship is actually inverse.

Correction: Always check first: does increasing

x

make

y

go up (direct) or down (inverse)? Verify by computing

y/x

xy

across your data.

Mistake: Forgetting that the constant

k

must be recalculated when the problem context changes.

Correction: The constant of proportionality depends on the specific scenario. Find

k

from the given pair of values before solving for unknowns.

Related Terms

Constant of Proportionality — The value k in both proportion formulas
Direct Variation — Another name for direct proportion
Inverse Variation — Another name for inverse proportion
Joint Variation — Extends proportion to multiple variables
Proportion — An equation stating two ratios are equal
Proportional — Describes quantities that maintain a constant ratio
Unit Rate — The constant k expressed per one unit