Direct and inverse proportionality

Late September 2026: in Year 1 at the science high school (liceo scientifico) physics has started with measurements — the International System of Units, scientific notation, significant figures, the calculation of errors. It is the chapter that causes the most worry at the start of the year, but it is not the one the first real test falls on: that comes when the syllabus moves on to graphs and proportionality, a topic that usually starts in October and on which the test concentrates between four and seven weeks from now, so between the end of October and November. It is a chapter that doesn’t stay shut inside physics: the same idea of “constant ratio” comes back identical in chemistry when you calculate density, and in maths when you study the linear function.

Direct and inverse proportionality: how to recognise them from a table

The teacher almost always starts from an experiment — a trolley on a track, a weight hanging from a spring, a gas in a cylinder — and gives you a table of measured data: two columns of numbers, an independent quantity and a dependent one. The task is twofold: first, draw the right graph; second, recognise what kind of relationship links the two quantities. They are two different skills and the test checks both, often in the same exercise.

From table to graph

First rule, trivial but the first thing a teacher checks: the axes must be labelled with the name of the quantity and the unit of measurement (e.g. “time $t$ (s)”), not just a letter. Second rule: the scale on each axis must be uniform — you can’t put 0, 1, 2, 5, 10 at equal distances on the page. Third: points are marked with a precise cross, never a big dot that covers half a square, because that is where you read the slope or draw the interpolation curve.

Once the points are marked, the shape of the graph already tells you a lot: a straight line through the origin, a straight line that doesn’t pass through it, a curve that falls ever more gently (a hyperbola) or a curve that rises ever more steeply (a parabola). They are four different shapes, and each corresponds to a different kind of relationship.

Direct proportionality

Two quantities $x$ and $y$ are directly proportional when their ratio is constant:

\frac{y}{x} = k

where $k$ is the constant of proportionality. Rewritten as $y = k x$ , this is the equation of a straight line through the origin — and indeed the fastest graphical test is exactly this: if the line you draw through the points passes through $(0, 0)$ , you have direct proportionality. The test on the table is even faster: divide each $y$ by the corresponding $x$ . If the ratio is always the same (apart from measurement errors), it is direct.

The constant $k$ is not a pure number: it has a unit of measurement, and that is what gives the graph its physical meaning. If $x$ is a volume in $c m^{3}$ and $y$ a mass in grams, $k$ is a density and is measured in $g/c m^{3}$ . On the graph, $k$ is exactly the slope of the line.

Linear dependence that is not proportional

Be careful not to confuse “proportional” with “linear”: a line that does not pass through the origin still describes a linear relationship, of the type

y = k x + q

but it is not direct proportionality, because the ratio $y / x$ changes from point to point (while the slope stays constant). It happens often in physics: a spring that starts already stretched, a thermometer that reads a non-zero value at rest. Calling “directly proportional” any straight line that rises is by far the most common mistake — see below.

Inverse proportionality

Two quantities are inversely proportional when it is their product that stays constant:

x \cdot y = k

that is $y = k / x$ . The graph is a branch of a rectangular hyperbola: it starts high near the $y$ axis and falls, getting closer and closer to the $x$ axis, without ever touching it. The test on the table: multiply each pair of values. If the product is constant, it is inverse. Physically, Boyle’s law for gases is the textbook example:

p \cdot V = constant (at constant temperature)

if the volume of a gas is halved, the pressure doubles.

Quadratic proportionality

There is a third case, less common but one that still appears in the Year 1 syllabus: quadratic proportionality, in which it is the ratio between $y$ and $x^{2}$ that is constant:

\frac{y}{x ^{2}} = k \Rightarrow y = k x^{2}

The graph is a branch of a parabola with its vertex at the origin. The most natural physical example is the distance covered in free fall (or more generally in uniformly accelerated motion starting from rest):

s = \frac{1}{2} g t^{2}

where the distance $s$ grows with the square of the time, not with the time. The test on the table: divide each $y$ by $x^{2}$ (not by $x$ ); if the result is constant, it is quadratic. If the $y$ - $x$ graph is a curve that rises ever more steeply and you can’t tell it by eye from an upside-down hyperbola, this is the way to settle the doubt without guessing.

Linearisation

A trick that teachers often ask for explicitly: if you suspect an inverse proportionality, instead of plotting $y$ against $x$ plot $y$ against $1/ x$ — if the relationship really is $y = k / x$ , that graph is a straight line through the origin, much easier to read than a hyperbola by eye. In the same way, if you suspect a quadratic, plot $y$ against $x^{2}$ : if it really is $y = k x^{2}$ , you again get a straight line through the origin. This technique is called linearisation and it is the way, even in the lab at university, that a physical law is checked without having to “guess” the shape of the curve.

The typical mistake: calling proportional anything that just grows

The most frequent mistake I see marked in tests is not one of calculation, it is one of physical vocabulary: the student sees two quantities growing together and writes “they are directly proportional” — even when the line doesn’t pass through the origin, even when the curve is a parabola. “Proportional” in physics has a precise meaning (constant ratio), it is not a synonym for “that increases when the other increases”. The twin second mistake: drawing by hand a line that “seems” to pass close to the points without checking whether it really passes through the origin — with a ruler and the point $(0, 0)$ marked, the difference shows at once.

Three worked exercises

Exercise 1 — recognising the type of relationship from three tables

Table A

$x$	1	2	3	4
$y$	2	4	6	8

I calculate the ratio $y / x$ for each pair: $2/1 = 2$ , $4/2 = 2$ , $6/3 = 2$ , $8/4 = 2$ . Constant ratio $\Rightarrow$ direct proportionality, with $k = 2$ .

Table B

$x$	1	2	4	5
$y$	20	10	5	4

The ratio $y / x$ is not constant ( $20$ , $5$ , $1.25$ , $0.8$ ): it is not direct. I try the product $x \cdot y$ : $1 \cdot 20 = 20$ , $2 \cdot 10 = 20$ , $4 \cdot 5 = 20$ , $5 \cdot 4 = 20$ . Constant product $\Rightarrow$ inverse proportionality, with $k = 20$ .

Table C

$x$	1	2	3	4
$y$	3	12	27	48

The ratio $y / x$ is not constant ( $3$ , $6$ , $9$ , $12$ : it grows). I try $y / x^{2}$ : $3/1 = 3$ , $12/4 = 3$ , $27/9 = 3$ , $48/16 = 3$ . Constant ratio $\Rightarrow$ quadratic proportionality, with $k = 3$ , so $y = 3 x^{2}$ .

Exercise 2 — finding the constant with its unit and predicting a value

An object of homogeneous material is measured with different volumes:

$V$ (cm³)	10	20	30
$m$ (g)	27	54	81

I check the ratio: $27/10 = 2.7$ ; $54/20 = 2.7$ ; $81/30 = 2.7$ . Constant $\Rightarrow$ direct proportionality between mass and volume, with

ρ = \frac{m}{V} = 2.7 g/c m^{3}

which is exactly the density of the material (it is aluminium). Typical question: how much does the same material weigh with $V = 50 c m^{3}$ ? I use $m = ρ V$ :

m = 2.7 g/c m^{3} \times 50 c m^{3} = 135 g

Exercise 3 — inverse-square proportionality: “if the distance triples…”

A light source produces an illumination of $180 lux$ at a distance $d_{1} = 2 m$ . The intensity of the light is inversely proportional to the square of the distance:

I = \frac{k}{d ^{2}}

If the distance triples, $d_{2} = 3 d_{1} = 6 m$ , what is the illumination $I_{2}$ ? I write the ratio between the two situations, knowing that $k$ doesn’t change:

\frac{I _{2}}{I _{1}} = \frac{d _{1}^{2}}{d _{2}^{2}} = (\frac{d _{1}}{d _{2}})^{2} = (\frac{1}{3})^{2} = \frac{1}{9}

I_{2} = \frac{180 lux}{9} = 20 lux

Note the key point of the exercise: when the distance triples, the illumination doesn’t become a third, it becomes a ninth — it is the signature of every relationship inversely proportional to the square, the same one you meet again in gravitation and the electric field.

You can practise exercises of this kind in the practice test on measurements, graphs and proportionality (questions in Italian): /simulazioni/verifiche-liceo/prima/misure-ed-errori/. The correct answer is free, and the guided step-by-step solution unlocks with your first lesson.

If the chapter on significant figures and conversions has stayed unclear to you, I covered it at length in Physics in Year 3: measurements and significant figures — the method is identical even though the class is different. On maths, the review of powers and fractions useful for calculating the constants is in Year 1: the whole maths syllabus in a formula sheet, and on the first test in general you’ll find The first maths test in Year 1 of high school: how to prepare. The full picture of the physics subjects is at /materie/fisica/.

Tables of measurements, with their error, are the next step: measurement errors and error propagation.

If after reading this post you still have doubts about a specific graph or table in your test, you can ask for a free introductory call at /contatti/.

Direct, inverse and quadratic proportionality in physics