Direct vs. Inverse Proportion: How to Tell Them Apart and Solve Each

Some quantities move together: buy twice as many apples and you pay twice as much. Others move in opposite directions: put twice as many workers on a job and it takes half the time. The first is a direct proportion, the second an inverse proportion, and they are solved differently.

Direct proportion

Two quantities are directly proportional when their ratio is constant.

y = k × x or y ÷ x = k

k is the constant of proportionality. If 3 kg of apples cost $7.50, then k = 7.50 ÷ 3 = $2.50 per kg, and 5 kg cost 5 × 2.50 = $12.50. You can also set up the proportion 7.50/3 = x/5 and cross multiply, as the proportion calculator does.

On a graph, a direct proportion is a straight line through the origin.

Inverse proportion

Two quantities are inversely proportional when their product is constant.

y = k ÷ x or x × y = k

If 4 painters take 6 days to paint a house, the job is 4 × 6 = 24 painter-days. With 8 painters it takes 24 ÷ 8 = 3 days; with 3 painters, 8 days. A simple cross multiplication would give the wrong answer here, because the ratio is not constant.

On a graph, an inverse proportion is a curve (a hyperbola) that falls steeply at first and then flattens, never touching either axis.

The quick test

Ask: "If I double one quantity, what happens to the other?"

It doubles → direct proportion. Multiply.
It halves → inverse proportion. Multiply the pair to find the constant, then divide.
Neither → not a proportion at all (for example, a fixed fee plus a rate).

Everyday examples

Direct	Inverse
Distance and fuel used at a steady speed	Speed and travel time for a fixed distance
Hours worked and pay at an hourly rate	Number of people and each person's share of a fixed bill
Recipe servings and ingredient amounts	Pipe flow rate and time to fill a tank
Map distance and real distance	Pressure and volume of a gas at constant temperature (Boyle's law)

Worked inverse example

A trip takes 3 hours at 60 mph. How long at 45 mph? Distance is the constant: 3 × 60 = 180 miles. Time at 45 mph is 180 ÷ 45 = 4 hours.

Proportional to a square

Some relationships follow a power. The area of a circle is proportional to the square of its radius, so doubling the radius makes the area four times larger. Light intensity falls with the square of distance (an inverse-square law). These still use a constant, but with x² in place of x.

Checking a table of data

Given pairs of values, divide y by x for each row. If the result is always the same, it's direct. If not, multiply x by y; if that product is constant, it's inverse. The checker in the proportion calculator compares cross products for direct proportions.

Step-by-step cross multiplication is covered in how to solve proportions. Price comparisons are a direct-proportion problem; see unit rates and best buys.