Direct Proportion Calculator

Give one known pair of values and the calculator finds the constant of proportionality, writes out the equation, and solves for whichever value you are missing. A table and a graph show the whole relationship, not just the single answer.

Formula y = kx

The known pair establishes the relationship — it is where k comes from. The new value is what you want converted using it. Optional labels just make the answer read as a sentence.

The known pair

The value to convert

Try an example
Method

Find k first, then everything else follows

Direct proportion problems all reduce to one division followed by one multiplication.

Confirm the relationship really is proportional

Ask whether doubling one quantity would double the other. If there is a fixed charge, a starting amount, or a threshold, the answer is no — the relationship is linear but not proportional, and y = kx will give wrong answers.

Divide the known pair to get k

k = y₁ ÷ x₁. This single number carries the whole relationship. Read it as "y per one x" — dollars per kilogram, miles per hour, cups per serving.

Write the specific equation

Substituting k into y = kx gives you an equation you can use repeatedly. With k = 1.5 the equation is y = 1.5x, and every future question about this relationship is one multiplication away.

Multiply or divide as needed

Going from x to y, multiply by k. Going from y to x, divide by k. That is the only decision left, and getting it the wrong way round is the most common mistake — which is why this calculator has separate modes rather than one ambiguous field.

Sanity-check the direction

A larger x must give a larger y (for positive k). If your answer moved the wrong way, you divided when you should have multiplied.

y = kx    k = y / x

Direct proportion. One constant, one multiplication. The constant k is the gradient of the line and the unit rate at the same time.

y₁ / x₁ = y₂ / x₂

The proportion form. Because k is the same for both pairs, you can skip calculating it and cross multiply instead — see the cross multiplication calculator.

Watch for the fixed charge

"$3 call-out plus $2 per mile" is y = 2x + 3, not a proportion. Doubling from 5 to 10 miles takes the cost from $13 to $23 — not double. Treating it as proportional would give $26, over by three dollars.

Worked examples

Direct proportion in practice

Cost from quantity

4 kg of apples cost $6. What do 7 kg cost?

Known pair
x₁ = 4, y₁ = 6
Find k
k = 6 ÷ 4 = 1.5
Equation
y = 1.5x
Substitute x = 7
y = 1.5 × 7
Check
10.5 ÷ 7 = 1.5 ✓

$10.50 — and $1.50 per kg

Distance from time

A train covers 150 miles in 3 hours. How far in 8 hours?

Known pair
x₁ = 3, y₁ = 150
Find k
k = 150 ÷ 3 = 50 mph
Equation
y = 50x
Substitute x = 8
y = 50 × 8
Check
400 ÷ 8 = 50 ✓

400 miles

Working backwards

At the same apple price, how many kilograms can you buy for $15?

Known k
k = 1.5 dollars per kg
Equation
15 = 1.5x
Divide by k
x = 15 ÷ 1.5
Check
1.5 × 10 = 15 ✓

10 kg

Fractional constant

5 litres of a liquid weighs ⅔ kg. What does 12 litres weigh?

Known pair
x₁ = 5, y₁ = 2/3
Find k
k = (2/3) ÷ 5 = 2/15
Equation
y = 2x/15
Substitute x = 12
y = 24/15 = 8/5
As a decimal
1.6 kg

8/5 kg = 1.6 kg

Comparison

Direct, inverse, and merely linear

Three relationships that are easy to confuse and behave completely differently.

Relationship Equation Constant Double x and… Graph
Direct proportion y = kx y / x y doubles Straight line through the origin
Inverse proportion y = k / x x · y y halves Hyperbola, never touching either axis
Linear, not proportional y = kx + c none constant y less than doubles Straight line missing the origin
Square proportion y = kx² y / x² y quadruples Parabola through the origin

Directly proportional

y = kx · ratio y/x is fixed

  • Cost of fuel against litres bought
  • Pay against hours at a fixed rate
  • Distance against time at constant speed
  • Circumference against diameter (k = π)
  • Mass against volume for one material

Not proportional

a fixed part breaks the ratio

  • Taxi fare with a flag-fall charge
  • Phone plan with a monthly base fee
  • Temperature in °F against °C (offset of 32)
  • Shipping with a first-item surcharge
  • Tax with a personal allowance
Background

What the constant actually tells you

The constant of proportionality is the most informative number in the problem, and it is easy to under-use. Once you have k = 1.5 for apples, you do not just have the answer to one question — you have the price per kilogram, which you can compare against another shop, use to check a receipt, or scale to any quantity without recalculating anything. Problems that look like they need a fresh proportion each time usually need k once.

It also has units, and reading them out loud catches errors. If x is in kilograms and y in dollars, then k is in dollars per kilogram. If you accidentally computed x ÷ y instead, you would have kilograms per dollar — a perfectly meaningful number, but not the one you wanted, and about 0.667 rather than 1.5. Checking that the units of your k match the phrase you would use in conversation is a reliable guard.

Why the graph must pass through the origin

Substituting x = 0 into y = kx gives y = 0, no matter what k is. So the point (0, 0) is on every direct-proportion graph. This is more than a technicality: it is the practical test for whether a real-world relationship is proportional. Zero kilograms of apples costs zero dollars, so that relationship qualifies. Zero miles in a taxi still costs the flag-fall, so that one does not.

Proportional constants that have names

Many of the constants in science are exactly this kind of k. Density is mass per unit volume. Speed is distance per unit time. Spring stiffness in Hooke's law is force per unit extension. Resistance in Ohm's law is voltage per unit current. In each case the law is the statement that the ratio stays constant, and the named constant is that ratio — which is why finding k is so often the actual goal rather than an intermediate step.

When you do not need k at all

If you only ever need one answer, you can skip k and cross multiply the two pairs directly: y₁/x₁ = y₂/x₂. That is fewer steps for a one-off, and the cross multiplication calculator handles it. Finding k pays off when you have several questions about the same relationship, or when the constant itself is what you were asked for. For the opposite behaviour — where the product rather than the ratio stays fixed — use the inverse proportion calculator.

Questions

Direct proportion — questions

Including the distinction that causes the most trouble: proportional versus merely linear.

What does directly proportional mean?
Two quantities are directly proportional when their ratio never changes. Double one and the other doubles; halve one and the other halves. Written as an equation that is y = kx, where k is a fixed number called the constant of proportionality. The graph is always a straight line through the origin — if the line does not pass through (0, 0), the relationship is linear but not proportional.
How do you find the constant of proportionality?
Divide y by x for any known pair: k = y ÷ x. If 4 kg of apples cost $6, then k = 6 ÷ 4 = 1.5, meaning $1.50 per kg. Because the ratio is constant, any pair from the relationship gives the same k — which is also how you can test whether a table of values really is proportional.
What is the difference between direct proportion and linear?
Every direct proportion is linear, but not every linear relationship is proportional. A linear relationship is y = kx + c; direct proportion is the special case where c = 0. A taxi fare of $3 plus $2 per mile is linear but not proportional, because doubling the miles does not double the fare — the fixed $3 does not double. Only when the line passes through the origin is the ratio y/x constant.
How do I know if a table of values is directly proportional?
Work out y ÷ x for every row. If all the quotients are identical, the table is directly proportional and that shared quotient is k. If they differ, it is not. This test is quicker than plotting the points and is what the calculator does internally when you enter a pair.
Can k be a fraction or negative?
Yes to both. A fractional k simply means y grows more slowly than x — k = 2/3 gives y = 2x/3. A negative k means the two quantities move in opposite directions along a straight line through the origin, which is still direct proportion in the strict sense even though it is unusual in everyday problems. Note that a negative k is not the same as inverse proportion: see the inverse proportion calculator.
What are real examples of direct proportion?
Cost against quantity at a fixed unit price; distance against time at constant speed; the circumference of a circle against its diameter (with k = π); mass against volume for one material (k is the density); pay against hours at a fixed hourly rate; and the amount of each ingredient in a recipe against the number of servings.
Is direct proportion the same as a unit rate?
They are two views of the same number. The constant of proportionality is the unit rate — the amount of y for one unit of x. Saying k = 1.5 and saying "$1.50 per kilogram" is the same statement. The unit ratio calculator approaches it from that direction if the per-unit phrasing is what you need.
What happens if x is zero?
In a true direct proportion, x = 0 forces y = 0, since y = k × 0. You cannot use a pair where x = 0 to find k, though, because that would mean dividing by zero — the pair (0, 0) is consistent with every possible k and so tells you nothing about which one applies.