# Ratio definition

My teacher gave me the definition of ratios as "the comparison between two like quantities"

And one of the example he gave for a popular ratio was that of Body mass index (BMI).

My question is that in the calculation of body mass index we use height and weight as the quantities to arrive at the number , however both of them are not "like" quantities, then why is this called a ratio ?

In that case Is there any better way to define ratio ?

• BMI may not be a good example: it is weight (kg) divided by the square of height (m^2) so is more of a quotient than a ratio. Commented Apr 11, 2023 at 8:00
• See also this: en.wikipedia.org/wiki/Dimensionless_quantity (And the main articles linked therein, like "list of..." and "dimensionless quantities in physics") Commented Apr 11, 2023 at 8:12

My teacher gave me the definition of ratios as "the comparison between two like quantities"

The ratio of any number of quantities is a quantitative comparison of their measures. There is no need for the quantities to be alike or to have the same units. An example is 2 boys : 1 girl : 3 cars : 0 chauffeur.

For numbers a ratio is a fraction.

So if $$a$$ and $$b$$ are non-zero numbers then the ratio $$a:b$$ is the same thing as the fraction $$\frac{a}{b}$$.

It's hard to argue that 2 : 1 : 3 is a fraction or that -2 : 3 and 2 : -3 are equivalent objects.

• If a quantity is a count I would agree the items need not be the same. Though one could argue that boys, girls, cars and chauffeurs are alike because they are all elements of finite sets. But if a quantity is a measure we have a problem. What is the ratio of 1 inch : 1 foot? 1:1? 1:12? Commented Apr 11, 2023 at 12:31
• @Peter 1 inch : 1 foot clearly equals 1:12 (the two quantities have the same base unit, metre). On the other hand, the ratio that I gave as example cannot be simplified or its units dropped. Commented Apr 11, 2023 at 16:32
• Interestingly, the article you referenced notes the use of rate for ratios that are not dimensionless. Clearly nomenclature varies. Commented Apr 12, 2023 at 4:09
• @Peter As pointed out by the comment under your answer, a ratio can accommodate zero anywhere whereas a rate cannot have a zero denominator; moreover, a ratio (3 men : 4 men : 1 man = 3:4:1) need not be binary whereas a rate can deal with only two quantities. All of this is already indicated in my above answer. -) Commented Apr 12, 2023 at 4:20
• I would regard 3 : 4 : 1 as two ratios, just as a < b < c is two inequalities. Commented Apr 12, 2023 at 5:26

I have always been uneasy about defining a ratio as a comparison, because a difference is also a type of comparison. For numbers a ratio is a fraction. So if $$a$$ and $$b$$ are non-zero numbers then the ratio $$a:b$$ is the same thing as the fraction $$\frac{a}{b}$$. Many people are quite happy dividing quantities, but some may argue that only numbers can be divided. In either case we can say that

If you have two like quantities, their ratio is the number that you multiply one by to get the other one.

where two like quantities are two quantities of the same thing, e.g. mass, distance, or pure numbers.

With this definition a ratio is a "pure number" and BMI is indeed not a ratio.

Some books distinguish between a ratio and a rate. A rate is similar to a ratio but has units. With this understanding the Body Mass Index is a rate, which is is defined to be in $$kg/m^2$$. If you know your measurements in pounds and inches you must convert before calculating, or (equivalently) multiply by a conversion constant.

In practice one can use ratio form or fraction form for numbers and quantities, as long as care is taken with units.

• A pedantic mathematician would say that a ratio $a:b$ is not the same as the quotient $\frac ab$, because you can have $1:0$ but you cannot have $\frac10$. Commented Apr 11, 2023 at 9:42
• @student91, thanks. Answer amended to avoid zero (anywhere). Commented Apr 11, 2023 at 12:17