Proportion

A proportion is a representation of one value as a fraction or multiple of another value. It is used to provide a sense of relative magnitude.

If some value \(a\) is proportional to another value \(b\), it means that \(a\) can be expressed as \(c \times b\), where \(c\) is some constant, also known as the constant of proportionality (or sometimes just the proportion) of \(a\) to \(b\).

For example, no matter how large a domino is, it is always twice as long as it is wide. Then, the proportion of the length compared to the width can be said to be \(2\), and we can write \(l = 2w\) where \(w\) is the width and \(l\) is the length.

Notation

To write that \(a\) is proportional to \(b\), the notation \(a \propto b\) is used. For example, the surface area of an object of a specific shape is proportional to the square of its length, so we write \(A \propto L^2\) where \(A\) is area and \(L\) is length.

Percentages

Percentages are a way of describing proportions relative to the number 100 to make them easier to understand. When people refer to a “percentage basis” for calculating something, they are specifically talking about this type of proportion.

To get a percentage from a proportion, simply multiply it by 100. In our domino example, we multiply \(2\) by \(100\) to find that the length is equal to \(200\%\) of the width.

Percentages only work with dimensionless proportions — that is, where the constant of proportionality has no units. For example, if a car requires \(6.7\) litres of gas to travel \(100\) kilometres, it makes no sense to say that the fuel efficiency of the car (which is a constant of proportionality for fuel used versus distance travelled) is \(6.7\%\) litres per kilometre. However, when comparing the fuel efficiencies of two cars, if one car has an efficiency of \(5.2\) L/​100km and another has one of \(6.8\) L/​100km, it makes sense to say that the first car is \(131\%\) as efficient as the second.

Common examples of proportions

The circumference of a circle is proportional to its diameter. We write \(C \propto d\), and the constant of proportionality is \(\pi = 3.14159265\ldots\).

The area of a circle is proportional to the square of its radius. We write \(A \propto r^2\), and the constant of proportionality is also \(\pi\).

The rate of growth of an exponential function is proportional to the current value of the function. We write \(\frac{df}{dt} \propto f\), or \(\Delta f \propto f\) if considering discrete values.

The gravitational potential energy of an object in a constant gravitational field is proportional to its height relative to some zero point. We write \(E \propto h\), and the constant of proportionality is the weight of the object, its mass multiplied by the gravitational acceleration in the field.

The amount of current flowing through a specific wire is proportional to the potential difference, or voltage, between the endpoints of the wire. We write \(I \propto V\), and the constant of proportionality is the resistance of that wire.

In an ideal gas, the pressure of the gas is proportional to its temperature, if the volume and number of particles in the gas is held constant. This is one property of the ideal gas law in physics. We write \(P \propto T\).

proportions as ratios