# proportion

Proportion is a statement that two ratios are equivalent,
written as *a*:*b* = *c*:*d*, as in the statement
2:3 = 6:9. Two functions are proportional
if for all *x*, *f *(*x*) = *kg*(*x*),
where *k* is the constant of proportionality.
If the value of *k* is not known, the statement may be written *f *(*x*) – *g*(*x*).

In the ratio-equation or proportion *a*:*b* = *c*:*d*, *a* and *d* are referred to as the outer terms, and *b* and *c* as the inner terms of the proportion. If the outer terms are interchanged and the inner terms are unchanged, the proportion remains valid. Inner terms can only be interchanged with outer terms if this is done simultaneously on both sides. The sides of a proportion may be interchanged. The product of the inner terms is equal to the product of the outer terms. *a*:*b* = *c*:*d* implies *a* × *d* = *b* × *c*.

In the relation *a*:*b* = *c*:*d*, *d* is called the **fourth proportional**. If *a*, *b*, and *c* are given, *d* can be calculated: *d* – (*b* × *c*)/*a*.

Suppose in a proportion with equal inner terms, *a*:*b* = *b*:*c*, *c* is to be calculated for given *a* and *b*. *c* is called the **third proportional** with respect to *a* and *b*. We have *c* *b *^{2}/*a*.

In the proportion *a*:*m* = *m*:*d*, *m* is the **mean proportional** or **geometric mean** of *a* and *d*; *m* = √(*a**d*).

If *a*:*b* = *c*:*d*, we have *a*/*b* = *c*/*d*. We can add ± to each side of this equation. Thus

*a*/*b* ± = *c*/*d* ±, so that (*a* ± *b*)/*b* = (*c* ± *d*)/*d*.

That is, if *a*:*b* = *c*:*d*, then *a* ± *b*:*b* = *c* ± *d*:*d*. These are known as **derived proportions**.

The pair of quantities

*a */*b* = *c */*d*, *c */*d* = *m */*n*

can also be written as a **continued proportion** in the form

*a*:*c*:*m* = *b*:*d*:*n*.