# 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

Parents:

• Mathematics

Math­e­mat­ics is the study of num­bers and other ideal ob­jects that can be de­scribed by ax­ioms.