Mathematics
Proportion calculator
Pick the term that is missing, enter the other three, and get the unknown with the check: the product of the means must equal the product of the extremes.
How a proportion is solved
A proportion is an equality between two ratios: a : b = c : d reads “a is to b as c is to d”. The first and last terms are called the extremes, the two middle ones the means. The fundamental property says that the product of the means equals the product of the extremes, a·d = b·c, and from that single equality any missing term follows: isolate it by dividing by the factor beside it.
The geometric mean is the case where the unknown appears twice, as in a : x = x : b. The fundamental property gives x² = a·b, so x = √(a·b): it is the geometric mean of the two extremes, not the arithmetic one. The difference is not cosmetic — between 4 and 9 the arithmetic mean is 6.5 while the geometric mean is 6 — and it is why the geometric mean turns up in Euclid's theorems and in growth rates.
Proportions are the tool for direct-proportionality problems: scaling recipes, maps and scales, percentages, currency conversion, dosages. The practical rule is always the same: write the two quantities in the same order in both ratios, and make sure they are expressed in the same unit.
Common mistakes
- Reversing the order in one of the two ratios: if the first is “kilometres to hours”, the second must be too. Writing “hours to kilometres” in the second ratio gives the reciprocal of the right answer.
- Mixing units: 30 minutes and 2 hours must be brought to the same unit before entering the proportion, otherwise the ratio is meaningless.
- Confusing the geometric mean with the arithmetic mean: the first is the root of the product, the second is half the sum. They coincide only when the two numbers are equal.
Frequently asked questions
How do you find the unknown term of a proportion?
With the fundamental property: the product of the means equals the product of the extremes. An unknown extreme is found by multiplying the two means and dividing by the other extreme; an unknown mean by multiplying the two extremes and dividing by the other mean.
What is the fourth proportional?
It is the fourth term of a proportion whose first three are known: in a : b = c : x, x = (b·c)/a. It is the form most direct-proportionality problems take.
What is the geometric mean?
It is the value that occupies both the second and third position, as in a : x = x : b. It equals √(a·b). It exists in the real numbers only when the product a·b is not negative.
Can a term of a proportion be zero?
No. If a term were zero so would the matching product be, and the proportion either becomes impossible or loses its meaning, because one of the two ratios would no longer be defined.
How this calculation works
Proportion: a : b = c : d, that is a/b = c/d. Fundamental property: a·d = b·c, the product of the extremes equals the product of the means. Missing term: a = (b·c)/d, b = (a·d)/c, c = (a·d)/b, d = (b·c)/a. Geometric mean, from a : x = x : b: x² = a·b, so x = √(a·b) (the positive root is taken; the negative one satisfies the equation but not the geometric reading). Third proportional, from a : b = b : x: x = b²/a. Addition property: (a+b) : b = (c+d) : d. Subtraction property: (a−b) : b = (c−d) : d.
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