Directly Proportional — Definition, Formula & Examples

Directly proportional describes two quantities that always maintain the same ratio. When one quantity doubles, the other doubles; when one is halved, the other is halved.

Two variables $x$ and $y$ are directly proportional if there exists a nonzero constant $k$ such that $y = kx$ . The constant $k$ is called the constant of proportionality, and the ratio $\frac{y}{x}$ remains equal to $k$ for all corresponding values.

Key Formula

y = kx

Where:

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

How It Works

To check whether two quantities are directly proportional, divide

y

x

for each pair of values. If every pair gives the same result, the quantities are directly proportional, and that result is your constant

k

. You can then use

y = kx

to find unknown values. On a graph, directly proportional relationships appear as a straight line passing through the origin.

Worked Example

Problem: A car travels at a constant speed. In 2 hours it covers 90 miles, and in 5 hours it covers 225 miles. Show that distance is directly proportional to time, and find how far it travels in 7 hours.

Check the ratio: Divide distance by time for each pair to see if the ratio is constant.

\frac{90}{2} = 45, \quad \frac{225}{5} = 45

Identify k: Both ratios equal 45, so the constant of proportionality is 45 miles per hour.

k = 45

Find distance at 7 hours: Substitute into the formula

d = kt

d = 45 \times 7 = 315 \text{ miles}

Answer: Distance is directly proportional to time with

k = 45

mph. In 7 hours the car travels 315 miles.

Why It Matters

Directly proportional relationships appear whenever you work with unit rates, such as price per item, speed, or currency exchange. Recognizing them lets you set up equations quickly in algebra, science, and everyday budgeting problems.

Common Mistakes

Mistake: Confusing directly proportional with any linear relationship, such as

y = kx + b

where

b \neq 0

Correction: A directly proportional relationship must pass through the origin. If there is a nonzero

y

-intercept, the quantities are linear but not directly proportional.